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2 Geothermics 44 (212) Contents lists available at SciVerse ScienceDirect Geothermics journa l h omepa g e: Vertical temperature profiles and borehole resistance in a U-tube borehole heat exchanger Richard A. Beier a,, José Acuña b, Palne Mogensen c, Björn Palm b a Department of Mechanical Engineering Technology, Oklahoma State University, Stillwater, OK 7478, United States b Department of Energy Technology, Royal Institute of Technology (KTH), Brinellvägen 68, 1 44 Stockholm, Sweden c Palne Mogensen AB, Emblavägen 29, Djursholm, Sweden a r t i c l e i n f o Article history: Received 29 March 211 Accepted 6 June 212 Available online 1 July 212 Keywords: Ground-source heat pump Borehole heat transfer Ground thermal conductivity Thermal response test Borehole thermal resistance a b s t r a c t The design of ground source heat pump systems requires values for the ground thermal conductivity and the borehole thermal resistance. In situ thermal response tests (TRT) are often performed on vertical boreholes to determine these parameters. Most TRT analysis methods apply the mean of the inlet and outlet temperatures of the circulating fluid along the entire borehole length. This assumption is convenient but not rigorous. To provide a more general approach, this paper develops an analytical model of the vertical temperature profile in the borehole during the late-time period of the in situ test. The model also includes the vertical temperature profile of the undisturbed ground. The model is verified with distributed temperature measurements along a vertical borehole using fiber optic cables inside a U-tube for the circulating fluid. The borehole thermal resistance is calculated without the need for the mean temperature approximation. In the studied borehole, the mean temperature approximation overestimates the borehole resistance by more than 2%. 212 Elsevier Ltd. All rights reserved. 1. Introduction Ground-source heat pump (GSHP) systems have been developed to increase energy efficiency for the heating and cooling of buildings. GSHP systems often use vertical boreholes as ground heat exchangers, which couple heat pumps to the ground. A common design uses a circulating fluid in a closed loop to carry heat between the heat pump and the ground heat exchanger with a U- tube (Fig. 1a). In the United States the space around the U-tube in the borehole is often filled with grout to prevent water and contaminants from migrating along the vertical borehole. In Sweden the common practice is to allow groundwater to fill the borehole around the U-tube (Gustafsson et al., 21). The design of the ground heat exchanger requires values for the ground thermal conductivity and the borehole thermal resistance, which is the thermal resistance between the borehole wall and the circulating fluid. An in situ thermal response test (TRT) on a borehole is a method to estimate both parameters as proposed and demonstrated by Mogensen (1983). Early analysis methods for determining the ground thermal conductivity are based on methods by Carslaw and Jaeger (1959) and Ingersoll and Plass (1948). Eklöf and Gehlin (1996) and Austin et al. (2) built early portable units for field tests. Gehlin and Spitler (23) and Sanner et al. Corresponding author. Tel.: ; fax: address: rick.beier@okstate.edu (R.A. Beier). (25) review the history of in situ TRTs. Hellström (1991) presented analytical solutions for determining the thermal resistance of the borehole if the placement of the U-tube is known. During a typical in situ thermal response test a pump circulates a fluid through a controlled heat source and the U-tube in a closed loop. The equipment on the ground surface includes the pump, electric heater as the heat source, flow meter, temperature sensors, and data acquisition system. In an ideal test, the electric heater supplies heat to the fluid at a constant rate. The temperatures of the fluid are measured at the inlet and outlet locations of the U-tube. Sometimes, as an alternative to the electric heater, a reversible heat pump is used to heat or cool the fluid (Witte et al., 22). In another variation of a TRT, Raymond et al. (21) place heating cables inside the pipes as the heat source. Thermistors are placed along the cables to measure temperatures at specified depths. Monitoring the temperatures continue after the heat injection is stopped. The ground thermal conductivity and borehole thermal resistance are determined by using data during both the heat injection and recovery periods (Raymond et al., 211a,b). Most analysis methods for the in situ test use the mean of the inlet and outlet fluid temperatures, (T in and T out, respectively) at each time increment to represent the average temperature along the length of the ground heat exchanger. This mean temperature is an approximation to the true average temperature of the circulating fluid. The difference between this average loop temperature and the undisturbed ground temperature, T s, represents the temperature difference driving the heat transfer between the circulating fluid /$ see front matter 212 Elsevier Ltd. All rights reserved.

3 24 R.A. Beier et al. / Geothermics 44 (212) Nomenclature a constants A constants B constant c volumetric heat capacity, J/(K m) C constants cosh 1 (x) inverse hyperbolic function with argument x d diameter, m h convective film coefficient, W/(K m 2 ) k thermal conductivity, W/(K m) L length of borehole, m N dimensionless thermal conductance p power in p-linear average Q heat input rate, W r radius, m R thermal resistance, (K m)/w T temperature, C w volumetric fluid flow rate, m 3 /s x s distance between center of pipe and center of borehole, m z vertical depth coordinate, m Greek symbols ij constants s thermal diffusivity, m 2 /s constant ı i constants T p p-linear average temperature, K T s temperature difference from undisturbed ground temperature, K T 1 temperature change along segment in pipe 1, K T 2 temperature change along segment in pipe 2, K density, kg/m 3 position along fluid path in U-tube, m Subscripts b borehole D dimensionless f circulating fluid in borehole entrance m mean temperature approximation out borehole exit po outside of pipe pi inside of pipe pw pipe wall rs reference value for ground or soil s ground or soil w water 1,2 pipe number Fig. 1. (a) Borehole geometry and (b) thermal resistance model. effects of the thermal storage of the circulating fluid, finite borehole diameter, location of the U-tube, and the grout (or water) outside the U-tube. Marcotte and Pasquier (28) argue that the use of the mean temperature under some conditions may lead to an overestimation of the borehole resistance. They introduce a p-linear approximation where the circulating fluid temperature varies linearly along the flow path between T in T s p and T out T s p. Marcotte and Pasquier suggest a value of p approaching 1 be used. They base their argument on calculations with a finite element model to generate the entire vertical temperature profiles of the circulating fluid. Du and Chen (211) and Lamarche et al. (21) modify the p-linear approximation by using the borehole wall temperature, T b, as a reference temperature instead of the undisturbed ground temperature, T s. Thus, the specific value of p used by Marcotte and Pasquier does not apply to the different equations used by Du and Chen, and Lamarche et al. The version by Marcotte and Pasquier is easier to apply to a TRT test data set, because T b is not usually measured. Du and Chen and Lamarche et al. use borehole models to estimate T b in their applications of the p-linear approximation. Until recently vertical temperature profiles have not been measured in the circulating fluid along the borehole to check such models with field data. Acuña et al. (29) and Acuña (21) have measured the entire vertical temperature profile of the circulating fluid during a TRT. In addition, they measure the vertical temperature profile in the undisturbed rock. The temperature measurements are made with fiber optic cables placed inside the U-tube. Laser light is guided down the cables, and an optical method based on Raman scattering gives the temperatures along the cables. Fujii et al. (29) have also applied this measurement technique to ground heat exchangers. To help interpret this new vertical temperature data set, the present paper develops an analytical model of the borehole to calculate the vertical temperature profiles of the circulating fluid and heat transfer rates to the ground. The vertical temperature profile and the undisturbed ground. The mean temperature approximation simplifies the calculations, but the method implicitly assumes the heat transfer rate is uniform along the length of the borehole, which does not strictly occur. The mean temperature during an in situ test is often graphed with the logarithm of time as shown in Fig. 2 for a data set from the borehole described in Section 3. The late-time data follow a linear trend, which is consistent with a model representing the borehole as a line-source (Carslaw and Jaeger, 1959; Ingersoll and Plass, 1948). The ground thermal conductivity is inversely proportional to the late-time slope. A change in the input heat rate causes the jump in the temperature at 2.1 h. Even without this jump the early-time data would not follow the late-time trend due to the Fig. 2. Circulating fluid temperatures during an in situ thermal response test.

4 R.A. Beier et al. / Geothermics 44 (212) of the undisturbed ground serves as input into the model. Unlike the proposed model in the present paper, previous models by Beier (211), Yang et al. (29), and Zeng et al. (23) assume a uniformly distributed undisturbed ground temperature. Furthermore, the theoretical temperature profiles in these earlier papers have not been verified by field data. The data set in the present paper provides such verification. The proposed model provides an estimate of the borehole resistance without requiring the usual mean temperature approximation. Many previous models of thermal response tests assume the U-tube legs are symmetrically placed with respect to the center of the borehole. The model does not require symmetry and can check if the assumption of symmetry provides a reasonable match to the measured vertical temperature profiles. 2. Model of borehole heat transfer During the TRT the heat transfer in the borehole is easier to model in the late-time period when the linear logarithmic trend applies. During this period the temperature at any point in the borehole, including the circulating fluid, increases at the same instantaneous rate, if the heat input and volumetric flow rate are constant. This period can be described as a steady-flux period. Beier (211), Yang et al. (29), and Zeng et al. (23) have presented analytical models for the vertical temperature profiles of the circulating fluid during this period. The models treat the undisturbed ground temperature as constant with depth. They ignore any heat conduction in the vertical direction within the pipe wall, the grout, and the ground. During the duration of a thermal response test, the assumption of no vertical heat conduction is reasonable based on the results presented by Diao et al. (24). The model developed in this paper adopts the steady-flux approach, which restricts the model to the late-time period. The restriction of uniform ground temperature with depth is relaxed so that the model can handle a known vertical temperature profile for the undisturbed ground. The thermal resistance model of the borehole in Fig. 1b is applied to quantify the heat transfer between the circulating fluid in each leg and the undisturbed ground. Also, heat exchange occurs between the two pipes. Heat transfer from the first pipe to the borehole wall must pass through a borehole resistance, R b1. The borehole resistance includes all the thermal resistances between the fluid and the borehole wall, such as the inner-pipe film resistance, pipe wall resistance, grout (or water) resistance, and any contact resistances. The temperature of the borehole wall, T b, is treated as being uniform around the borehole circumference at a given depth. The heat transfer from the borehole wall to the distant, undisturbed ground passes through the ground resistance, R s1. Corresponding resistances apply for the second pipe. The heat transfer between the fluids in the two pipes passes through the thermal resistance R 12. The inlet fluid temperature at any given time is measured in the test and is an input to the model. The ground thermal resistance does change with time slowly, but this resistance can be evaluated at any given time the model is applied. The thermal resistances do not change with depth. The model calculates the vertical temperature profile at a given time under the above assumptions. An energy balance on an elemental length of the first pipe (where the fluid enters) gives an equation for the fluid temperature in the pipe at depth z as presented by Hellström (1991). The energy balance is written as: wc dt 1 dz = 1 1 (T 2 T 1 ) + (T s T 1 ) (1) R 12 R b1 + R s1 where w is the fluid volumetric flow rate, c is the fluid volumetric heat capacity, T 1 and T 2 are the fluid temperatures in the first and second pipes, respectively. The term on the left side of the equation takes into account the difference in thermal energy of the fluid entering and the fluid leaving the element. The first term on the right side accounts for the shunt heat transfer between the fluids in the pipes. The second term on the right side takes into account the heat transfer between the fluid and the surrounding ground. A similar energy balance on the second pipe (from which the fluid exits at the surface) gives: wc dt 2 dz = 1 1 (T 1 T 2 ) + (T s T 2 ) (2) R 12 R b2 + R s2 Note that the direction of the circulating flow is important in determining the algebraic sign on each term. The temperature of the entering fluid, T in, is one boundary condition: T 1 () = T in (3) At the bottom of the U-tube (z = L), the temperatures of the circulating fluid in both pipes are set equal. That is: T 1 (L) = T 2 (L) (4) Dimensionless variables and parameters are introduced to form a set of equations with the least number of independent parameters. Dimensionless temperature, T D, and depth, z D, are defined as: T D = T T rs T in T rs (5) z D = z (6) L In general, the undisturbed ground temperature, T s, is a function of the depth, z. The temperature, T rs, is a reference ground temperature. The introduction of these variables into Eqs. (1) and (2) gives the dimensionless equations: dt D1 = (T D2 T D1 ) N s1 T D1 + N s1 T Ds dz D (7) dt D2 = (T D2 T D1 ) + N s2 T D2 N s2 T Ds dz D (8), N s1, and N s2 are dimensionless conductances as explained below. The vertical profile of the dimensionless ground temperature is represented by a third-degree polynomial in z D with the coefficients A 1, A 2, and A 3 obtained from distributed temperature measurements along an existing borehole. (See next section for details about our test case.) The equation is written as: T Ds (z D ) = T s(z D ) T rs = A 1 z D + A 2 z 2 T in T D + A 3 z 3 D (9) rs The reference ground temperature, T rs, is set equal to the ground temperature at z =, T s (). The difference between T in and T rs provides a temperature difference for the dimensionless profile in Eq. (9). As a reference temperature, T rs may be set equal to any value within the ground temperature profile. Three dimensionless thermal conductances appear in Eqs. (7) and (8) and are defined as: = N s1 = N s2 = L wcr 12 (1) L wc(r b1 + R s1 ) L wc(r b2 + R s2 ) (11) (12) The conductance corresponds to the heat transfer between the two legs of the U-tube. The conductances N s1 and N s2 correspond to the heat exchange between the ground and the circulating fluid

5 26 R.A. Beier et al. / Geothermics 44 (212) in the respective leg. The boundary conditions in terms of dimensionless variables are: T D1 () = 1 (13) T D1 (1) = T D2 (1) (14) Eqs. (7) and (8) are two first-order ordinary differential equations. Solving the equations simultaneously (Spiegel, 1967), one identifies the solution as: T D1 (z D ) = C 1 e a 1 z D + C 2 e a 2 z D z D + 12 z 2 D + 13 z 3 D (15) T D2 (z D ) = C 3 e a 1 z D + C 4 e a 2 z D z D + 22 z 2 D + 23 z 3 D (16) where a 1 = (Ns1 N s2) + [(N s1 N s2) 2 + 4[( + N s1)( + N s2) N 2 12 ]]1/2 2 (17) a 2 = (Ns1 N s2) [(N s1 N s2) 2 + 4[( + N s1)( + N s2) N 2 ]]1/ (18) C 3 = [N s1 + + a 1 ] C 1 (19) C 4 = [N s1 + + a 2 ] C 2 (2) The constants written as ij are listed in Appendix A. To evaluate the remaining two constants C 1 and C 2 in Eqs. (15) and (16), one applies the boundary conditions in Eqs. (13) and (14) to find: C 1 = (1 1 )(ı 2 1) + ı 3 e a 2 (21) e (a 1 a 2 ) (ı 1 1) + (ı 2 1) C 2 = 1 C 1 1 (22) where ı 1 = N s1 + + a 1 (23) ı 2 = N s1 + + a 2 (24) ı 3 = (25) The dimensionless temperature profiles in Eqs. (15) and (16) are dependent on 3 dimensionless parameters: N s1, N s2, and. If the 2 U-tube legs are not symmetrically placed from the center of the borehole, N s1 and N s2 have different values, because R b1 does not equal R b2. Previous studies, Hellström (1991), Yang et al. (21), often make the simplifying assumption that the 2 U-tube legs are located in symmetric positions. In the symmetric case R b1 equals R b2, and R s1 equals R s2. Then, N s1 is equal to N s2, and the dimensionless temperature profiles depend on the 2 dimensionless parameters N s1 and. Under the symmetry assumption, one can write R b1 = R b2 = 2R b. Here R b represents the thermal resistance between the circulating fluid in both pipes taken together and the borehole wall. With the temperature profiles in Eqs. (15) and (16), one can find an expression for the heat transfer rate between the circulating fluid in the U-tube and the surrounding ground. The heat transfer rate is found by integrating along the vertical borehole and using the thermal resistances in Fig. 1. The heat transfer rate, Q, is written as: L L 1 1 Q = (T 1 T s )dz + (T 2 T s )dz (26) R b1 + R s1 R b2 + R s2 In terms of dimensionless variables Eq. (26) becomes 1 Q D = (T D1 T Ds )dz D + N 1 s2 (T D2 T Ds )dz D (27) N s1 Table 1 Parameters used in verification case of borehole. Parameter Symbol Value Borehole radius r b 7 mm Active U-tube length L m Pipe outer radius r pi 2 mm Pipe inner radius r po 17.6 mm Pipe wall thermal conductivity k pw.4 W/(K m) Thermal conductivity of water in borehole k w.57 W/(K m) Ground thermal conductivity k s 3.8 W/(K m) Ground density s 27 kg/m 3 Ground specific heat C s 83 J/(kg K) Fluid volumetric heat capacity c 4,26, J/(K m 3 ) Fluid volumetric flow rate w.5 l/s Convective heat transfer coefficient inside pipes h i 12 W/(K m 2 ) Heat input rate Q 945 W Reference ground surface temperature T rs 1.5 C Average ground temperature T s 9.1 C where Q D = Q(R b1 + R s1 ) (28) (T in T s )L Substituting Eqs. (15) and (16) for the dimensionless temperatures into Eq. (27) and integrating, ones finds: Q D = C 5 + C 6 + C 7 (29) where C 5 = C 1 [ 1 + N s2 N s1 ( N12 + N s1 + a 1 )] (e a1 1) a 1 (3) C 6 = C 2 [ 1 + N s2 N s1 ( N12 + N s1 + a 2 )] (e a2 1) a 2 (31) C 7 = A A N [ s A A 2 N s A A 3 4 ] (32) Like the dimensionless temperature profiles the dimensionless heat transfer rate also depends on the 3 dimensionless parameters: N s1, N s2, and. Again, if the U-tube is symmetrically placed in the borehole, R b1 equals R b2, and R s1 equals R s2. Then, N s1 is equal to N s2, and the dimensionless heat transfer rate depends on the 2 dimensionless parameters N s1 and. 3. Experimental setup The borehole model has been applied to a Distributed Thermal Response Test (DTRT) in a vertical borehole with a polyethylene U-tube, i.e., a distributed temperature sensing system (DTS) has been installed in the tested borehole to obtain vertical temperature profiles during the thermal response test (Acuña et al., 29). The borehole is located in Stockholm, Sweden. The borehole is filled with groundwater, which is common practice in Sweden. The total U-tube length is 257 m, but the groundwater level is about 5.5 m below the surface. Thus, the active U-tube length is m. Parameters for the tested borehole are listed in Table 1. The circulating fluid in the U-tube is a water solution with 2% ethanol by volume to provide a freezing point of 8 C based on the properties listed by Melinder (27). Through an optic fiber cable installed along the depth of the borehole, DTS provides the temperature profile without the need of many individual temperature sensors. A cable is installed as a loop inside both legs of the U-tube to measure the fluid temperature profile of the circulating fluid with depth. The same cable provides the undisturbed ground temperature profile, measured

6 R.A. Beier et al. / Geothermics 44 (212) m sec on 1 3 m sec on 2 5 m sec on 3 7 m sec on 4 9 m sec on 5 11 m sec on 6 13 m sec on 7 15 m sec on 8 17 m sec on 9 19 m sec on 1 21 m sec on m sec on m F4 F5 F7 F8 F9 F13 F23 F19 F16 Fig. 3. Sections along borehole depth. before the test was carried out during undisturbed ground conditions. Short light pulses from a laser are directed through the optic cable. A nonlinear part of the back-scattered light has a different frequency from the input light and travels back from the temperature measurement location to the input location. The light scattering process that produces the frequency shift is called Raman scattering. The temperature and the position of any measured section are estimated by analyzing the ratio between the intensities of the upshifted and down-shifted light over a time window corresponding to the delay time for the light to travel to the measured section and back. The measurements have been taken with an instrument of the type HALO from Sensornet which, according to the specifications from the manufacturer, has a minimum spatial resolution of 2 m and temperature accuracy depending on the averaging time and distance from the instrument. The instrument data sheet presents temperature uncertainties within the range.5.45 K corresponding to measurement times from 6 min to 15 s. It is important to keep in mind that this type of measurement depends on integration time, the measuring length interval, the laser features, the distance between the measured section and the instrument (cable length) and the calibration procedure, among others. For this specific test, these optic cables provide temperature measurements at 52 segments (1 meter long each) that were later simplified into 26 locations, which are delimited and numbered F1 through F26 and illustrated in Fig. 3. Temperature measurements are averaged over a segment of the cable for each numbered temperature location. The measurements are averaged over a time window of 5 min and the standard deviation during a 3-day period of measurements under undisturbed ground conditions is.3 K (the instrument data sheet gives a maximum deviation of.1 K for a 5 min integration time). Regarding systematic errors, the cable was carefully calibrated using an ice bath and the signal offset was corrected, allowing to adjust for accurate absolute temperature values along the whole cable length (±.1 K). Since most calculations are based on temperature differences, no significant errors are expected. Also, the influence of the unknown lateral position of the fiber optic cable inside the U-pipe was considered in the analysis of systematic errors. The pattern along the U-pipe pipe is turbulent during the test (Reynolds number between 65 and 89), and the boundary layer thickness is between.3 and.4 mm (heat conduction is the only heat transfer mechanism in this region). The temperature difference between the pipe wall and the fluid bulk temperature is calculated to be about.15 K for the conditions of this test, and the temperature drop in the laminar layer is about.14 K, meaning it occurs mainly in this thin layer and that the rest of the temperature profile is flat. Given the diameter of the fiber optic cable of 3.8 mm, it can be stated that the systematic error due to the cable position inside of the pipe is negligible. Details of these measurements have been reported earlier by Acuña et al. (29) and Acuña (21). Additional equipment for the DTRT consisted of a circulation pump, an inductive flow and energy meter, flow regulation valve and an electric heater with an adjustable heating power between 3 and 12 kw. The DTRT lasted approximately 16 h and consisted of 4 phases Acuña et al. (29) and Acuña (21). The first phase of the DTRT focused on measuring the undisturbed ground temperature with no fluid circulation in the U-tube. This period lasted 65 h. During the second phase, the fluid was circulated through the U-tube for 24 h without any heating. The temperatures along the entire borehole length become nearly uniform due to the circulation, with the mean value equal to the mean temperature of the undisturbed ground temperature profile (9.1 C, shown in Table 1). During the third phase of 48 h, a nearly constant heat input rate to the circulating fluid was maintained. The rising temperature response of this period can be used to estimate the ground thermal conductivity and borehole resistance. Finally, during the fourth phase, temperature measurements continued without any heating or circulation in an effort to observe the temperature recovery, and to determine the ground thermal conductivity, while taking advantage of the small temperature gradients inside the borehole during this period. The resulting thermal conductivity value was 3.8 W/(K m), as tabulated in Table Vertical temperature profiles The detailed temperature measurements of the DTRT provide a data set to test the vertical profile model. As mentioned earlier, the model is valid for the late-time (steady-flux period) during the constant heating phase (phase 3). For this reason the model is compared to the measured vertical temperature profiles near the end of the constant heating phase. Before applying the model, information is needed about the undisturbed ground temperature. For convenient input into the model, the ground temperatures are normalized using Eq. (9) with T rs set equal to the ground temperature (1.5 C) at 5.5 m (the groundwater level) below the surface. In the model, the vertical coordinate z is set equal to zero at 5.5 m below the surface. Fig. 4 shows a graph of the normalized ground temperature. A third-order polynomial fit to the temperature profile obtained from the data measured during the DTRT is also shown. This polynomial fit serves as input into the model and assigns values to the coefficients in Eq. (9). The average temperature of the undisturbed ground is 9.1 C, with the minimum temperature at 11 m below the surface. The applications of the model below use either the average temperature uniformly applied along the depth or the polynomial fit to the detailed temperature profile. As Fig. 4 indicates, the measured temperature data are extrapolated (over a 5 m distance) to estimate the surface (z D = ) and bottom-tube (z D = 1) temperatures. Conventional analysis methods based on a line-source model (Ingersoll and Plass, 1948) provide an estimate for the ground

7 28 R.A. Beier et al. / Geothermics 44 (212) Depth (m) Measured temp. Tubes together 25 Bottom temp. fit Tubes apart Temperature (ºC) Fig. 6. Calculated temperatures of the circulating fluid compared to measured temperatures. In the calculations the U-tube is symmetrically placed in the borehole and the undisturbed ground temperature is uniform. Fig. 4. Vertical profile of normalized undisturbed ground temperature. thermal conductivity and borehole resistance. The model uses the average of the inlet and outlet fluid temperatures at a given time, and applies this mean temperature along the entire borehole. According to Acuña et al. (29), the application of this technique to the test data, gives a ground thermal conductivity of 3.8 W/(K m) and a borehole resistance of.79 (K m)/w. The line-source model does not match the early-time data, because the model does not fully account for the thermal storage of the circulating fluid, location of the U-tube, and the thermal properties of the water-filled borehole. For this reason, the first 15 h of the data are ignored in the analysis. The average temperature of 9.1 C is used for the undisturbed ground temperature. Within the line source model, the mean temperature approximation implicitly assumes the heat transfer rate is uniform along the length of the borehole. The resulting temperature profiles within the U-tube legs appear as straight lines, which do not agree with the measured temperatures in Fig. 5. These measured temperatures correspond to 48 h into the constant heating rate phase of the test. Indeed, the estimated bottom-loop temperature of the circulating fluid is substantially larger than the measured bottom-loop temperature. As a more general approach, the vertical temperature profile model is applied to estimate the borehole resistances corresponding to each leg, R b1 and R b2. Eq. (29) for the heat transfer rate is a nonlinear equation with three unknowns: R b1, R b2 and R 12. The heat input rate, Q, and the temperatures T in and T s (or the ground Depth (m) Measured temp. 25 Mean temp. approx. Unifrom ground temp. Ground temp. profile Temperature (ºC) Fig. 5. Calculated temperatures of the circulating fluid compared to measured temperatures. In the calculations the U-tube is symmetrically placed in the borehole. temperature profile) are measured and are input to the model. The rock thermal conductivity estimate from the line-source model is also used as input. Below Eq. (29) is used to estimate the borehole resistances R b1 and R b2 over a possible range of values for R 12. The exact locations of the U-tube legs within the borehole are unknown and probably vary with depth. A typical assumption is a symmetrically placed U-tube within the borehole where the position does not vary with depth. A logical starting approach is to apply Eq. (29) assuming a symmetrically placed U-tube, and the average ground temperature applied uniformly over the depth, to see if a reasonable fit can be obtained. Under these conditions R b1 = R b2. Using one equation and two unknowns (R b1 and R 12 ), one needs to determine the value of R 12 independently of R b1. The practical range of R 12 is generated using the equations in Appendix B. The measured vertical temperature profiles of the circulating fluid in the U-tube are shown in Fig. 6 with the model results for a symmetrically placed U-tube after 48 h into the constant heating rate phase. The proposed model has been validated with measured temperature profiles at other times during the steadyflux period of the TRT with matches similar to the shown profiles at 48 h. The profiles corresponding to the minimum and maximum values of R 12 are shown for reference. For the minimum R 12 the tubes are next to each other with no separation. For maximum R 12 the tubes are spread apart and touching opposite sides of the borehole. The bottom temperature is matched for a value of R 12 equal to.37 (K m)/w. The corresponding value of R b1 is.129 (K m)/w. Note the value of the borehole resistance R b is equal to R b1 /2 for a symmetrically placed U-tube. Thus, the corresponding estimate of R b based on the vertical temperature profile model is.64 (K m)/w. The earlier value of.79 (K m)/w by conventional TRT methods is 23% greater, based on the mean temperature approximation. A previous analysis (Beier, 211) indicates that the error in the conventional estimate increases if the circulating fluid flow rate decreases or the U-tube length increases. Furthermore, the mean temperature approximation systematically overestimates the thermal resistance. Proposing an alternative to the mean temperature approximation, Marcotte and Pasquier (28) assume a fluid temperature variation at power p, T() p, varies linearly within the U-tube between T in p and T out p. Here T represents the temperature difference from the undisturbed ground temperature, T s. Under this assumption the vertical temperature profiles are given by: { ( ) 1/p T() = T in p + ( T out p T 2L in )} p (33)

8 R.A. Beier et al. / Geothermics 44 (212) Depth (m) Measured temp. p = 1 p p = -2 p = Temperature (ºC) Fig. 7. Calculated temperatures of the circulating fluid based on the p-linear average approximation compared to measured temperatures. Note the variable represents the position along the full pathway in the U-tube from the inlet to outlet connections. Temperature profiles for various values of p are compared to the measured profiles in Fig. 7. The profile with p = 1 is identical to the mean temperature approximation. Marcotte and Pasquier recommend using p 1, based on comparisons with numerical models of a borehole. However, the corresponding profile does not agree with the measured data in Fig. 7. The profile with p = 3 matches the bottom temperature, but the shape of the profile does not match the other measured temperatures. For a calculation of borehole resistance, Marcotte and Pasquier (28) recommend the use of the p-linear average temperature along the U-tube. This average temperature is obtained from integration of Eq. (33) along the U-tube path to give: T p = p( T in p+1 T out p+1 ) (1 + p)( T in p T out p (34) ) This average temperature is used in the line-source model in place of T m. The borehole resistance with p 1 is.74 (K m)/w, which is a smaller estimate than the value stated above from the mean temperature approximation. The estimates for borehole resistance with p = 2 and p = 3 are smaller values of.68 (K m)/w and.63 (K m)/w, respectively. These last 2 estimates are close to the.64 (K m)/w estimate from the proposed model based on Eq. (29). The best choice of p is different from the value of p 1 suggested Marcotte and Pasquier. The proposed model using Eq. (29) also allows one to study how R b1 varies as a function of the input value for R 12 as shown in Fig. 8. Again, the smallest value of R 12 corresponds to the tube legs R b1 ((Km)/W) Uniform ground temperature Ground temperature profile R 12 ((Km)/W) Fig. 8. Borehole resistance, R b1, corresponding to each leg of a symmetrically placed U-tube as a function of the shunt resistance, R 12. R b1 or R b2 (Km)/W) Uniform ground temperature Ground temperature profile R b2 R b R 12 ((Km)/W) Fig. 9. Borehole resistances, R b1 and R b2, corresponding to separate legs of an asymmetrically placed U-tube as a function of the shunt resistance, R 12. touching each other, and the largest value corresponds to the legs against opposite walls of the borehole. At first glance, one may expect R b1 to decrease in Fig. 8 as R 12 increases (shanks spreading apart), which is opposite of the results. However, simultaneously increasing R 12 (smaller shunt effects) and decreasing R b1 would increase the heat transfer rate to the rock and the measured heat input rate to the circulating fluid. Because all the calculations for Fig. 8 are made with the measured heat input rate fixed, a different trend appears. The estimated range of R 12 has a substantial uncertainty, and the range used is probably too wide because convection effects for the groundwater in the borehole are not taken into account. The vertical temperature model has been applied using a uniform ground temperature (9.1 C) as one case, and the measured ground temperature profile (Fig. 4) as a second case. The temperature profiles are nearly identical for both cases as shown in Fig. 5, where the bottom-tube temperature is forced to match the measured value. Also, the estimated borehole resistances in Fig. 8 are nearly the same. For this borehole, using the average ground temperature applied along the entire borehole depth gives good results. Of course, it is not known if the uniform average ground temperature approximation gives such good results in all other applications. If the measured bottom temperature is used as input, the model handles the case of an asymmetrically placed U-tube, where R b1 may differ from R b2. Using Eq. (15), one sets the circulating fluid temperature at the bottom of the loop to the measured temperature. Again the value of R 12 is varied over a range determined by the equations in Appendix B. The range of calculated values of R b1 and R b2 are graphed in Fig. 9. For each value of R 12 the values of R b1 and R b2 are unique. If the model uses the ground temperature profile (Fig. 4) as input, the results are nearly the same as the results with the averaged uniform ground temperature. For an asymmetrically placed U-tube, the calculated temperature profiles in Figs. 1 and 11 have a rather narrow range as R 12 is varied with the bottom-loop temperature fixed at its measured value. The average error between the calculated and measured temperatures is shown in Fig. 12 as the value of R 12 is varied. The average error is the average magnitude of the differences between these temperatures along the borehole. The symmetric case (with R b1 = R b2 =.129 K m/w) has an average error of.27 C, while the minimum error of.22 C occurs at a slightly larger value of R 12. In practice, these differences are beyond the resolution of the measured temperatures. Thus, the symmetric case fits the measured temperatures, as well as any of the other cases, within the uncertainty of the measured temperatures.

9 3 R.A. Beier et al. / Geothermics 44 (212) Depth (m) Measured temp. 25 Tubes together Symmetric tubes Tubes apart Temperature (ºC) Fig. 1. Calculated temperatures of the circulating fluid compared to measured temperatures. In the calculations the U-tube is asymmetrically placed in the borehole, the bottom-loop temperature is fixed, and the undisturbed ground temperature is uniform. Depth (m) Measured temp. 25 Tubes together Symmetric tubes Tubes apart Temperature ( C) Fig. 11. Calculated temperatures of the circulating fluid compared to measured temperatures. In the calculations the U-tube is asymmetrically placed in the borehole, the bottom-loop temperature is fixed, and the undisturbed ground temperature follows the measured vertical profile. et al. (29) and Acuña (21) have applied the line-source model (Ingersoll and Plass, 1948) to each section. The heat rate input, Q, into a segment can be estimated using the sum of the heat rates at each of the U-pipe legs as: Q = wc( T 1 + T 2 ) (35) The fluid temperature difference across the segment, T 1 or T 2, is evaluated from the measured fluid temperatures at the entrance and exit of the section. This detailed DTRT analysis by Acuña et al. (29) provides another comparison for the vertical temperature model. From the DTRT analysis the borehole thermal resistances over the 2 m sections vary between.54 and.78 (K m)/w. This variation suggests the pipe positions in the borehole change along the borehole length. The mean value of R b over all sections is.62 (K m)/w. The vertical profile method with a symmetrically placed U-tube gives a borehole resistance of.64 (K m)/w, which is nearly the same value as found with the DTRT analysis. For the DTRT analysis, the thermal conductivity values range from 2.6 to 3.62 W/(K m) among the sections. The mean thermal conductivity is 3.1 W/(K m), which agrees with the estimate from the conventional analysis [3.8 W/(K m)]. The above agreement between the proposed model and the DTRT analysis is obtained by using the inlet temperature and the bottom-loop temperature as input in the proposed model. Thus, adding the bottom-loop temperature measurement during the thermal response test enhances the estimate of borehole resistance. Without the bottom-loop measurement in the proposed model, one could estimate the value of R b by dividing the median value of R b1 in Fig. 8 by 2 to estimate R b as.68 (K m)/w. In Fig. 8 the value of R b1 (and thus R b ) varies by ±12% from its median value when R 12 is varied. In a previous study of thermal response tests on groutfilled boreholes, Beier (211) found the estimate of R b changes by ±2% from its mean value when R 12 is varied. The much wider range of the R b estimate in the water-filled borehole is partially due to the difficulty in representing the convection effects and to a lower U-tube pipe wall resistance, which determine the lower bound for R 12 in Fig Analysis on each vertical section The temperature history of each section in the DTRT provides sufficient data to evaluate the rock thermal conductivity and borehole resistance in each 2 m section of the borehole (Fig. 3). Acuña Average difference (ºC) Uniform ground temperaturre Ground temperature profile R 12 ((Km)/W) Fig. 12. Average difference between the calculated and measured temperatures as the shunt resistance, R 12, varies. 6. Borehole resistance in design tools Often the results from TRT analysis are used as input into design software for a GSHP system. Estimates for both ground thermal conductivity and borehole thermal resistance are required by the design calculations. The mean temperature approximation, the p- linear approximation and the proposed vertical temperature profile model have been used in this paper to estimate borehole resistance from TRT data set. Each method is based on different assumed vertical temperature profiles in the borehole fluid. In the ideal case, the assumption about the vertical temperature profile should be the same in both the TRT analysis and the design software. Under limited circumstances, an overestimated borehole resistance from the mean temperature approximation in the TRT data analysis may have little impact, if the design program is based on the same approximation. Then both calculations are using the same equation for the heat transfer rate. As an example, little error is expected to be carried over to the design, if the circulating fluid rate and borehole length are nearly the same in the TRT and the design. On the other hand, the proposed vertical temperature profile model allows the borehole resistance from a test to be rigorously carried over to the design software under different circulating rates and borehole depths. Of course, the design software must include the same vertical profile model in order to be consistent.

10 R.A. Beier et al. / Geothermics 44 (212) Conclusions A steady-flux model has been developed for the vertical temperature profile of the circulating water through a U-tube in a borehole heat exchanger. This analytical model applies during the steadyflux period of an in situ thermal response test. Most current analysis methods use the mean of the inlet and outlet temperatures of the circulating water, which is assumed to be a representative temperature for the average fluid temperature along the depth of the borehole. The proposed model does away with the need for this mean temperature approximation. The proposed model also handles a vertical temperature profile for the undisturbed ground, if known from independent measurements. The model has been verified using measured vertical temperature profiles in a borehole equipped with a distributed temperature sensing (DTS) system. The system uses optic fiber cables installed inside the U-tube pipes along the length of the borehole. These temperature measurements provide much more information than typically obtained from measuring only the inlet and outlet fluid temperatures. The analytical model provides a method to estimate the borehole thermal resistance taking into account the vertical temperature profile instead of relying on the mean temperature approximation. For the borehole studied, the borehole resistance is 23% less than the value estimated from the mean temperature approximation. In this borehole the proposed model gives nearly identical results whether the average undisturbed ground temperature or the actual measured vertical ground temperature profile is used as input. The p-linear approximation gives a better estimate of borehole resistance in the field case than the mean temperature approximation. However, the best value to use for the exponent p in Eqs. (33) and (34) is unknown for a specific case unless the fluid temperature at the bottom of the U-tube is measured. Then, the bottom temperature can be used as an additional constraint in Eq. (33) to set the value of p. The proposed model fits the measured vertical temperature profiles, if the U-tube legs are treated as if they are symmetrically placed around the borehole center. This simplified approach gives good results even though a previous DTRT analysis indicates the pipe positions inside the borehole change with depth. The results suggest that on average each pipe may be about the same distance from the borehole wall. Acknowledgements The International Ground Source Heat Pump Association (IGSHPA) provided support to R.A. Beier. The Swedish Energy Agency and all sponsors within the EFFSYS2 and EFFSYSPLUS research programs are also greatly acknowledged for financing the project at KTH. 23 = B = B 22 3(N s1 N s2 ) = B 21 2(N s1 N s2 ) = B 2 (N s1 N s2 ) The additional constants and the B ij have values of: (A.5) (A.6) (A.7) (A.8) = (N s1 + )(N s2 + ) + (A.9) C 8 = N s1 N s2 + N s1 + N s2 (A.1) B 1 = A 1 N s1 (A.11) B 11 = 2A 2 N s1 C 8 A 1 (A.12) B 12 = 3A 3 N s1 C 8 A 2 (A.13) B 13 = C 8 A 3 (A.14) B 2 = A 1 N s2 (A.15) B 21 = 2A 2 N s2 C 8 A 1 (A.16) B 22 = 3A 3 N s2 C 8 A 2 (A.17) B 23 = C 8 A 3 (A.18) Appendix B. Thermal resistances The thermal shunt resistance between the two legs is evaluated based on an approach similar to the one used earlier by Kavanaugh (1985). Due to convection effects in the water-filled borehole and the complicated geometry, no rigorous analytical estimate of R 12 is available. The intent is to provide an expected range of R 12, while seeking a better estimate from the model fits in Figs The shunt resistance is estimated as the sum of two pipe wall resistances, R pw, the film resistances, R f, and the water (or grout) resistance, R w : R 12 = 2R pw + 2R f + R w (B.1) The water (or grout) resistance between the two pipes is evaluated based on a heat conduction shape factor between two isothermal cylinders in an infinite medium. From Schneider (1985) the conduction heat transfer rate per unit length between two parallel cylinders of equal size is: Q L = 2k w T s cosh 1 ((2x 2 s /r 2 po) 1) (B.2) Appendix A. Expressions for constants The expressions for the constants ij are listed below as: 13 = B = B 12 3(N s1 N s2 ) = B 11 2(N s1 N s2 ) = B 1 (N s1 N s2 ) (A.1) (A.2) (A.3) (A.4) The shank distance, x s, is the distance between the center of the pipe and the center of the borehole. Eq. (B.2) is an approximation to our case, because the equation does not take into account the interaction of the two pipes with the borehole wall or any convection effects. Then, the water (or grout) resistance is approximated as: R w = cosh 1 ((2x 2 s /r2 po ) 1) 2k w (B.3) In this water-filled borehole, the thermal conductivity k w is set equal to the thermal conductivity of water. This approach ignores the convection effects in the water-filled borehole and overestimates the resistance R w.

11 32 R.A. Beier et al. / Geothermics 44 (212) The pipe wall resistance is calculated for only one-half of the tube wall area, which represents the section of wall area facing the neighboring pipe: R pw = ln(r po/r pi ) (B.4) (1/2)2k pw Similarly, the convective film resistances are calculated for only one-half of the pipe wall area, which represents the section facing its neighbor. Thus, 1 R f = (B.5) (1/2)d pi h pi Eqs. (B.2) to (B.5) are substituted into Eq. (B.1) to get an expression for the shunt resistance, R 12. To obtain a minimum (lower bound) value for R 12, the value of R w is set equal to zero in Eq. (B.1) to represent the two pipe walls touching. For a maximum (upper bound) value of R 12, the value of R w is evaluated using Eq. (B.3) at the maximum shank spacing, where each tube is against opposite sides of the borehole wall. References Acuña, J., Mogensen, P., Palm, B., 29. Distributed thermal response test on a U-pipe borehole heat exchanger. 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