Soil Thermal Conductivity Tests

Transcription

1 Soil Thermal Conductivity Tests Richard A. Beier Mechanical Engineering Technology Department Oklahoma State University Stillwater, OK In-Situ Borehole Tests BACKGROUND The design of ground loops for heat pump systems depends on the thermal conductivity of the soil and rock that surrounds the ground heat exchanger. Thermal conductivity is a measure of the capacity of a material to conduct heat. A larger soil thermal conductivity allows heat to be exchanged with the soil at a faster rate for a given ground loop geometry and size. For a given heat input rate, a larger soil thermal conductivity reduces the required depth of a vertical borehole, which decreases installation costs. Because soil thermal conductivity is such an important parameter, insitu tests are routinely performed to estimate it (Figures 1-1 and 1-2). If the uncertainty of soil thermal conductivity is large, the engineer will tend to design the system more conservatively and possibly cause the project to be more expensive than needed. An in-situ test will provide the thermal conductivity value needed for a more accurate design with reduce uncertainty. It will also provide drilling information, which is valuable knowledge that promotes competitive bidding. In addition, the test borehole may be used as one of the heat exchangers in the field, which makes the economics of the test more feasible. HISTORY OF TESTING METHODS The method for in-situ tests on boreholes follows an earlier method of estimating thermal conductivity on a small soil sample in the laboratory by placing a cylindrical probe in the sample. The probe has an internal electrical heat source. Jaeger (1956, 1959) and de Vries and Peck (1958a, 1958b) developed methods to analyze the rising temperature of the probe for a steady heat input. Using the temperature curve they estimated soil thermal conductivity. The concepts from tests on laboratory samples were carried over to the geometry of in-situ field tests on vertical boreholes. Mogensen (1983) proposed an in-situ test as a viable method to determine soil thermal conductivity for ground-source heat pump systems. Beck et al. (1956) described early attempts at in-situ tests. Early portable test rigs were described by Eklöf and Gehlin (1996) and Austin et al. (2000). Reviews of the history and status of in-situ thermal conductivity tests were written by Gehlin and Spitler (2003), Sanner et al. (2005, 2008), and Mattsson et al. (2008). 1

2 TEST SETUP AND DATA An in-situ tests is typically performed on a vertical borehole with approximately the same diameter and depth as the heat exchangers planned for the site. A vertical ground-loop heat exchanger has a U-tube inserted into a borehole, as illustrated in Figure 1-3. Grout is placed in the borehole to fill the space that is not occupied by the U-tube. The equipment for an in-situ test is illustrated in Figures 1-4 and 1-5, where an electric heater at the surface serves as a controlled heat source. Water is pumped through the U- tube and exchanges heat with the ground. All the field test results reported here have been taken with this type of instrumentation. Although an electrical heater is usually used as the heat source, in-situ tests have been performed with other equipment. Witte et al. (2002) use a reversible heat pump to heat or cool the circulating fluid through the ground loop. In the ideal test the heat input rate is constant during the test (Figure 1-6). The fluid flow rate and power input into the electrical heater are recorded (Figure 1-4). Transient temperatures of the circulating fluid are also recorded at the supply and return connections of the ground loop. The average of these two fluid temperatures is used to approximate the average temperature of the loop. This average loop temperature is often plotted versus time on a linear scale or logarithmic scale as shown in Figures 1-6 and 1-7. Because the temperature rises very quickly, the early time data are difficult to view with a linear time scale (Figure 1-6), unless we zoom in on the early-time period. A logarithmic scale for time spreads the early-time data out (Figure 1-7), which helps to detect any early-time abnormalities. In practice, making both graphs and zooming in on different periods is worthwhile. Still, the graph with the logarithmic time scale is the more useful plot for estimating soil thermal conductivity. For a given heat input rate, the recorded temperature rise will be steeper for soil with lower thermal conductivity, because the soil does not conduct the heat away from the borehole as quickly as in the case of higher soil thermal conductivity. Thus, the transient temperature of the ground loop together with the heat input rate measurements contains information about the soil thermal conductivity. As explained later, analysis methods demonstrate the late-time data should follow a linear trend in the semilog plot, and the soil thermal conductivity is inversely proportional to the slope of the trend. Another choice for the horizontal scale is the natural logarithm (base e) of time, where the ln(time) is plotted on a linear scale (not shown). In general, the common logarithmic scale of time (Figure 1-7) is used here, because the values on the scale correspond to our decimal number system. In addition to soil thermal conductivity the loop temperature curve is also influenced by the borehole parameters such as grout thermal conductivity, borehole diameter, and the location of the U-tubes in the borehole. All of these effects can be lumped together into a borehole thermal resistance. As explained later, the borehole resistance can also be evaluated by the borehole test. The heating rate with time is of primary importance and can be estimated from both the electrical power and the fluid temperature difference across the heater. The electric power may be plotted in Watts or Btu/hr (Figure 1-8). The conversion factor is Watts for each Btu/hr. In this test the standard deviation of the electric power is only 0.7% of the average rate (7260 Btu/hr), which indicates good control over the 2

3 power. A few spikes get about 9% below the average power. Variations in power are never completely eliminated during field tests. The heat input rate estimate from the temperate difference of the circulating fluid across the heater is q = m& cp (Tsup Trtn ) (1-1) where q m& c p T sup T rtn = heat input rate, Btu/hr (W) = mass flow rate, lbm/hr (kg/hr) = specific heat of circulating fluid, Btu/lbm-ºF (kj/kg-ºc) = supply water temperature to ground loop, ºF (ºC) = return water temperature from ground loop, ºF (ºC) For the standard (English) units and water as a circulating fluid, the estimate of q becomes q = 500 Q (T T ) (1-2) sup rtn where q is in Btu/hr, Q is the volume flow rate in gpm, and temperatures are in ºF. Comparing the heat input rate from the temperature difference across the heater to the electric power input is a good quality control check. The heat input rate from Equation 1-2 agrees closely with the electric power input to the heater in Figure 1-8. The percent difference between the two estimates is plotted in Figure 1-9. The percent difference stays within ±2% except for brief spikes. High frequency power variations with amplitudes of 100 to 200 Btu/hr are observed throughout the test. A zoom of one portion of the power curve shows the period of these high frequency variations is 4 to 6 minutes (not shown). Such variations are expected from an electric generator or electric power from a local utility. If the period of these variations is not greater than the circulation cycle time through the ground loop, the variations generally do not distort the temperature data. Note the higher frequency variations do not appear in the loop temperature plots in Figures 1-6 and 1-7. The thermal storage of the circulating fluid and heat conduction into the grout and soil tend to dampen out these high frequency variations. However, the lower frequency variations seen in Figure 1-8 do distort the loop temperatures and are suspected of causing the scatter in the late-time temperature curve in Figure 1-7. The undisturbed soil temperature is required to calculate the borehole resistance from the field test data. Gehlin and Nordell (2003) discuss in detail methods to determine the undisturbed ground temperature by inserting a temperature probe into the liquid-filled heat exchanger before starting the test. An alternative method is recording the temperature of the exiting loop fluid immediately after starting the pump. 3

4 SPECIFICATIONS FOR THERMAL CONDUCTIVITY TESTS The American Society of Heating, Refrigerating and Air-Conditioning Engineers ASHRAE (2007) lists a set of specifications for thermal conductivity tests on vertical boreholes following the recommendations of Kavanaugh (2000) and Kavanaugh et al. (2001). A summary of these specifications is given at a website ( Kavanaugh, 2008). The specifications include: Test duration should be 36 to 48 hours. Heat input rate should be 15 to 25 W for each foot of borehole depth. The input electric power to the heater should have a standard deviation of less than ±1.5% of the average value and the peaks less than ±10% of the average. After the ground loop is installed with grout, a five-day waiting period should elapse before starting a test in low-thermal conductivity soils [k<1.0 Btu/hr-ft-ºF (1.7 W/m- ºC)]. In more conductive soils a three-day waiting period is recommended. At the end of the waiting period the ground temperature should be measured by inserting a probe inside the liquid-filled ground heat exchanger at three vertical locations to get an average. An alternative measurement is recording the temperature of the liquid as it exits the loop immediately following startup of the pump to circulate the fluid. Other specifications cover the accuracy for flow rate and temperature measurements, range for differential loop temperatures, frequency of temperature measurements, insulation of surface equipment and a waiting period for retesting a borehole. A suggestion is made to use multiple software programs to analyze test data for thermal conductivity. A working group has developed guidelines for thermal conductivity tests for the International Energy Agency (IEA), Energy Storage and Implementation Agreement, Annex 13. Sanner et al. (2005) list these guidelines. Austin et al. (2000) and Witte et al. (2002) have described the primary sources of experimental errors associated with borehole tests. They suggest the determined soil thermal conductivity from a typical test has an uncertainty of about ±10%. Calculations by Kavanaugh (2000) indicate a 10% error in soil thermal conductivity and diffusivity results in a 4.5% to 5.8% error in the design borehole length and a 1% change in the cooling capacity of a geothermal heat pump system. BOREHOLE TEMPERATURE PROBE The in-situ tests described above yield an average soil thermal conductivity that is averaged over the entire depth of the borehole. No information is obtained about the variation of thermal conductivity with depth. Rohner et al. (2005, 2008) describe a small wireless probe that is placed in a completed but not working borehole heat exchanger. By it own weight the probe sinks to the bottom of the liquid filled U-tube. The probe records temperature and pressure at pre-set time intervals during its descent. Then, the probe is flushed back to the surface using a small pump. Analysis of the data gives a 4

5 vertical temperature profile along the borehole. In order to extract thermal conductivity from the data, one must have a good estimate of the local terrestrial heat flow value. Then a thermal conductivity profile with depth is calculated. If information about the local heat flow is not available, the method still provides information about the vertical variation of soil thermal conductivity. Drury et al. (1984) describe how the temperature profile can be used to identify depths where groundwater flow occurs. REFERENCES ASHRAE ASHRAE Hnadbook: HVAC Applications. Chapter 32. ASHRAE Inc., Atlanta, GA. Austin, W. A., C. Yavuzturk, and J. D. Spitler Development of an in-situ system for measuring ground thermal properties. ASHRAE Transactions 106(1): Beck, A., J. C. Jaeger, and G. Newstead The measurement of the thermal conductivities of rocks by observations in boreholes. Australian Journal of Physics 9: de Vries, D. A., and A. J. Peck, 1958a. On the cylindrical probe method of measuring thermal conductivity with special reference to soils: I. Extension of theory and discussion of probe characteristics. Australian Journal of Physics 11: de Vries, D. A., and A. J. Peck, 1958b. On the cylindrical probe method of measuring thermal conductivity with special reference to soils: II. Analysis of moisture effects. Australian Journal of Physics 11: Drury, M. J., A. M. Jessop, and T. J. Lewis The detection of groundwater flow by precise temperature measurements in boreholes. Geothermics 13(3): Eklöf, C., and S. Gehlin TED A moble equipment for thermal response test. Master s Thesis 1996:198E. Lulea University of Technology, Sweden. Gehlin S., and J. D. Spitler Thermal response test for BTES applications State of the art Proceedings of Futuretock th International Conference of Thermal Energy Storage, Warsaw, Poland. Gehlin, S. E. A., and Nordell, B Determining undisturbed ground temperature for thermal response test. ASHRAE Transactions 109(1): Jaeger, J. C. 1956, Conduction of heat in an infinite region bounded internally by a circular cylinder of a perfect conductor. Australian Journal of Physics 9:

6 Jaeger, J. C The use of complete temperature-time curves for determination of thermal conductivity with particular reference to rocks. Australian Journal of Physics 12: Kavanaugh, S. P Field tests for ground thermal properties Methods and impact on GSHP system design. ASHRAE Transactions 106(1): Kavanaugh, S. P., L. Xie, and C. Martin, Investigation of methods for determining soil and rock formation thermal properties from short-term field tests, ASHRAE TRP. Kavanaugh, S. P Ground source heat pump design, Thermal conductivity testing suggested specifications. Mattsson, N., G. Steinmann, and L. Laloui Advanced compact device for the insitu determination of geothermal characteristics of soils. Energy and Buildings 40: Mogensen, P Fluid to duct wall heat transfer in duct system heat storages. Proceedings of the International Conference on Subsurface Heat Storage in Theory and Practice. Swedish Council for Building Research. June 6-8. Rohner E., L. Rybach, and U. Schärli A new, small wireless instrument to determine ground thermal conductivity in-situ for borehole heat exchanger design. Proceedings World Geothermal Congress, Antalya, Turkey, April Rohner, E., L. Rybach, T. Mégel, and S. Forrer New measurement techniques for geothermal heat pump borehole heat exchanger quality control. Proceedings 9 th International IEA Heat Pump Conference. May 20-22, Zürich, Switzerland. Sanner, S., G. Hellström, J. Spitler, and S. Gehlin Thermal response test Current status and world-wide application. Proceedings World Geothermal Congress. Antalya, Turkey, April Sanner, S., E. Mands, M. K. Sauer, and E. Grundmann Thermal response test, a routine method to determine thermal ground properties for GSHP design. Proceedings 9 th International IEA Heat Pump Conference. May 20-22, Zürich, Switzerland. Witte, H. J. L., G. J. van Gelder, and J. D. Spitler In situ measurement of ground thermal conductivity: a Dutch perspective. ASHRAE Transactions 108(1):

7 Soil Thermal Conductivity Tests Figure 1-1. Borehole test equipment setup. Soil Thermal Conductivity Tests Design of ground loops requires value of soil thermal conductivity In-situ test on vertical borehole Portable test unit Data acquisition Figure 1-2. Soil thermal conductivity needed for design of ground-source heat pump systems. 7

8 Vertical Ground Loop Grout Soil T s,i T f T f Borehole Wall Figure 1-3. Cross section of vertical borehole with ground loop. Heat Exchanger Pipes Borehole Test Setup Heat Rate Flow Rate Hot Temp T sup T rtn Cold Temp Ground Loop Figure 1-4. Typical borehole test setup. 8

9 Test Equipment Figure 1-5. Test equipment at borehole site. Ground Loop Temperature Curve Step Change In Heat Rate Input q Temperature (ºF) Time (hr) Figure 1-6. Ground loop temperature curve after heater is started. 9

10 Temperature Curve: Log Time Scale Temperature (ºF) late-time linear trend 1 k s ~ slope Time (hr) Figure 1-7. Semilog plot of temperature curve. Heat Input Rate Heat Rate (Btu/hr) Electric Power Thermal Balance Time (hr) Figure 1-8. Variations in heat input rate based on electrical power and a heat balance on circulating fluid. 10

11 Differences in Measured Rate 8 Percent Difference Time (hr) Figure 1-9. Percentage difference between the two calculations of heat input rate in Figure

12 LINE-SOURCE MODEL 2. Interpretation of Test Data A variety of models have been used to interpret soil thermal conductivity tests on vertical boreholes. These models use the transient loop temperature (Figure 2-1) and the heat input rate to estimate the soil thermal conductivity and the borehole thermal resistance. To gain an understanding of the basic concepts, it is useful to start with a simple model and then consider more sophisticated models. The simplest approach is to treat the borehole as a vertical line source of heat into the ground, which is initially at a uniform temperature. As in geometry, the line is infinitesimally thin. Heat travels radially away from the vertical line source (Figure 2-2). This line source represents the entire borehole including the U-tube and the grout. In practice, the grout has different thermal properties than the soil, but this simple model ignores any variation in thermal properties. The thermal storage of the circulating fluid in the U-tube is also ignored. The motivation for the line-source model is that if the borehole test has sufficient duration, all the details of the borehole geometry and grout properties have little effect on the shape of the late-time loop temperature curve. At early times the heat transfer mechanisms within the borehole dominant the shape of the loop temperature curve. As test time increases, conduction through the soil transports heat radially outward from the borehole (Figure 2-2). With sufficient test time, heat conduction into the soil dominates the shape of the late-time loop temperature curve. Such an approach is consistent with the major objective of estimating the soil thermal conductivity. Based on this line-source model (Carslaw and Jaeger, 1959) the loop temperature, T, can be represented as q 4α st T T s,i ln 2 (2-1) 4πk sl γ r Other parameters in the above equation are k s L q r t T s,i α s γ = soil thermal conductivity, Btu/hr ft ºF (W/m ºC) = borehole depth, ft (m) = heat input rate, Btu/hr (W) = radial position from line source, ft (m) = time, hr = undisturbed soil temperature, ºF (ºC) = soil thermal diffusivity, ft 2 /hr (m 2 /hr) = constant In Equation 2-1, ln(x) represents the natural logarithm of x, and γ is a constant that is equal to approximately More details about the line-source model equations are in Appendix A. 12

13 Then, Equation 2-1 may be written with common logarithms with ln(x) = log(x). (2.303) q 4α st T T s,i log 2 (2-2) 4πk sl γ r The line-source approximation in Equation 2-2 suggests the late-time slope of the loop temperature should have a linear trend, if a plot is made with the common logarithm of time (Figure 2-1). The slope of the trend, m, is the multiplier in front of the logarithm on the right-hand side of Equation 2-2. Then the soil thermal conductivity, k s, is inversely related to the slope, m, of the trend as k s ( 2.303) q = (2-3) 4πm L If the natural logarithm of time is used on the horizontal axis, the estimate of k s becomes q k s = (2-4) 4π m L Despite its simplicity and all the near borehole heat transfer mechanisms it ignores, the line-source model can give valid results, if the appropriate late-time trend is identified. To define a unique late-time straight line, one needs to specify not only a slope, but also an intercept. (Alternatively, one can specify the slope and one point, or just two points.) This suggests the linear trend in Figure 2-1 contains more information than just the soil thermal conductivity. There should be additional information associated with the intercept value. Indeed, the borehole resistance, R b, is related to the intercept of the linear trend (Witte et al., 2002; Beier and Smith, 2002). The following paragraphs describe a modification of the line-source model as a way to include the borehole resistance. In the modified model the U-tube inside the borehole is represented by a single pipe with the diameter of the actual borehole, as illustrated in Figure 2-3. In the model the circulating fluid fills this single pipe. An infinitely-thin thermal resistance layer, or skin, is located on the borehole wall. This thin layer must take into account all the thermal resistances associated with the borehole. These resistances are the convective heat transfer resistance between the fluid and inner-pipe wall of the U-tube pipe, pipe resistance, grout resistance, and any contact resistances. Because the skin has no thermal storage capacity, the heat capacity of the circulating fluid and grout are ignored. The thermal model for the borehole consists of the thermal resistance of the borehole and the thermal resistance of the soil, as illustrated in Figure 2-4. We break the temperature drop between the circulating fluid temperature, T, and the initial soil temperature, T s, i into two pieces to account for these two resistances. The borehole 13

14 resistance (skin layer) is the thermal resistance between the loop temperature, T, and the temperature of the borehole wall, T b, T Tb R b = (2-5) q / L where R b T b = borehole resistance, hr ft ºF/ Btu (m ºC/W) = borehole wall temperature, ºF (ºC) The soil resistance is represented by the line-source solution (Equation 2-2) applied between the borehole wall temperature and the far-field soil temperature T b Ts,i α st R s = log 2 (2-6) q / L 4πk s γ rb where R s r b = soil resistance, hr ft ºF/ Btu, (m ºC/W) = borehole radius, ft (m) Note that Equation 2-6 is applied at the borehole radius, r b, to capture the borehole wall temperature. The borehole resistance and soil resistance are added to represent the total thermal resistance between the circulating fluid temperature and the initial soil temperature. With some algebraic manipulation (in Appendix A), the resulting equation gives an expression for the borehole resistance as T 1hr Ts,i 4α st1hr R b = log 2 (2-7) 4πk s m γ rb In Equation 2-7 the symbol T 1hr represents the loop temperature at one hour and t 1hr is equal to one hour. If the horizontal axis of the loop-temperature graph in Figure 2-1 was chosen to be log(t) on a linear scale, when t equals one hour, log(t) equals zero. Thus, temperature at one hour, T 1hr, can be viewed as the intercept value. If natural logarithms are used instead of common logarithms for the graph to estimate the late-time slope, the expression for the borehole resistance is 1 T 1hr Ts,i 4α st1hr R b = ln 2 (2-8) 4πk s m γ rb 14

15 One may choose any value of time and the corresponding temperature in the application of Equation 2-7 (or 2-8) and get identical results. The value of one hour is just a convenient choice. Equations 2-7 and 2-8 suggest the borehole resistance is linearly proportional to the temperature difference between the intercept temperature value and the undisturbed soil temperature, T 1hr T s,i. The modified line-source model has a linear temperature response, with a large jump when the heater is turned on (Figure 2-5). This initial temperature jump is associated with the borehole resistance. In the model the jump is instantaneous, because no thermal heat capacity is associated with the circulating fluid or the borehole. SOIL THERMAL CONDUCTIVITY The thermal conductivity estimated by a borehole test represents an average along the vertical borehole. The simplest geological model is a stack of horizontal layers, which is penetrated by the borehole. Each layer represents a different soil or rock type with different thermal properties. If one could easily measure the thermal conductivity at onefoot intervals along the borehole, then one would take the arithmetic average of the measured values as the appropriate thermal conductivity over the entire borehole depth. Indeed, the appropriate thermal conductivity for heat conduction through parallel layers is the arithmetic average (Holman, 1997), if vertical heat conduction among the layers is negligible. The borehole test is doing a similar averaging. The following example problem illustrates how to estimate soil thermal conductivity using Equation 2-4. Example 2-1. Estimate the soil thermal conductivity from the loop temperature curve displayed in Figures 2-6 and the following data: Heat Input Rate, q = 7260 Btu/hr Borehole Depth = 250 ft Step 1. The start of the late-time linear trend must be identified in Figure 2-6. One method is to draw a graph in Microsoft Excel by using a scatter plot. The time and loop temperature should be in the spreadsheet as separate columns. Create a scatter plot with time for x and loop temperature for y. Select a logarithmic scale for the horizontal axis. Choose a line from the draw toolbar and overlay the line onto the late-time trend in the graph. Adjust the line to fit over the data to provide the best visual fit to the curve. Identify the starting time of the late-time trend by the intersection of the data and the straight line. This method indicates 2.5 hours is a reasonable starting time in Figure 2-6. Step 2. A plot is made with only the data after 2.5 hours in Figure 2-7. Step 3. A trendline or fit is made to the data after 2.5 hours. In Excel 2003 click the right mouse button while the cursor is on one of the points and choose Add Trendline. Choose Logarithm type of trend. Under options, select to display 15

16 equation on the chart. Figure 2-8 has a fit to the data. The coefficient on ln(t) in Figure 2-8 is the slope, m = 1.59 ºF/cycle. Alternatively, one can use the built-in function Slope to find the slope of the data, but this function will not display the fit line through the data. Step 4. Estimate the soil thermal conductivity using Equation 2-4. k q 7260 Btu / hr Btu s = = = π m L 4 π( F / cycle) ( 250 ft) hr ft F It is important to review the graph and visually judge the goodness of fit to the data. In any fit line, it is better to have data scatter above and below the line, everywhere along the line. If portions of data lie above the line for an extended period of time, then the plot suggests a different type of trend exists, which is not captured by the linear fit. BOREHOLE THERMAL RESISTANCE The borehole thermal resistance is affected by the grout thermal conductivity, the borehole diameter, and the exact placement of the U-tube in the borehole, among other things. Grouts with higher thermal conductivities have been development to reduce borehole resistance. In addition, sometimes spacers are used to spread the U-tube legs out to reduce borehole resistance and enhance heat transfer. Borehole tests provide a method to evaluate the effectiveness of enhanced grout and spacers. Example 2-2. Estimate the borehole resistance from the loop temperature curve displayed in Figure 2-6 and the following data: Soil Thermal Conductivity, k s = 1.45 Btu/(hr-ft-ºF) (from Example 2-1) Slope, m = 1.59 ºF/cycle (from Figure 2-8) Initial Soil Temperature, T s,i = 64.0 ºF Soil Thermal Diffusivity, α s = ft 2 /hr Borehole Radius, r b = 2.25 in = ft Step 1. In Figure 2-8, the intercept of 73.1 is the extrapolated temperature at one hour, because ln(1)=log(1)=0. Therefore, T 1hr = 73.1 ºF. Step 2. Estimate the borehole resistance using Equation

17 R b 1 = 4πk s T f,1hr T m s,i 4αst ln 2 γ rb 1hr 1 2 ( )( F) 4( ft / hr)(1.0 hr) = ln 2 4 π (1.45 Btu / hr ft F) 1.59 F / cycle (1.78) (0.188 ft) = ft hr F / Btu As mentioned earlier, the borehole resistance can be used to quantify the effects of spacers to spread the U-tube legs apart and toward the borehole wall. The loop temperature curves from two boreholes in Figure 2-9 are used to illustrate this point (Beier and Smith, 2002). The boreholes have surface locations within a 25 ft (7.6 m) by 80 ft (24.4 m) area. Therefore, the soil thermal conductivity should be approximately the same for each borehole. Indeed, the late-time slope in both curves is nearly the same. Both boreholes have a diameter of 4.5 inches and bentonite grout. The effect of spacers can be clearly seen as a vertical offset between the loop temperature curves in Figure 2-9. Because the heat input rates are not identical among the tests, a dimensionless temperature is plotted on the vertical axis to remove the effects of different heat input rates. The dimensionless temperature is given by TD πksl(t Ts,i ) = 2 q (2-9) The borehole with no spacers shows a larger dimensionless temperature rise corresponding to a larger borehole resistance (0.372 hr-ft-ºf/btu). In the other borehole, pipe spacers push the U-tube legs closer to the borehole wall, which reduces the temperature rise and the borehole resistance (0.296 hr-ft-ºf/btu). These results are consistent with Equation 2-8, where a larger temperature rise at one hour corresponds to a larger borehole resistance. LIMITATIONS OF THE LINE-SOURCE MODEL The simple line-source model is easy to apply, but some important limitations become apparent from its application to field tests. At least two drawbacks exist. First, the start of the linear trend is not always apparent for every temperature curve. Second, the method ignores the effects caused by variable heat input rates. Because the model does not fit the early-time data, the user must make a judgment about the starting time of the linear trend and the required duration of the test. Choices for different starting times will give different estimates of soil thermal conductivity and borehole resistance. For instance, variable heat input rates (Figure 2-10) during a test on a borehole cause significant fluctuations in the late-time temperature curve in Figure Typically one will choose a fit line that passes through several cycles of these variations. The linear fit starting at 20 hours does this. But other choices in the starting time will change the estimate for thermal conductivity. Figure 2-12 shows 17

18 large changes in the estimated thermal conductivity depending on the selected starting time. It s true most people will not select a shallow slope like the shorter line in Figure 2-11, which leads to very high thermal conductivity. The short line sees only part of a heat input rate cycle. Still, one has no guarantee that the correct linear trend has been selected. Thus, there are ample reasons for looking at more sophisticated models, which are topics in the next section. REFERENCES Beier, R. A. and M. D. Smith Borehole thermal resistance from line-source model of in-situ tests. ASHRAE Transactions 108(2): Carslaw, H. S. and J. C. Jaeger Conduction of heat in solids. Oxford University Press, New York. Holman, J. P., 1997, Heat Transfer, McGraw-Hill, New York. Witte, H. J. L., G. J. van Gelder, and J. D. Spitler In-situ measurement of ground thermal conductivity: A Dutch perspective. ASHRAE Transactions 108(1):

19 Ground Loop Temperature 80 Temperature (ºF) Time (hr) Soil Thermal Conductivity Heat Rate (Btu/hr) Heat Input Rate Electric Power Thermal Balance Borehole Resistance Time (hr) Figure 2-1. Ground loop temperature curve and heat input rate curve are used to determine soil thermal conductivity and borehole resistance. Line-Source Model Figure 2-2. Line-source model showing radial heat flow. 19

20 Actual Borehole Skin Model Grout (a) Soil Circulating Fluid Infinitely thin resistance layer (b) Soil Figure 2-3. Model of borehole resistance as a thin skin layer. Thermal Resistances of Borehole and Soil Infinitely thin resistance layer Circulating Fluid T T b T s,i Soil R b R s T T b T s,i T Tb Tb Ts,i Rb = R s = q / L q / L Figure 2-4. Thermal resistances of borehole and soil add in series. 20

21 Line-Source Model With Skin Circulating Fluid Infinitely thin resistance layer Soil Temperature Rise Figure 2-5. Line-source model is a straight line in semilog plot. Resistance-layer temperature rise Log(Time) Ground Loop Temperature Curve 80 Temperature (ºF) Time (hr) Figure 2-6. Linear fit of late-time temperature curve. 21

22 80 Late-Time Data Temperature (ºF) Time (hr) Figure 2-7. Enlarged view of late-time temperature curve. Temperature (ºF) Late-Time Data With Line Fit T = 1.59 ln(t) Time (hr) 1 k s ~ slope T 1hr R b ~ (T 1hr -T s,i ) Figure 2-8. Enlarged view of linear fit of late-time temperature curve. 22

23 Effect of Spacers To Spread U-tube Legs Dimensionless Temperature Rise, T D T D,1hr Without Spacers Time (hr) With Spacers 10 ft Apart Figure 2-9. Loop temperature curves of two boreholes with and without spacers. Heat Input Rate Heat Rate (Btu/hr) Time (hr) Figure Fluctuations in heat input rate with time. 23

24 Late-Time Slope 22 Temperature Rise (ºF) Time (hr) Figure Late-time slope obscured by scatter in loop temperature curve. Scatter is partially caused by variations in heat input rate. Soil Thermal Conductivity Soil Thermal Conductivity (Btu/hr-ft-F) Starting Time (hr) Figure Estimated soil thermal conductivity corresponding to various linear fits. Starting time of data used in fit is on the horizontal axis. 24

25 3. More Advanced Thermal Models INTRODUCTION More advance thermal models of borehole tests (Figure 3-1) have been developed in order to overcome some of the shortcomings of the simple line-source model. No attempt is made here to give an exhaustive study of every model applied to a borehole test. Instead, we focus on the more commonly used models that are relatively easy to implement or for which software is available. While discussing the performance of various thermal models, it is helpful to have a reference borehole test. For this reason, we first discuss a test on a laboratory sandbox, which serves a useful reference (Figure 3-2). Unlike a borehole in the ground, the laboratory sandbox has known values of soil thermal conductivity and borehole resistance from independent measurements. Thus, we have known values to compare with the estimates from borehole test models. Using the sandbox test as a reference data set, we describe thermal models to take into account the finite size of the borehole, the grout and soil thermal properties, and variations in heat input rate. Some limitations of the line-source model are addressed by discussing methods to handle variable heat input rates and identifying the minimum duration for a borehole test. SANDBOX THERMAL TEST A reference borehole test is helpful in appraising the usefulness of various thermal models. Tests on a laboratory sandbox at Oklahoma State University are particularly helpful, because independent measurements of soil thermal conductivity and borehole resistance are available. The sandbox has dimensions of 6 ft x 6 ft x 60 ft (1.8 m x 1.8 m x 18 m). A 5 inch (0.13 m) inner diameter aluminum pipe is centered along the length of the sandbox. The thermal resistance of the pipe is negligible. Spacers keep the U-tube centered inside the aluminum pipe. Bentonite grout surrounds the U-tube. Heat probe measurements along the length of the sandbox estimate the average thermal conductivity of the soil to be 1.63 Btu/hr ft ºF (2.82 W/m ºC). Thermistors have been placed at the grout/soil interface to measure temperature at this radial location. The temperature difference between the circulating fluid and the grout/soil interface provides an independent estimate of hr ft ºF/Btu (0.173 m ºC/W) for borehole resistance. The application of the line-source model to a sandbox test is shown in Figure 3-3 The late-time linear trend and Equation 2-4 estimate the soil thermal conductivity to be 1.72 Btu/hr ft ºF (2.98 W/m ºC), which is within 6% of the independent measurement. From Equation 2-8 the borehole resistance is hr ft ºF/Btu (0.163 m ºC/W), which is 6% lower than the independent measurement. 25

26 CYLINDRICAL-SOURCE MODEL A cylindrical-source model (Deerman and Kavanaugh, 1991) takes into account the finite radius of the borehole. As illustrated in Figure 3-4, the model treats a heat source with a radius equal to the borehole radius, r b. The grout and U-tube within the borehole are not explicitly taken into account, but are incorporated into a borehole thermal resistance. The detail equations for the model are given in Appendix B. Kavanaugh et al. (2001) estimate the values of soil thermal conductivity, k s, and borehole resistance, R b, using equations in Appendix B to calculate a loop temperature curve and then by compare the computed loop temperatures with measured temperatures from a borehole test. Because the cylindrical model will not match the early-time data, they usually use only test data after some arbitrary test time. The algorithm may be represented by the following steps: Step 1. Guess values for k s and R b. Choose an estimate for the volumetric heat capacity of the soil, (ρ c p ) s, between 20 to 45 Btu/ft 3 ºF (1300 to 3000 kj/ m 3 ºC). Calculate the loop temperature for the cylindrical-source model using Equation B- 9 for all test times after a certain time. Step 2. Calculate the error between the measured loop temperature and the model temperature for each of these times. Then calculate the sum of the squares of these errors (SSE). Step 3. Steps 1 and 2 are repeated over a range of values for k s and R b. The estimated values of k s and R b are identified by the least SSE. Step 4. Steps 1 through 3 may be repeated for other guesses for volumetric heat capacity to cover the entire range between 20 to 45 Btu/ft 3 ºF. Then, the values of k s and R b associated with the least SSE are chosen. The comparison of the cylindrical model to the sandbox data set is shown in Figure 3-5. The model cannot match the early time data, because the model does not take into account the thermal properties of the grout, which differ from those of the soil. Also, the thermal storage of the circulating fluid is ignored. The cylindrical model matches the same late-time linear trend as the line-source model. Indeed, the estimated soil thermal conductivity of 1.60 Btu/hr ft ºF (2.78 W/m ºC) and borehole resistance of hr ft ºF/Btu (0.153 m ºC/W) compare well with the independent measurements and line-source model estimates as listed in Table 3-1. The cylindrical source model takes into account the finite radius of the borehole, but the model does not match the early-time data. Similar to the line-source model, the differences in the grout thermal conductivity, along with all other factors contributing to the borehole resistance, are represented by an infinitesimal skin at the borehole/soil 26

27 interface. Still, the cylindrical model matches the late-time trend well. Like the linesource model, one needs to identify a late-time interval in the data to fit. COMPOSITE MODEL Shonder and Beck (1999), Beier and Smith (2003a), and Wagner and Clauser (2005) use a composite model of the borehole to take into account the different thermal properties of the grout and soil. An illustration of the model is shown in Figure 3-6, where the actual borehole geometry is represented by a simplified, radially symmetric geometry. The U-tube is replaced by a single pipe with an effective radius of r p. The model does not explicitly account for the thermal resistance due to the U-tube pipe walls or any contact resistances at the pipe/grout or grout/soil interfaces. Instead, these resistances are implicitly rolled into the value of r p. Shonder and Beck (1999) have included a fluid film to represent the thermal resistance between the fluid and inside wall of the pipe. Beier and Smith (2003a) do not explicitly account for the film resistance, but instead the value, is implicitly taken into account in the value of r p. In both studies the thermal storage of the circulating fluid is taken into account, which may have an effect on the early-time loop temperature during a test. The model captures the important heat transfer mechanisms. The model is onedimensional (radial coordinate), which allows the model to be evaluated quickly by numerical methods on a computer. Gu and O Neal (1995) also used a composite model for the borehole, but their solution will not be used here, because they neglected the thermal storage of the circulating fluid. Shonder and Beck (1999, 2000) use this borehole model in their parameter estimation method to estimate the soil thermal conductivity, along with the borehole resistance. They solve the equations numerically using finite-difference techniques with a computer program called Geothermal Properties Measurements (GPM). The model match to the sandbox data set is shown in Figure 3-7. The model matches the entire test data set. The composite model successfully matches the early-time loop temperature curve, which is affected by the lower grout thermal conductivity. The estimated soil thermal conductivity of 1.64 Btu/hr ft ºF (2.84 W/m ºC) is within 1% of the independently measured value. The estimated borehole resistance of hr ft ºF/Btu (0.187 m ºC/W) is within 8% of the independent value. The GPM computer model also treats the initial soil temperature as an unknown. The model estimates the soil temperature to be 68.1 ºF (20.1 ºC), which is lower than the measured soil temperature of 71.5 ºF (21.9 ºC). The soil temperature is input to all the other models. This lower estimated initial temperature would tend to raise the estimated borehole resistance for the GPM model. Note the GPM model match to the temperature curve is the poorest at early times (Figure 3-7). Beier and Smith (2003a) developed an analytical solution to the composite model. The match of their solution is given in Figure 3-8. Again, the estimates of soil thermal conductivity and borehole resistance in Table 3-1 agree well with the independent estimates. All the model fits to the sandbox data set give estimates of soil thermal conductivity within 6% of the independent estimate. All the models work reasonably well on this data set. 27

28 OTHER NUMERICAL MODELS Researchers have developed several detailed numerical models to match the earlytime data, which are influenced by borehole effects. Yavuzturk et al. (1999), Mei (1985), Muraya et al. (1996), Rottmayer et al. (1997), and Signorelli et al. (2007) have developed models for a single borehole and its surroundings. Most of these models require more details than the composite model, such as the exact placement of the U-tube relative to the borehole wall, which is generally not known. These models have served as research tools, but are not widely used to analyze routine borehole tests as the previously discussed models. VARIABLE HEAT INPUT RATE EFFECTS A decreasing or increasing heat-rate trend throughout the test will distort the transient, ground-loop temperature curve. For example, the heat rate data in Figure 3-9 have a decreasing trend as time increases. These data have been smoothed to remove the high-frequency variations. To demonstrate the decreasing trend, the variations about a mean heat rate in the test data have been amplified by a factor three before plotting Figure 3-9. For these amplified variations, a corresponding loop temperature curve has been calculated by numerical methods and plotted as the solid curve in Figure For comparison a curve based on the average and constant heat input rate is shown with the dashed curve. Simply smoothing out the fluctuations in the variable-rate loop temperature curve will not recover the correct shape of the dashed line that corresponds to the average heat input rate. The late-time slope for calculating the soil thermal conductivity will be in error if simple smoothing is used on the solid curve in Figure Shonder and Beck (1999, 2000) and Wagner and Clauser (2005) have taken into account variations of the heat input rate in their application of the composite model. The other numerical models listed earlier also handle variable heat input rates. In a different approach, Beier and Smith (2003b) apply a deconvolution technique to remove the effects of variable heat input rates on the loop temperature curve. The line-source and cylindrical models are often applied without any account for rate variations. One should look out for long term trends in the heat input rates, which indicate variable input heat rate effects may be significant for estimating soil thermal conductivity. If significant long term trends exist, one of the methods for variable rates can be used. MINIMUM DURATION OF TEST Because the cost of a test increases with increasing duration, there is an economic incentive to decrease the duration of a field test. On the other hand, the test duration must be sufficient to provide a valid estimate for soil thermal conductivity. Although past authors agree for the need for an estimate of the required duration, their recommendations for the duration do not agree. Austin et al. (2000) recommend a minimum duration of 50 hours based on their experiences with field data sets. Kavanaugh et al. (2001) recommend 28

29 test durations of 36 to 48 hours. Gehlin (1998) suggests a minimum duration of 60 hours but recommends using 72 hours. Smith and Perry (1999) suggest that 12 to 20 hours may sometimes be sufficient, partly because if the test duration is too short, the resulting underestimate of soil thermal conductivity is a conservative estimate for the design of ground heat exchangers. Beier and Smith (2003a) and Signorelli et al. (2007) argue no simple rule for minimum duration applies to all cases. Calculations based on the composite model (Beier and Smith, 2003a) indicate the required test duration increases significantly as the grout thermal conductivity decreases below the soil thermal conductivity. Also, the minimum test duration increases as the borehole thermal resistance increases. Because enhanced (high thermal conductivity) grout or spacers between U-tube legs decrease the borehole resistance, they also tend to reduce the required test duration. Procedures for analyzing filed test data generally do not check if the test duration is sufficient to give an accurate estimate for soil thermal conductivity. Beier and Smith (2003a) developed a method to carry out such a check. Typically one would perform this check once after estimating the thermal conductivity and borehole resistance with any of the available models. The method is based on the analytical composite model and has been incorporated into a spreadsheet. The spreadsheet is available from R. A. Beier ([email protected]). COMPARISON OF THE DIFFERENT MODELS So far the sandbox data set has been used to compare different models. Results in Table 3-1 from the different models are similar. The sandbox data set is of better quality than many field tests. One may argue that field tests are more representative of typical applications and more challenging for the models. Researchers have conducted studies on sets of field tests to compare models and their resulting estimates of soil thermal conductivity. Gehlin and Hellström (2003) found the estimates of soil thermal conductivity from both line-source models and a composite numerical finite-difference model were within 5% of each other for three data sets. A cylindrical-source model gave values about 10% to 15% higher. They used a parameter estimation technique with each model to find the best fit to the measured data. Shonder and Beck (2000) report good agreement between their parameter estimation composite model and the line-source method for several field tests. Kavanaugh et al. (2001) performed an extensive comparison of soil thermal conductivity estimates from linesource models, cylindrical models, and numerical models (Shonder and Beck, 1999; Austin et al. 2000). When test data were good they report agreement among the methods and recommend applying multiple methods to any field test data set. SOIL THERMAL DIFFUSIVITY Can additional parameters be estimated from the composite model or detailed numerical models? After all, the simple line-source model estimates soil thermal conductivity and borehole resistance from the late-time linear trend. Parameter sensitivity studies with the composite model indicate the late-time trend also largely 29

30 determines its estimates of soil thermal conductivity and borehole resistance. Can more information be extracted from the early-time data? In addition to soil thermal conductivity, the design of ground-loop heat exchangers requires an estimate of the soil volumetric heat capacity. Although the design is less sensitive to soil heat capacity than thermal conductivity, the heat capacity is a parameter worth trying to estimate. However, detailed sensitivity studies with the composite model (analytical model) indicate the estimated soil heat capacity is not unique if heat capacity is solved along with the soil thermal conductivity and borehole resistance. Large changes in soil volumetric heat capacity, along with relatively small changes in the borehole resistance, produce many reasonable fits to the loop temperature data. The goodness of the fit is relatively insensitive to the choice of soil heat capacity. The fit becomes even more nonunique if small changes are allowed in the value of the initial soil temperature. These results are consistent with published numerical modeling results by Yavuzturk et al. (1999) and Shonder and Beck (1999), who solved for soil thermal conductivity and borehole resistance (grout thermal conductivity), but fixed the value of soil heat capacity. In applying a parameter estimation technique, Wagner and Clauser (2005) solve for the soil heat capacity and the soil thermal conductivity, but the borehole thermal resistance and all other parameters are fixed. Although the study indicates soil heat capacity may be determined from the loop temperature curve if all the borehole properties are known, in a field test the unknown position of the U-tube in the borehole usually makes the borehole resistance unknown. In conclusion, the composite and numerical models fit the entire loop temperature curve, but the models are not able to independently estimate the soil volumetric heat capacity if the soil thermal conductivity and borehole resistance are also treated as unknowns. INTERRUPTED TESTS In-situ borehole tests are sometimes interrupted by electric power outages or other unexpected events. In such cases, the length of the test prior to the interruption is often inadequate to determine the value of soil thermal conductivity. If the test is restarted immediately after the power is restored, large swings in the heat input rate to the groundloop complicate the analysis of the test. Nearly all the models assume a spatially uniform ground temperature at the start of the test. In cases where the field test is immediately restarted after an interruption, this assumption of uniform ground temperature is typically invalid at the time of restart. Some guidelines are available in the technical literature for handling interrupted tests. After a complete 48-hour test has been conducted, Martin and Kavanaugh (2002) recommend a ten- to fourteen-day waiting period before retesting a borehole in formations with medium to high thermal conductivity. The waiting period allows the heat to dissipate around the borehole as the nearby ground temperature approaches the undisturbed temperature. If the initial test was shorter, they suggest the waiting period can be reduced in proportion to the reduced test time (Kavanaugh et al., 2001). 30

31 If the interruption is a few hours or less a reasonable approach is to resume the test as soon as possible. Consider the temperature curve (open symbols) in Figure 3-11, which is taken from a test in the laboratory sandbox with a two-hour interruption. The heat input from an electrical heater in Figure 3-12 illustrates the power interruption between 9 and 11 hours. The temperature rise in a previous test without any interruption (with the same sandbox setup) is given by the solid symbols in Figure The interrupted temperature rise (open symbols) eventually overlays on the uninterrupted test curve (solid symbols). The late-time slopes are nearly the same, which give comparable estimates of soil thermal conductivity. For this two-hour interruption, a reasonable approach is to resume the test as soon as the power is restored. Cumulative test time, including the interruption period, is 51 hours. Therefore, in some cases restarting the test immediately after power is restored makes sense. In theory parameter estimation methods (Shonder and Beck, 1999, 2000; Austin et al., 2000; Wagner and Clauser, 2005) should be able to handle interrupted tests as a generalization of taking into account a variable heat input rate schedule. When applying such numerical methods one must keep in mind that the loop temperature data are missing during the interruption. Such gaps in temperature data and the abrupt loss of heat input present additional challenges for the stability of numerical methods. If a test is restarted immediately after the interruption, Beier and Smith (2005) describe a method to estimate the required testing time (or recovery time) when the effects of the interruption dissipate sufficiently so that the estimated thermal conductivity is changed by 10% or less. After the power is restored, the method can be used to estimate the required recovery time. Because the test duration using line-source methods can be prohibitively long following the interruption, some analysis techniques have been developed that shorten the required test duration for a valid estimate of thermal conductivity (Beier and Smith, 2005; Beier, 2008). These methods have been validated using data sets from the laboratory sandbox. GROUNDWATER EFFECTS All the above analysis methods are based on the assumption that heat conduction is the dominate mechanism of heat transfer within the soil. The movement of groundwater has been ignored. Groundwater effects can change the characteristic shape of the transient loop temperature curve. In Figure 3-13 the late-time loop temperature curve becomes horizontal and flat. Such a shape suggests groundwater effects dominate the late-time temperature curve. Groundwater movement enhances heat transfer between the circulating fluid in the loop and the ground. Thus the required length of a ground loop becomes smaller if groundwater movement is present. Numerical models have been used to study the effects of groundwater on vertical boreholes (Chiasson et al., 2000; Signorelli et al., 2007; Fujii et al., 2005). Gehlin and Hellström (2003b) studied the influence of groundwater flow in fractures. Analytical studies based on line-source models of the borehole have been made by Diao et el. (2004) and Sutton et al. (2003). Sutton et al. (2003) developed expressions for the thermal resistance of the ground as a function of groundwater velocity and soil properties. 31

32 The (late-time) total thermal resistance, T tot, between the circulating loop fluid and the undisturbed ground temperature, T i, can be evaluated from the temperature curve in Figure 3-13, if the late-time temperature plateau, T plat, is reached. The total resistance is R tot Tplat Ts,i = (3-1) q / L Evaluating the individual components of borehole resistance and soil resistance from a borehole test requires more sophisticated analysis with numerical models. REFERENCES Austin, W. A., C. Yavuzturk, and J. D. Spitler Development of an in-situ system for measuring ground thermal properties. ASHRAE Transactions 106(1): Beier, R. A. and M. D. Smith. 2003a. Minimum duration of in-situ tests on vertical boreholes. ASHRAE Transactions 109(2): Beier, R. A. and M. D. Smith. 2003b. Removing variable heat-rate effects from borehole tests. ASHRAE Transactions 109(2): Beier, R. A. and M. D. Smith Analyzing interrupted in-situ tests on vertical boreholes. ASHRAE Transactions 111(1): Beier, R. A Equivalent time for interrupted tests on borehole heat exchangers. HVAC&R Research. 14(3): Chiasson, A., S. J. Rees, and J. D. Spitler, A preliminary assessment of the effects of ground-water flow on closed-loop ground-source heat pump systems, ASHRAE Transactions, 106(1): Deerman, J. D. and S. P. Kavanaugh. Simulation of vertical U-tube ground-coupled heat pump systems using the cylindrical heat source solution. ASHRAE Transactions 97(1): Diao, N., Q. Li, Z. Fang, Heat transfer in ground heat exchangers with groundwater advection. International Journal of Thermal Sciences. 43: Fujii, H., R. Itoi, J. Fujii, and Y. Uchida Optimizing the design of large-scale ground-coupled heat pump systems using groundwater and heat transport modeling. Geothermics. 34:

33 Gehlin, S Thermal response test, in-situ measurements of thermal properties in hard rock. Licentiate Thesis, Lulea University of Technology, Department of Environmental Engineering, Division of Water Resources Engineering, 1998:37. Gehlin, S. E. A. and G. Hellström. 2003a. Comparison of four models for thermal response test evaluation. ASHRAE Transactions 109(1): Gehlin, S. E. A. and G. Hellström. 2003b. Influence on thermal response test by groundwater flow in vertical fractures in hard rock. Renewable Energy 28: Gu, Y. and D. L. O Neal An analytical solution to transient heat conduction in a composite region with a cylindrical heat source. Journal of Solar Energy Engineering 117(8): Kavanaugh, S. P., L. Xie, and C. Martin Investigation of methods for determining soil and rock formation thermal properties from short-term field tests. ASHRAE TRP. Martin, C. A., and S. P. Kavanaugh Ground thermal conductivity testing Controlled site analysis. ASHRAE Transactions 108(1): Mei, V. C. and C. J. Emerson, New approach for analysis of ground-coil design for applied heat pump systems. ASHRAE Transactions 91(2B): Muraya, N. K., D. L. O Neal, and W. M. Heffington Thermal interference of adjacent legs in a vertical U-tube heat exchanger for a ground-coupled heat pump. ASHRAE Transactions 102(2): Rottmayer, S. P., W. A. Beckman, J. W. Mitchell 1997, Simulation of a single vertical U- tube ground heat exchanger in an infinite medium, ASHRAE Transactions 103(2): Shonder, J. A. and J. V. Beck Determining effective soil formation thermal properties from field data using a parameter estimation technique. ASHRAE Transactions 105(1): Shonder, J. A. and J. V. Beck Field test of a new method for determining soil formation thermal conductivity and borehole resistance. ASHRAE Transactions 106(1): Signorelli, S., S. Bassetti, D. Pahud, T. Kohl, Numerical evaluation of thermal response tests. Geothermics, 36, Smith M. and R. Perry In-situ testing and thermal conductivity testing. Proceedings of the Geoexchange Technical Conference and Exposition, Oklahoma State University, Stillwater, Oklahoma, May

34 Sutton, M. G., D. W. Nutter, and R. J. Couvillion A ground resistance for vertical bore heat exchangers with groundwater flow. Journal of Energy Resources Technology. 125: Wagner, R. and C. Clauser Evaluating thermal response tests using parameter estimation for thermal conductivity and thermal capacity. Journal of Geophysics and Engineering 2: Yavuzturk, C. J. D. Spitler, and S. J. Rees A Transient two-dimensional finite volume model for the simulation of vertical U-tube ground heat exchangers 105(2):

35 TABLE 3-1 Results From Model Fits to Sandbox Test Independently Measured Values Line-Source Model Cylindrical-Source Model Composite Model GPM (Shonder and Beck, 1999) Composite Model Analytical Solution (Beier and Smith, 2003) Soil Thermal Conductivity Btu/hr ft F (W/m C) 1.63 (2.82) 1.72 (2.98) 1.60 (2.78) 1.64 (2.84) 1.70 (2.84) Borehole Resistance hr ft F/Btu (m C/W) (0.173) (0.163) (0.153) (0.187) (0.152) 35

36 More Advanced Models Diameter of borehole Grout properties Variable heat input rates Duration of test Figure 3-1. List of topics for more advanced thermal models. Laboratory Sandbox Borehole Test Dimensions of 6 ft x 6 ft x 60 ft Length of 60 ft has horizontal orientation Spacers keep U-tube centered in aluminum pipe (5-inch diameter) Bentonite grout surrounds U-tube Figure 3-2. Description of laboratory sandbox for borehole tests. 36 5/28/2012

37 Temperature (ºF) Line-Source Model Fit k s ~ 6% high R s ~ 6% low Time (hr) Figure 3-3. Line-source model applied to sandbox test. Cylindrical-Source Model r b Figure 3-4. Cylindrical-source model showing radial heat flow. 37 5/28/2012

38 Cylindrical-Source Model Temperature (ºF) Raw Sandbox Data Cylindrical Model k s ~ 2% low R s ~ 11% low Time (hr) Figure 3-5. Cylindrical-source model applied to sandbox test. Actual Borehole Composite Model Film Circulating Fluid r p Grout Grout Soil (a) (b) Soil Figure 3-6. (a) Geometry of actual borehole. (b) Composite model of borehole. r b 38 5/28/2012

39 Parameter Estimation Composite Model 105 Temperature (ºF) k s ~ 1% high R s ~ 8% high Sandbox Data 80 GPM Model Time (hr) Figure 3-7. Parameter estimation and composite model fit (GPM) to sandbox test. Analytical Composite Model 105 Temperature (ºF) k s ~ 4% high R s ~ 12% low Sandbox Data Analytical Model Time (hr) Figure 3-8. Analytical composite model fit of borehole test. 39 5/28/2012

40 Decreasing Heat Input Rate Field Test 1.2 Normalized Heat Rate Time (hr) Figure 3-9. Normalized heat input rate from field test after the fluctuations are amplified by a factor of three. Loop Temperature Rise 20 Temperature Rise (ºF) Variable Rate Constant-Rate Model Time (hr) Figure Loop temperature rise for decreasing heat input rate and constant (average) heat input rate. 40 5/28/2012

41 Temperature Rise (ºF) Interrupted Test in Laboratory Sandbox Uninterrupted Test Interrupted Test Time (hr) Figure Loop temperature curves from uninterrupted and interrupted tests in laboratory sandbox Temperature Rise (ºC) Electric Power to Heater Electric Power (Btu/hr) Electric Power (W) Time (hr) Figure Electric power to heater during interrupted test. 41 5/28/2012

42 Groundwater Effects Loop Temperature Without Groundwater With Groundwater Time Figure Loop temperature curve for test with and without groundwater effects. 42 5/28/2012

43 Appendix A Line-Source Model BASIC EQUATIONS Consider a loop surrounded by soil initially at a uniform temperature. From Carslaw and Jaeger (1959) the temperature, T, at a radial distance, r, from a line source of heat is given by 2 q r T T s,i = Ei (A-1) 4πk sl 4αst where T s,i represents the initial soil temperature, and Ei(-x) represents the exponential integral, Ei( x) = x e u u du ln 1 x ln () γ (A-2) Some authors denote Ei(-x) as E 1 (x). Other parameters in the above equations are defined below Equation 2-1 in the main text. For small values of x, the natural logarithm approximation, ln(x), of the exponential integral is valid in Equation A-2. In our application, the logarithm approximation is accurate within 5%, if (4α s t/r 2 )> 11. Then, substitution of this approximation (Equation A-2) into Equation A-1 gives Equation 2-1 in the main text. BOREHOLE RESISTANCE The thermal model for the borehole consists of the thermal resistance of the borehole and the thermal resistance of the soil, as illustrated in Figure 2-4. The borehole resistance (Equation 2-5) and soil resistance (Equation 2-6) are added to represent the total thermal resistance between the circulating fluid temperature and the undisturbed soil temperature. With some algebraic manipulation, the resulting equation gives an expression for the borehole resistance as T T s,i q / L = ( T T ) ( T T ) q / L b + b q / L s,i = R b + R s = R b 1 + 4πk s 4α ln γ r s 2 b t (A-3) Equation A-3 may be rearranged to give an expression for T-T s,i as 43 5/28/2012

44 (2.303) q 4αst T T s,i log + 4πk π 2 sr b (A-4) 4 k sl γ rb where r b is the borehole radius. In applying Equation A-4, we must choose a value for time, t, and a value for the loop temperature, T, which corresponds to the chosen time. If the horizontal axis in Figure 2-6 is rescaled to be the log(t), when t equals one hour, log(t) equals zero. Thus, temperature at one hour, T 1hr, can be interpreted as the intercept value. Then, Equation A-4 (with the use of Equation 2-4 for m) may be rearranged to give the borehole resistance as T 1hr Ts,i 4α st1hr R b = log 2 (A-5) 4πk s m γ rb If natural logarithms are used instead of common logarithms for the graph to estimate the late-time slope, the expression for the borehole resistance is 1 T 1hr Ts,i 4α st1hr R b = ln 2 (A-6) 4πk s m γ rb One may choose any value of time and the corresponding temperature in the application of Equation A-5 (or A-6) and get identical results. The value of one hour is just a convenient choice. REFERENCE Carslaw, H. S. and J. C. Jaeger Conduction of heat in solids. New York: Oxford University Press. 44 5/28/2012

45 Appendix B Cylindrical-Source Model Ingersoll et al. (1954) treats the borehole as a cylinder as illustrated in Figure 3-4. In their formulation they introduce a G-factor, G(Fo, r D ). The G-factor, G(Fo, r D ), is a dimensionless temperature that depends on the Fourier number, Fo, and a dimensionless radius, r D. The G-factor is related to the loop temperature by q T Ts,i = G(Fo, rd ) (B-1) ks L where k s L q T s,i = soil thermal conductivity, Btu/hr ft ºF (W/m ºC) = borehole depth, ft (m) = heat rate, Btu/hr (W) = initial soil temperature, ºF (ºC) The Fourier number is a dimensionless time and is given by Fo = ρ s k c t s 2 p,srb (B-2) where c p,s r b = soil specific heat, Btu/lb ºF (kj/kg ºC) = borehole radius, ft (m) t = time, hr ρ s = soil density, lb/ft 3 (kg/m 3 ) In a borehole test, the G-factor is evaluated at a radial distance equal to the borehole radius. That is, r = r b. Then, the dimensionless radius, r D, is equal to one, because r r D = (B-3) r b A plot of the G-factor (for r D =1) is shown in Figure B-1, where the horizontal axis is Fourier number or dimensionless time. For Fourier numbers greater than 30 the G- factor is linear in the plot. In this region the cylindrical model and the line-source model overlay upon each other. 45 5/28/2012

46 To apply the cylindrical source model, one can use a closed form solution for the G-factor instead of the graphical solution in Figure B-1. Ingersoll et al. (1954) write the G-factor as G(Fo, r D 1 e ) = 2 π β J (β) + Y Fo (β) [ J (r β) Y (β) J (β) Y (r β) ] 0 D D dβ β 2 (B-4) where J 0 (β), J 1 (β), Y 0 (β), and Y 1 (β) are Bessel functions. Press et al. (1992) give algorithms and computer routines for evaluating these functions. To represent the loop temperature in borehole tests, Equation B-4 is evaluated at r D = 1, and the G-factor corresponds to the radial distance r equal to the borehole radius, r b. To take into account for the thermal resistance of the borehole, Kavanaugh and Rafferty (1997) apply the cylindrical source model to borehole test by adding the borehole resistance to the soil resistance, in much the same way as is done for the linesource model. From Equation B-1, the soil resistance for the cylindrical-source model is R Tb Ts,i 1 = = G(Fo, ) (B-5) q / L k s 1 s Similar to Equation A-3, the sum of the borehole and soil resistances account for the temperature difference between the circulating fluid temperature and the initial soil temperature. That is, T T q / L s,i = ( T T ) ( T T ) q / L b + b q / L s,i = R b + R s = R b 1 + k s G(Fo, 1) Equation B-6 can be rearranged to give and expression for R s as (B-6) R s L(T Ts,i ) R b q = (B-7) Kavanaugh and Rafferty (1997) introduce the factor F sc as a short-circuit heat loss factor that accounts for performance degradation due to heat losses between legs of the U-tube. They estimate the degradation is about 4% when water rates of 3 gpm per ton are applied. Then, the corresponding value of F sc is With this factor, Equation B-7 is rewritten as R s L(T Ts,i ) 1 R b q Fsc = (B-8) If Equation B-8 is solved for the loop temperature, one finds 46 5/28/2012

47 q Fsc T R b + G(Fo, ) + Ts,i L ks = 1 (B-9) Kavanaugh et al. (2001) use Equation B-9 to calculate a loop temperature curve and compare it to the measured temperatures from the borehole test. An iterative process is used to find a match. REFERENCES Ingersoll, L. R., Zobel, O. J., and Ingersoll, A. C., Heat Conduction With Engineering, Geological, and Other Applications, Revised Edition, University of Wisconsin Press. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., Numerical Recipes, Second Edition, Cambridge University Press. Kavanaugh, S. P. and Rafferty, K., Ground-Source Heat Pumps: Design of Geothermal Systems for Commercial and Institutional Buildings, ASHRAE, Atlanta. Kavanaugh, S. P., L. Xie, and C. Martin, Investigation of methods for determining soil and rock formation thermal properties from short-term field tests, ASHRAE TRP. 47 5/28/2012

48 Cylindrical-Source Model G(Fo,1) Fo Figure B-1. Dimensionless temperature (G-function) for cylindrical-source model. 48 5/28/2012