A SIMPLIFIED MODEL FOR MEASURING THERMAL PROPERTIES

EHT 17(2) #14553

Experimental Heat Transfer, 17:119–130, 2004 Copyright © Taylor & Francis Inc. ISSN: 0891-6152 print/1521-0480 online DOI: 10.1080/08916150490271845

A SIMPLIFIED MODEL FOR MEASURING THERMAL PROPERTIES OF DEEP GROUND SOIL M. Z. Yu and X. F. Peng Department of Thermal Engineering, Tsinghua University, Beijing 100084, China

X. D. Li and Z. H. Fang Shangdong Institute of Architecture and Engineering, Jinan 250014, China

A simplified heat transfer model is proposed for conveniently determining the thermal properties of deep ground soil at worksites and other associated engineering applications. The model does not need exact information of the U-tube location, spacing between up and down tubes, the properties of the tube and the backfill, and other parameters in the borehole as usual cases and greatly eliminates the errors caused by measuring the parameters mentioned. A practical site test was conducted to obtain heat flux imposed on the buried loop, water flow rate, and temperatures at the outlet and inlet of the loop by an in-situ test apparatus. The simplified model, together with optimal estimation algorithm, was used to determine the thermal conductivity of the deep ground soil in the worksite.

INTRODUCTION Compared with outdoor environment, deep ground temperature remains constant annually, being much lower in summer and higher in winter than that of air. Employing deep ground as a heat source/sink, ground source heat pumps (GSHP) effectively breakthrough some limits of air source heat pumps, thereby significantly improving their cycle efficiency. They also boast other advantages such as low noise, less occupation, less pollution, low operation and maintenance cost, and long life [1, 2]. GSHP systems are extensively used in North America, Europe, and Asia [3–7]. Thermal properties of the deep ground soil are important parameters for designing ground loops of GSHP systems. In-situ tests combined with parameters estimation algorithm is an accepted method to determine soil thermal properties [8–14]. Available heat transfer models for the measurements usually require detailed information of a ground loop, including thermal properties of buried tubes, diameter and wall thickness of U-tubes, spacing between up and down pipes (also refers to shank spacing), and thermal properties of the grout. In practical applications, it is very difficult to obtain and/or measure all these parameters with adequate precision. It is impossible to accurately determine the shank Received 18 April 2003; accepted 18 August 2003. The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (No.50136020). Address correspondence to Mr. M. Z. Yu, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China. E-mail: [email protected] 119

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NOMENCLATURE cs cw db di do De Dei H kg kp ks kw N ql Ro Rs

specific heat of soil, J/kg·K specific heat of water, J/kg·K diameter of the borehole, m inner diameter of buried pipe, m outer diameter of buried pipe, m equivalent pipe diameter, m inner diameter of equivalent pipe, m convection coefficient, W/m2 ·K thermal conductivity of grout, W/m·K thermal conductivity of buried pipe wall, W/m·K thermal conductivity of soil, W/m·K thermal conductivity of water, W/m·K the total number of data points heat injection rate per unit borehole depth, W/m thermal resistance of borehole per depth, K·m/W conductive heat resistance of soil, K·m/W

S Tm,i

S Tm,i

Tm2D Tf Tff Tw ρs ρw τ u νw

U-tube shank space, m average water temperature at the ith time point which was measured during test (assumed equals to the mean temperature of inlet and outlet), K mean water temperature at the ith time point which was calculated by the simplified heat transfer model, K mean water temperature was calculated by the QTD heat transfer model, K average temperature of fluid flowing in buried U-tube, K far field soil temperature, K borehole wall temperature, K soil density, kg/m3 water density, kg/m3 time, s velocity of circulating water, m/s kinematical viscosity, m2 /s

spacing once the U-tube is inserted into the borehole, and the thermal properties of the grout are seldom measured correctly at worksite. Too many uncertainties in obtaining correct information may result in errors in the final soil thermal properties. All models for engineering simulations and tests are mathematical formulations of the practical processes on the basis of a series of assumptions and simplifications. Though a heat transfer model containing more parameters may depict a process in more detail, the reliability of the parameters greatly restricts the accuracy of its predictions. On the contrary, a simpler model with fewer parameters focuses on the impacts of these factors, and may be more satisfactory, as well as useful, in certain practical applications, especially in parameter estimations. In this article the authors make an attempt to propose a simplified model to measure ground soil thermal properties. This technology has only very few parameters to measure at worksite and is quite convenient for practical engineering applications. SIMPLIFIED HEAT TRANSFER MODEL For a typical buried loop of the geothermal heat exchanger shown in Figure 1, the following assumptions are introduced. (1) The soil around the borehole is a kind of homogenous medium (since the thermal properties for designing a GSHP system are usually average values). (2) The heat exchange between the buried U-tube and soil is considered as that between an infinite long line heat source at the borehole center and soil. Heat transfer along vertical direction is neglected, for the ratio of the borehole diameter to its depth is only about 0.001–0.003. (3) Heat flux imposed on the buried loop maintains constant. This can be obtained by adjusting the input power. In addition to these assumptions, a total heat resistance of the borehole is introduced as a parameter to substitute all unknowns in the borehole. Figure 2 shows the simplified model.

A MODEL OF DETERMINING SOIL PROPERTIES

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Figure 1. Ground heat exchanger configuration.

From the above considerations and assumptions, the governing equation of outside borehole zone is, ∂T ks = ρs cs ∂τ

∂ 2T 1 ∂T + 2 r ∂r ∂r

,

db ≤ r < ∞, 2

τ >0

(1)

with initial condition T = Tff ,

db < r < ∞, 2

τ =0

(2)

and boundary conditions ∂T −ks = ql , ∂r r= db

τ >0

(3)

T = Tff ,

τ > 0.

(4)

2

r → ∞,

Figure 2. Simplified borehole cross section.

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The temperature of the borehole wall is obtained as, Tw = Tff + ql ·

2 π 2 ks d b

∞

ks

(e− ρs cc β

2τ

− 1)

0

J0 (βdb /2)Y1 (βdb /2) − Y0 (βdb /2)J1 (βdb /2) · dβ β 2 [J12 (βdb /2) + Y12 (βdb /2)]

(5)

where J0 , J1 , Y0 , Y1 are primary and secondary Bessel functions. Liu’s research [15] indicated that, when time was longer than about 10 hours, Eq. (5) could be simplified as 1 Tw = Tff + ql · · Ei 4π ks

db2 ρs cs 16ks τ

(6)

∞ −S where Ei(x) = x e S dS is an exponential integral function. Let heat resistance of the borehole be Ro , then the difference between line source temperature (mean temperature of the fluid in the ground loop), Tf , and borehole wall temperature, Tw , is deduced, Tf − Tw = ql · Ro .

(7)

From Eqs. (5) and (6), the mean temperature of the fluid in the ground loop is expressed as [10, 11],

1 Tf = Tff + ql · Ro + · Ei 4π ks

db2 ρs cs 16ks τ

.

(8)

Equation (8) contains three unknown parameters, ks , Ro , and ρs cs . Here, volumetric specific heat capacity, ρs cs , is considered as one unknown parameter. Equation (8) is a simplified heat transfer model that uses one parameter Ro to replace complex expression of the borehole heat resistance. The only difference between this model and other analytical models is the borehole heat resistance, Ro . For cylindrical source model [3], the U-tube was simplified to an equivalent pipe having diameter [16], De =

√ 2Dp

(9)

where Dp is the specified pipe diameter. In this model, Ro is composed of the convection resistance inside the pipe, conduction resistance of the pipe wall, and conduction resistance of the grout. Thus, Ro can be represented by do db −1 do ln db ln di do 0.4 kw di Ro = 0.023Re0.8 + + Dei Pr Dei do 4kp 4kg

(10)

where di , do , and Dei are inner diameter, outer diameter, and equivalent pipe inner diameter of the buried pipe, respectively. kw , kp , and kg are conductivity of water, pipe wall, and grout, respectively.


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For a 2-D model [11], the borehole heat resistance is calculated on the assumption of steady heat transfer inside the borehole [17]:

db4 k g − ks db db 1 + ln + ln · ln Ro = 0.5 · do 2S kg + k s 2πkg db4 − S 4 (11) do 1 1 + + · ln πdi h 2π kp di where S is the distance between axes of up and down branches of the U-tube. For the above heat transfer models, unknown parameters can generally be obtained with parameter estimation algorithm. The simplified model has three unknown parameters, while the other two models have only two. Though cylindrical and above 2-D heat transfer models had fewer parameter to estimate, borehole heat resistance should be determined from a number of measured parameters; this would cause greater errors. In practice, parameters like the distance between axes of up and down pipes and grout conductivity are very difficult to determine. DESCRIPTION OF TEST For the simplified heat transfer model, parameter estimation algorithm is used to find the soil conductivity, ks , borehole heat resistance, Ro , and volumetric specific heat capacity, ρs cs , from Eq. (8). This method makes Ro determined by estimating rather than the detailed information inside the borehole, even though it needs one more parameter to be estimated than other techniques. Figure 3 shows the inner configuration of the measuring apparatus, while Figure 4 is the sketch of the test system. The main components of the measuring apparatus include an electrical heater, a circulating pump, two thermocouples, a flow meter, a data logger, and data transition equipment. The measuring apparatus was connected to the buried loop of the ground heat exchanger at worksite. To reduce heat loss, all exposed pipes were coated with insulation materials.

Figure 3. Measuring apparatus.

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Figure 4. Test system.

During tests, being heated by the electrical heater and driven by the circulating pump, water circulated in the buried loop and released heat to the ambient grout and soil. The U-tube inlet and outlet water temperatures and flow rate were measured and transmitted to a data processor. Tests usually take scores of hours, typically 50 to 80. Data of ground loop inlet and outlet temperatures and the water flow rate were collected at regular intervals, in the range of 10 to 20 minutes. Based on the measured data, soil conductivity ks , borehole heat resistance Ro , and volumetric specific heat capacity ρs cs were found by inverse heat transfer analysis. By comparing the mean water temperatures variation with time calculated from the heat transfer model with those data measured during the test and adjusting the values ks and Ro contained in Eq. (8), the final soil thermal conductivity and borehole heat resistance were obtained if the deviation between calculation and measurement reached a minimum. The sum of the squares of the errors between the heat transfer model and the experimental results is f =

N i=1

s (Tm,i − Tm,i )2

(12)

The minimum sum of the squares of the errors can be obtained by an optimization method. The complex method was used here as described in Shi and Dong [18]. TEST RESULTS AND DISCUSSION A test was carried out at Shandong Institute of Architecture and Engineering, China. The HVAC (Heating, Ventilating, and Air Conditioning) system installed in its library is the first GSHP system in China with a vertical ground heat exchanger in practical application. The borehole diameter was 115 mm, the local mean initial ground temperature 14.5◦ C, power imposed on the buried loop 46 W/m, and test duration 68 hours. Figure 5 illustrates the measured mean water temperatures. As mentioned above, the model treats the U-tube in the borehole as a line source, and then fails to describe the temperature variation properly in the beginning hours of the test, which is strongly influenced by the finite dimension and complicated configuration


125

Figure 5. Measured and calculated mean water temperature.

of the heat source. As a consequence, the recorded temperature variations in the beginning period play a negative role in the parameter estimation procedure, and should be abandoned. Great improvement in the estimated parameters can be seen in Figures 6 to 8 with certain temperature data recordings in initial hours discarded. These figures illustrate that, when the data from the initial 8 hours are abandoned, estimated values of Ro , ks , and ρs cs converge to narrow ranges of 0.065–0.072 K·m/W, 1.205–1.213 W/m·K, and 2.012–2.322 × 106 J/m3 ·K, respectively. To ignore data of an even longer initial period seems to be of limited impact on the estimated ground thermal properties. Therefore, the following discussions are based on the condition that the temperature recordings of the initial ten hours are ignored in data processing.

Figure 6. Influence of initial hours data on borehole heat resistance.

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Figure 7. Influence of initial hours data on soil thermal conductivity.

The influence of test duration on estimated soil thermal conductivity ks is illustrated in Figure 9. After 35 hours, thermal conductivity became relatively steady and varied from 1.15 to 1.33 W/m·K. When a 2-D heat transfer model was used, in contrast with this, the value of estimated thermal conductivity became steady after about 50 hours and maintained at about 1.53 W/m·K [11]. This might mean that test duration could be greatly shortened with the simplified heat transfer model. Further studies and more in-situ tests are needed to explain this difference. One possible reason is that some parameters and data of the borehole inner configuration estimated may be quite different from their “true” values when the 2-D model was adopted.

Figure 8. Influence on initial hours data on volumetric specific heat capacity.


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Figure 9. Influence of test time on soil thermal conductivity.

Figures 10 and 11 show similar trend of variation. Estimated borehole heat resistance Ro and volumetric specific heat capacity ρs cs became relatively steady and varied not much after 35 hours of testing. Ro ranges in 0.059–0.076 K·m/W, and ρs cs in 1.738– 2.455 × 106 J/m3 ·K. According to above analyses, to obtain more exact properties of soil and diminish the influence of the borehole configuration, it is appropriate to ignore some initial part of the data and to adopt long enough test durations. With the simplified heat transfer model, the soil thermal conductivity was obtained as ks = 1.22 W/m·K, borehole heat resistance per meter depth as Ro = 0.072 K·m/W, and volumetric specific heat as ρs cs = 2.21 × 106 J/m3 ·K, and Eq. (8) was used to calculate

Figure 10. Influence of test time on borehole heat resistance.

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Figure 11. Influence of test length on soil volumetric specific heat capacity.

mean buried loop water temperature variation with time, Tms , as shown in Figure 5. The calculated water temperatures are in very good agreement with the measured results. The goodness of fit R 2 is R2 =

N i=1

S Tm,i 2

N i=1

Tm2 = 1.0132.

(13)

Figure 12 shows temperature residuals. Temperature residuals for longer than 10 hours vary randomly except for 42–58. During this period, temperatures measured

Figure 12. Temperature residuals.


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are abnormally lower than their original developing tendency due to some unknown reason (see Figure 5). The 2-D model was also used to calculate mean buried loop water temperature variation with time, Tm2D , which is also plotted in Figure 5. Some parameters and data were measured and chosen as the following: di = 0.025 m, do = 0.032 m, db = 0.115 m, kp = 0.33 W/m·K, kg = 1.49 W/m·K, and u = 0.625 m/s. Properties of circulating water were taken at 29 K, i.e., ρw = 995.95 kg/m3 , kw = 0.616 W/m·K, cw = 4186.8 J/kg·K, and νw = 0.985 × 10−6 m2 /s. Values of soil thermal conductivity and soil volumetric specific heat calculated by the 2-D model are ks = 1.53 W/m·K and ρs cs = 2.019 × 106 J/m3 ·K, respectively. Figure 5 shows that TmS fits Tm quite well; meanwhile, Tm2D only has a good fitness after about 20 hours. One reason may be that the heat transfer in the borehole was assumed a steady process when the 2-D model was adopted. In fact, this assumption is reasonable only after longer times. Another possible reason may be that the 2-D model needs many parameters and data to be determined when the borehole resistance is calculated accordingly. Exact values of some parameters and data are quite difficult to obtain, and errors are inevitable in calculating Ro . Similar situations also exist when other available models are adopted. CONCLUSIONS In this article a simplified heat transfer model was introduced to estimate ground thermal properties at worksite, greatly simplifying the parameters measurement during practical application and calculation. Detailed information on the borehole configuration is unnecessary in estimating ground thermal properties. An in-situ test was conducted and the simplified heat transfer model was used for data processing. Mean circulating water temperature calculated with estimated soil thermal properties has a high goodness of fit with data collected in the test. Calculation results also indicate the possibility that the test duration may be greatly shortened with the simplified model. REFERENCES 1. M. T. Sulatisky et al., Ground-Source Heat Pumps in the Canadian Prairies, ASHRAE Trans., vol. 97, part 1, pp. 374–385, 1991. 2. R. L. D. Cane et al., Modeling of Ground-Source Heat Pump Performance, ASHRAE Trans., vol. 97, part 1, pp. 909–925, 1991. 3. S. P. Kavanaugh, Simulation and Experimental Verification of Vertical Ground-Coupled Heat Pump Systems, Ph.D. thesis, Oklahoma State University, Stillwater, Oklahoma, 1984. 4. S. P. Kavanaugh and K. Rafferty, Ground Source Heat Pumps-Design of Geothermal Systems for Commercial and Institutional Building, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1997. 5. M. Z. Yu, N. R. Diao, D. C. Su, Y. J. Ge, P. C. Wang, and Z. H. Fang, A Pilot Project of the Closed-Loop Ground Source Heat Pump System in China, Proc. 7th Int. Energy Agency Heat Pump Conf. 2002 Proc., Architecture Industrial Press, Beijing, pp. 356–364, 2002. 6. P. Cui, N. R. Diao, and Z. H. Fang, Analysis on Discontinuous Operation of Geothermal Heat Exchangers of the Ground-Source Heat Pump Systems, J. Shandong Institute of Architecture and Engineering, vol. 16, no. 1, pp. 52–57, 2001. 7. H. Y. Zeng and Z. H. Fang, A Fluid Temperature Model for Vertical U-Tube Geothermal Heat Exchangers, J. Shandong Institute of Architecture Engineering, vol. 17, no. 1, pp. 7–11, 2002.

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8. T. D. Wei, M. M. Hu, Y. Ding, and X. Y. Liu, Measurement and Heat Transfer Modeling of a Shallow Buried Ground Source Heat Pump in Winter Heating Operation, Heating Ventilating & Air Conditioning, vol. 30, no. 1, pp. 12–14, 2000. 9. S. P. Kavanaugh, Field Tests for Ground Thermal Properties-Methods and Impact on GroundSource Heat Pump Design, ASHRAE Trans., vol. 98, no. 9, pp. 607–615, 1992. 10. M. Z. Yu and Z. H. Fang, A Method for the On-Site Testing of Average Thermal Physical Parameters of Underground Rock Soil, J. Engineering Thermal Energy Power, vol. 17, no. 5, pp. 489–492, 2002. 11. M. Z. Yu and Z. H. Fang, A Method for In Situ Determining the Thermal Properties of Deep Ground, J. Eng. Thermophysics, vol. 23, no. 3, pp. 354–356, 2002. 12. W. A. Austin, Development of an In-Situ System for Measuring Ground Thermal Properties, Ph.D. thesis, Oklahoma State University, Stillwater, Oklahoma, 1998. 13. J. A. Shonder and J. V. Beck, Field Test of a New Method for Determining Soil Formation Thermal Conductivity and Borehole Resistance, ASHRAE Trans., vol. 106, part 1, pp. 843–850, 2000. 14. J. A. Shonder, P. J. Hughes, and J. V. Beck, Determining Effective Soil Formation Thermal Properties from Field Data Using a Parameter Estimation Technique, ASHRAE Trans., vol. 105, no. 1, pp. 458–466, 1999. 15. X. L. Liu, D. L. Wang, and Z. H. Fang, Modeling of Heat Transfer of a Vertical Bore in Ground-Source Heat Pumps, J. Shandong Institute of Architecture and Engineering, vol. 16, no. 1, pp. 47–51, 2001. 16. J. E. Bose, Closed-Loop Ground-Coupled Heat Pump Design Manual, Stillwater, Oklahoma State University, Engineering Technology Extension, 1984. 17. H. Y. Zeng, N. R. Diao, and Z. H. Fang, Thermal resistance Inside Boreholes of Vertical Geothermal Heat Exchangers, Gas & Heat, vol. 13, no. 3, pp. 134–138, 2003. 18. G. Y. Shi and J. L. Dong, Optimization Methods, High Education Express, Beijing, 1999.