HEAT TRANSFER EXPERIMENT IN THE GROUND WITH GROUND WATER ADVECTION T. Katsura, K. Nagano, S. Takeda, K. Shimakura Graduate School of Engineering, Hokkaido University Sapporo, 060-8628 Japan Tel: +81-11-706-7597
[email protected]
1.
INTRODUCTION
The authors have developed a design tool for the ground source heat pump (GSHP) system based on the heat conduction theory (Nagano et al, 2006, Katsura et al, 2006). However, there are a lot of regions where have plenty ground water in Japan. There are several reports that when the ground is used as heat source, amount of heat extraction (for heating) or injection (for cooling) is increased by ground water flow even in a case where the ground thermal energy is used indirectly (For example, Iwata et al, 2005). Therefore, it is important to consider the ground water flow for the design of the GSHP system in order to scale down the GSHP system and reduce initial cost. The design tool for the GSHP system developed by the authors applies superposition of the thermal response of the cylindrical heat source theory in order to calculate temperature of the ground surrounding the ground heat exchangers (Nagano et al, 2006). Thus, it is expected that superposing thermal response for heat source (line or cylindrical) in the ground with ground water advection and applying superposition of the thermal response are effective method for calculation of the ground temperature. In this paper in order to investigate thermal response for heat source in the ground with ground water advection, laboratory experiment, in which temperatures are measured under the condition that constant heat is generated from a thermal probe in the ground with uniform ground water flow of constant velocity, is carried out at first. On the other hand, there are some solutions of moving source of heat in the infinite medium (Carslaw and Jeager, 1959). An unsteady state analytical solution for the moving line source of heat is obtained and applied for calculation of temperature of the ground surrounding ground heat exchangers (Diao et al, 2004). Applying superposition of the thermal response for the moving line source of heat is also effective for calculation of the ground temperature. For this reason, temperatures calculated by the moving line heat source theory are compared with each one measured during the experiments in order to validate reproducibility of the calculated value for measured one. In addition, measuring ground water velocity more accurately is important in order to consider the ground water flow for the design of the GSHP system. But since observation wells to measure ground water velocity is required apart from boreholes used as ground heat exchangers, making the observation wells takes a lot of time and cost. A method applying the moving line heat source theory, which can estimate of the ground water velocity more simply, is proposed. The outlines and examples of the estimation are shown in this paper. 2. LABORATORY EXPERIMENT OF THERMAL RESPONSE IN THE SOIL WITH WATER FLOW 2.1. Outlines of Laboratory Experiment Figure 1 shows a diagram of the laboratory experiment apparatus. The apparatus features to imitate grand water flow in a sand filled layer into the apparatus. A thermal probe buried in the sand filled layer can generate constant heat. Relation between the ground water velocity and thermal response for heat source (line or cylindrical) was investigated by measuring temperatures of the probe and each point in the sand shown in Figure 2. The cross sections of the apparatus are shown in Figure 2.The apparatus is made from an acrylic cylinder with internal diameter of 300 mm. The sand filled layer and water layers under and over the sand filled layer are separated by nonwoven fabrics and acrylic perforated panels. Silica sand with average particle diameter of approximately 0.2 mm is filled in the cylindrical tank and the thickness of the sand filled layer is 200 mm. Water is supplied from top of the cylinder. By keeping difference of water levels ∆H constant shown in Figure 1, which is
between outlet of the over flow pipe and the water outlet, water flow rate through the sand filled layer is kept constant. The flow rate is measured and velocity of the Darcy flow is calculated from the measured flow rate. The probe is buried at the center of the sand filled layer. Composition of the thermal probe is shown in Figure 3. A line heater and Pt-100 sensor are sheathed with a stainless steel pipe with length of 200 mm and external diameter of 3.2 mm. The temperature sensor is fixed at the center of the probe where is the representative point. The heater is connected to constant voltage device and generates heat continuously. Temperatures in the sand filled layer are measured by thermo couples placed at the each point shown in Figure 2. All of the experimental apparatus was equipped in the constant temperature chamber. The experiment is carried out by changing velocity of the imitated ground water flow. Table 1 indicates the experimental conditions, which are measured volume and calculated velocity according to each experiment.
ON/OFF
∆H
Over Flow Pipe
Water Nonwoven fabric + Perforated Panel
Constant Voltage Device
Thermal Prove Sand filled layer (Silica sand) Nonwoven fabric + Perforated Panel
To each temperature measured point
Data logger
PC
Water
P
Water flow direction
Acrylic cylinder
Water outlet
Temperature is kept at
Temperature measured point (Thermo couple)
20oC
Figure 1 Diagram of laboratory experiment apparatus Thermal Probe
50 50 40
A
300
Sand
Temperature measurement point (Thermo couple) Used for comparison with calculated value, Point A, B, C
40
200
Water
Acrylic Cylinder
50
B Water
C Water Outlet
Front view
2.2. Results and Discussions
Side view
Elevation view
Figure 2 Cross sections of apparatus Temperature measurement point (Pt-100) 200 100
To Data Logger
3.2
Figure 4 shows temperature variations ∆T in the thermal probe according to elapsed time t in each experimental condition when the heating rate from the heater is kept at 6.6 W/m constant. The temperature variation in CASE1 (u = 0) increases linearly according to logarithm t. On the other hand, when the ground water velocity is larger, the time to achieve at steady state of the temperature variation is shorter and the temperature variation is smaller. In CASE6 (u = 6.22×10-5 m/s), the temperature variation achieves at steady state after elapsed time of approximately 1000 s and the temperature variation ∆T is approximately 1.4 oC. In addition, the soil effective thermal conductivity λs is estimated by the following equation applying the line heat source theory with the temperature variations.
Heater
Stainless Steel Pipe
Heater Code
To Constant Voltage Device
Figure 3 Composition of thermal probe
Table 1 Experimental condition Measured water flow rate [ml/min] Water velocity through the sand layer [m/s] Water velocity through the sand layer [m/year]
CASE1
CASE2
0
36
CASE3
CASE4
51 -6
CASE5
112 -5
0
8.39× 10
1.20× 10
2.64× 10
0
265
377
833
q' λ s = (4πk )
Here, k is a gradient of temperature obtained from an approximated equation of T according to t.
CASE6
186 -5
4.39× 10 1383
264 -5
6.22× 10-5 1963
(1)
T s = k ln (t ) + l
(2)
5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
25
CASE1, CASE2, CASE3, CASE4, CASE5, CASE6 from top to bottom
0
Heating rate from thermal prove:6.6W/m
10
100
1000
10000
t [s]
Ground water velocity [m/year] 500 1000
1500
Period of elapsed time used for estimation of the effective thermal conductivity
20 λs [W/m/K]
Δ Ts [ºC]
If heat in the soil is transferred by only conduction, k is almost constant with large value of t (t ≥ 5r2/a). However, since the heat in the soil with the ground water flow is transferred by advection, k varies depending on the elapsed time. Figure 5 shows effective thermal conductivities λs, which are obtained by Equation (1) with temperature variations in each range of the elapsed time in the experiments. It is clear that the difference of λs is small in the short range for every flow rate even for large flow rate. However, the variations of λs in the long time range differ depending on the ground water velocity. The effective thermal conductivity λs become larger when the ground water velocity is larger. Thus in cases where performance of the GSHP system is evaluated by changing effective thermal conductivity for ground water flow, it is anticipated that error occurs. Also, since the temperature variation achieves at steady state and k is equal to zero for the large value of t in the condition where u is large, λs obtained by Equation (1) is infinity. From above, it is expected that it is possible to estimate the ground water velocity by investigating variation of λs obtained by Equation (1) for the elapsed time.
50s～100s
15
100s～200s
200s～500s
500s～1000s
1000s～5000s
5000s～10000s
10 5 0 0
1×10-5 2×10-5 3×10-5 4×10-5 5×10-5 6×10-5 7×10-5
u [m/s]
Figure 4 Temperature variations ∆T in the thermal probe from start of the experiment for elapsed time t
Figure 5 Variations of λs for each different period of the elapsed time in the experiments
3. COMPARISON THERMAL RESPONSE BETWEEN MEASURED IN EXPERIMENT AND CALCUALTED USING THE MOVING LINE HEAT SOURCE THEORY
3.1 Calculation of Thermal Response with Ground Water Advection Applying the Moving Line Heat Source Theory. In order to apply the moving line heat source theory for calculation of temperature of the ground surrounding the heat source (line or cylindrical), the ground is treated as infinite medium and the heat source is regarded as a line heat source with infinite length in the infinite medium. When a uniformly ground water flow, whose velocity is u, in the direction of x-axis is occurred and constant heat q’ is generated from the heat source, the thermal response of the ground can be calculated by the following solution (Diao et al, 2004). 4 ast
⎛ 1 U 2r 2β ⎛ Ur ⎞ r2 1 ∆Ts = exp⎜⎜ cos ϕ ⎟⎟ ∫ exp⎜⎜ − − 2 4πλs ⎝ 2a s ⎠ 0 β ⎝ β 16a s q'
⎞ ⎟dβ ⎟ ⎠
(3)
3.2 Results and Discussions of Comparison between Thermal Responses of Measurement and Calculation The thermal response calculated by the moving line heat source theory is validated by comparison to one measured in experiment. As calculated conditions, the effective thermal conductivity of the saturated silica sand of 1.85 W/m/K obtained by Equation (1) with temperature measured in the experiment of CASE1 (u = 0) is given. The heat
capacity of the sand is as 2869 kJ/m3/K, which is estimated by giving the density of the sand particle of 2533 kg/m3, porosity of the filled sand of 36.7 %, specific heat of the sand particle of 0.84 kJ/kg/K. Figure 6 and Figure 7 shows comparisons between thermal responses at points of A-C shown in Figure 2 of the measurement and the calculation in CASE2 and CASE5, respectively. From these results, since the variations of the thermal responses calculated by moving line heat source theory produce good agreement with measured one, it is confirmed that applying moving line heat source theory is effective method for calculation of the ground temperature with ground water flow. 3.0
Measured
3.0
Calculated
Measured
A
2.0 1.5 1.0
B
0.5
C
o Δ Ts [ C]
o Δ Ts [ C]
Calculated
2.5
2.5
2.0 1.5
A
1.0 B
0.5
C
0.0
0.0 0
2000
4000
6000
8000
10000
t [s]
Figure 6 Comparisons between thermal responses at points of A-C (CASE2)
0
2000
4000
6000
8000
10000
t [s]
Figure 7 Comparisons between thermal responses at points of A-C (CASE5)
4. METHOD OF ESTIMATION OF GROUND WATER VELOCITY APPLYING THE MOVING LINE HEAT SOURCE THEORY 4.1 Method From Section 2, the variation of the thermal response with ground water advection isn’t constant according to logarithm t. Thus it is expected that ground water velocity can be estimated by investigating gradient of the thermal response. Here, the gradient of the temperature kwf (t) is expressed by Equation (4) when the temperature Ts for a very short time dt according to certain radius from the heat source r can be shown by Equation (2). k wf (t ) =
Ts (t + dt ) − Ts (t ) ln{(t + dt ) t}
(4)
Additionally, gradient of the non-dimensional temperature kwf* (t*) can be obtained by substituting non-dimensional number R*, non-dimensional temperature T*, and Fourier number t*. Introducing non-dimensional time Fo, which is function of ground water velocity, Figure 8 shows variations of gradients of non-dimensional temperature kwf* according to Fo. When the value of t is enough large, the gradients are almost the same for non-dimensional number R*. From these results, it is confirmed that kwf* can be approximated by a function representing intention of the ground water flow.
k wf ≅ 0.5e *
F − o 4
= k wof e *
F − o 4
(5)
The parameter kwof*is gradient of non-dimensional temperature in a case where the heat in the ground is transferred by only conduction without advection. Provided dimension, gradient of temperature kwf is expressed from Equation (6). k wf ≅ k wof e
−
U 2t 4as
(6)
Here, a constant number n is introduced. n=
U2 4a s
(7)
The number n can be provided by the exponential approximate equation of kwf according to t. The gradient of temperature kwf (t) is calculated by Equation (4) from a temperature variation measured in the actual experiment. The gradient of temperature without ground water advection kwf is obtained from the temperature variation in the short range of elapsed time since the gradient hardly changes for every flow rate in the short range as shown in Figure 4. Then the ground water velocity can be calculated by using the constant number n. u = 2 as n ×
cs ρ s cw ρ w
(8)
With regard to applicable range of Equation (5), relative error between kwof* calculated by Equation (4) and approximated by Equation (5) is investigated. The non-dimensional time Fo according to R* in which the relative error is equal to 1% is shown in Figure 9. Figure 9 also shows the actual time t according to u on conditions that radius of the probe of 1.6×10-3 m, soil effective thermal conductivity of 1.85 W/m/K, and heat capacity of 2869 kJ/m3 are given (the same as the experiment in this paper). From these, if the conditions are the same as the experiment, the range of data for t ≥ 100 s is applicable for Equation (5). u [m/s] R*=0.005、0.01、 0.02、0.1、0.2
3.5
1.5×10-4
2.0×10-4
2.5×10-4
R=
1.6×10-3
140
m, λs = 1.85 W/m/K, csρs =2869
120
KJ/m3
2.5
R*=1.0
kwf
*
2.5×10-5
Condition for calculation of t
3.0
0.1
5.0×10-5
0
Fo
Approximated equation of kwf * -F /4 in range of Fo=3~20 kwf * = 0.5e
0.01
100
2.0
o
80 Fo
1.5
60
1.0 0.001 0
3
40
0.5 5
10
15
Fo
Figure 8 Variations of gradient of non-dimensional temperature kwf* according to Fo
20
t [s]
1
20
t
0.0
0 0
0.2
0.4
0.6
0.8
1
R*
Figure 9 Fo according to R* in which relative error between kwof* calculated by Equation (4) and approximated by Equation (5) is equal to 1% and the actual time t according to
4.2 Examples Examples of estimation of the ground water velocity are demonstrated by using the temperature variations of the actual experiments in Section2. Figure 10 shows variations of kwf according to t on a condition that the time interval dt given in Equation (4) is as 100 s. The data of temperatures measured in the range of t ≥ 100 s is used for estimations. However, the data in the range where the difference between kwf calculated by Equation (4) and approximated by Equation (6) is large for small value of t and the amplitude of temperature variation is large for large value of t is excluded. Table 2 indicates actual measured water velocities, the ranges of time for estimation, the constant number n of approximated equations in Figure 10 and estimated water velocities for the each case. Relative errors between the velocities of estimation and measurement are also included in Table 2. From these results, it was indicated that the method is effective for the estimation of the ground water velocity. 5.
CONCLUSIONS
1. 2. 3.
From probe experiments, it was identified that when the effective thermal conductivity of the ground is estimated by applying the line heat source theory, the variation of the effective thermal conductivity for elapsed time depends on the ground water velocity. Since the variations of the thermal responses calculated by moving line heat source theory produced good agreement with measured one, it was confirmed that applying moving line heat source theory is effective method for calculation of the ground temperature with ground water flow. Method to calculation the ground water velocity applying the moving line heat source theory was proposed. Then the examples indicated that the method is useful for the estimation of the ground water velocity. Table 2 Comparisons of ground water velocities between estimation and measurement
1 CASE2: kwf = 0.283e-5.89×10
-5t
CASE3: kwf = 0.283e-1.78×10
-4t
kwf
CASE2
0.1
-8.16×10-4t
CASE4: kwf = 0.283e CASE5: kwf = 0.283e-1.93×10 t -3
CASE6: kwf= 0.283e
-3.57×10-3t
0.01 0
1000
2000
3000
4000
t [s]
Measured water velocity [m/s] Range of time for estimation [s] n in Figure 10 Estimated water velocity [m/s] Relative error [%]
CASE3 -6
8.39× 10
1.20× 10
CASE4 -5
2.64× 10
CASE5 -5
4.39× 10
CASE6 -5
-5
6.22× 10
500 ~ 4000 400 ~ 3000 300 ~ 2000 150 ~ 1000 130 ~ 700 -5
5.89× 10
1.78× 10
-4
8.16× 10
-4
1.93× 10
-3
-3
3.57× 10
8.45× 10-6 1.40× 10-5 3.14× 10-5 4.83× 10-5 6.57× 10-5 0.7
17.0
19.1
10.2
5.7
Figure 10 Variations of gradient of temperature kwf according to t in the experiments ACKNOWLEDGMENTS The authors would like to express appreciation to the laboratory of ground thermal energy system of corporate donated chair in Hokkaido University. Part of this study is supported by the 21st century COE program of the Ministry of Education, Culture, Sports, Science and Technology “Sustainable Metabolic Systems of Water and Waste for Area-Based Society” (Project representative: Prof. Y. Watanabe) that provided financial participation in this research. NOMENCLATURES a: Thermal diffusivity [m2/s] c: Specific heat capacity [kJ/kg/K] Fo: Non-dimensional time (= U2t / a) [-] k: Gradient of temperature [K] k*: Gradient of non-dimensional temperature [-] q’: Amount of heat per length [W/m] R*: Non-dimensional number (=Ur/a) [-] r: radius [m] T: Temperature [ºC] T* ：Non-dimensional temperature (=2πλ∆T/r/q’) [-] t: Time [h] t*: Fourier number (= ast/r2) [-] U: Revised ground water velocity ( = ucwρw / cs / ρs ) [m/s] u: Ground water velocity [m/s] Greek letters β: Integration parameter λ: Thermal conductivity [W/m/K] Subscripts p: Ground heat exchanger s: Soil
st: Steady statement w: Water wf: With ground water flow wof: Without ground water flow ∞: At infinity REFERENCES H. S. Carslaw, and J. C. Jaeger: Conduction of Heat in Solid, Oxford University Press, 1959 Chiasson, A.D. , S.J. Rees, J.D. Spitler. 2000. A Preliminary Assessment of the Effects of Ground-Water Flow on Closed-Loop Ground-Source Heat Pump Systems. ASHRAE Transactions. 106(1):380-393. Claesson, J., G. Hellstrom. 2000. Analytical Studies of the Influence of Regional Groundwater Flow on the Performance of Borehole Heat Exchangers. Proceedings of Terrastock 2000, Vol. 1, Stuttgart, August 28September 1, 2000, pp. 195-200. Nairen Diao, Qinyun Li, Zhaohon Fang: Heat Transfer in Ground Heat Exchangers with Groundwater Advection, International Journal of Thermal Sciences 43, pp.1203-1211, 2004 Per Eskilson: Thermal Analysis of Heat Extraction Boreholes, Univ. Lund, 1986-1987 N. Iwata, T. Kobayashi, G. Fukaya, K.Yokohara, Y. Niibori: Performance Test of the Geothermal Heat Pump System, Considering Underground water, the Journal of the GRSJ Vol.27-4, pp.307-320, 2005 (In Japanese) T. Katsura, K. Nagano, S. Takeda: Development of a Design and Performance Prediction Tool for the Ground Source Heat Pump System, Proceedings of IEAs 10th Energy Conservation Thermal Energy Storage Conference. Ecostock’2006, New Jersey, 2006 K. Nagano, T. Katsura, S. Takeda: Development of a Design and Performance Prediction Tool for the Ground Source Heat Pump System, Applied Thermal Engineering, 2006 Witte, H.J.L. 2001. Geothermal response tests: The design and engineering of geothermal energy systems. Europäischer workshop über Geothermische Response Tests, EPFL, Lausanne, 25th and 26th of October 2001
