2.5 Ground Effects
2.5.5 Models that includes natural convection inside the borehole
During operation of boreholes filled with groundwater, the heat extraction or injection will induce a temperature gradient in the borehole. This leads to a density gradient in the groundwater, which results in a convective heat flow. For groundwater filled boreholes the natural convection will affect the heat transfer between the ground and the working fluid, reducing the thermal resistance compared to stagnant water. The convective flow of the groundwater depends on the temperature gradient between the borehole wall and the collector wall, and it will therefore depend on the ground temperature, the amount of heat injected or extracted and whether heat is injected or extracted. A thermal response test where heat is injected into the borehole might therefore result in a different borehole resistance than a borehole heat exchanger in operation which extracts heat. Figure 2-20 shows a velocity profile of water between borehole wall and pipe wall induced by a tem-perature gradient between the pipe wall and the borehole wall.
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Figure 2-20 Velocity profile of natural convection induced flow, Gustafsson and Westerlund (2011) A multi-injection rate thermal response test was performed by Gustafsson and Wester-lund (2009) with no influence from regional groundwater flow. Results showed that a larger heat injection results in a decrease in the borehole thermal resistance. The large heat injection triggers more convective heat transfer, thus lowering the resistance in the borehole. The test also showed that the length of the collector did not influence the natu-ral convection in the borehole. Without any influence from groundwater flow the thermal resistance decreased from 0.12 to 0.065 mK/W with an increase of heat injected from 21 W/m to 83 W/m, due to an increase in the convective flow in the borehole. This results supports that higher flow velocities of the borehole water due to the larger density differ-ences in the borehole water enhances the heat transfer in the borehole.
The thermal effects of natural convection are highly dependent on the given geometry and orientation. A simplification of the borehole geometry is often used to make it possi-ble to have a numerical solution. Studies on the heat transfer effects of natural convection have been done with a numerical model, approximating a BHE u-pipe geometry to an equivalent radius geometry with constant heat flux or constant temperature at the inner wall and constant temperature at the outer wall as boundary conditions.
Since the natural convection influence on the borehole resistance is dependent on the relationship between the changes in water density in the radial direction several Nusselt number relationships have been developed, with possibilities of varying the boundary conditions by implementing a Rayleigh number, the radius ratio and the aspect ratio.
The Nusselt number can be used to investigate the heat flow increase due to convective heat transfer compared to stagnant water with only conductive heat transfer, but there
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are none published Nusselt number correlations for BHE u-pipe geometry applications.
Approximating the u-pipe geometry to an equivalent radius model is therefore necessary to be able to use Nusselt number correlations for natural convection induced heat trans-fer.
The Nusselt number is the ratio of convective to conductive heat transfer across the boundary
Nu = Rb conduction
Rb convection
(2-25) and
Rb convection= Rb conduction
Nu (2-26)
With Nusselt relations given in equation (2-25) and (2-26) an effective thermal conductiv-ity can be derived, when Rb conduction and Nu is known. By implementing an effective thermal conductivity a water filled borehole can be simulated with the effects of an ap-proximated natural convection inside the borehole.
Gustafsson and Westerlund (2011) made a 3D model to study the effects of convection and phase change in a water filled borehole. The model was made with an equivalent diameter approximating a u-pipe collector with a specified thickness and a fluid inlet at the bottom and outlet at the top. The height of the model was only one meter, so it prac-tically has the same restrictions as a two-dimensional model. In the model the fluid enter the inlet with a fluid flow velocity at 1 m/s and the fluid inlet temperature varied, start-ing at 15°C and decreasstart-ing linearly with 0.67°C per simulated hour. The undisturbed ground temperature was 6°C during the simulation. With the heat flow in the pipe wall calculated numerically, the borehole resistance was calculated by
Rb=Tbhw−(Tf,in+ Tf,out) 2⁄ qpw
(2-27)
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Figure 2-21 Borehole resistance with change in mean water temperature, Gustafsson and Westerlund (2011)
Figure 2-21 show how the borehole thermal resistance varies with mean water tempera-ture which is the mean value of the collector pipe temperatempera-ture and the borehole wall temperature. As the inlet temperature decreases the borehole resistance peaks at tempera-tures around 4°C because the water has its highest desist around 4°C as shown in Figure 2-22
.
When the mean water temperature in the borehole is around 4°C the convective flow is almost negligible, and the borehole resistance is close to the stagnant water case.Figure 2.19 shows that the borehole resistance has a large variation for positive mean water temperatures inside the borehole, and it is constant for a mean water temperature lower than -2°C because the water inside the borehole is frozen. Ice has a thermal conduc-tivity approximately three times higher than for the stagnant water case which results in a lower borehole resistance for meant temperatures around -2°C than 4°C. Since the tem-perature is given as a mean water temtem-perature, the ice forming before the mean water temperatures reaches 0°C.
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Figure 2-22 Change in water density with change in temperature 997
997.5 998 998.5 999 999.5 1000 1000.5
0.0 5.0 10.0 15.0 20.0
Water Density [kg/m^3]
Temperature [°C]
Water Density [kg/m^3]
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