Chapter 3. Methodology
3.2. Thermal aspects of single piles
The experimental and analysis methods are described, after definitions of the main terms, are introduced.
3.2.1. DEFINITIONS
The average fluid temperature Tf [°C] circulating through the ground-loop is one of the main parameters required to choose the most adequate heat pump for a GSHP installation. The average fluid temperature Tf is defined as:
Tf= T0+ q
2πλsGg+ qRcGc+ qRpipe (1)
where T0 [°C] is the undisturbed soil temperature, q [W/m] is the heat transfer rate per metre length of pile heat exchanger, λs [W/m/K] is the thermal conductivity of the soil, Gg is the g-function describing the ground temperature response, Rc [K∙m/W] is the steady state concrete thermal resistance, Gc is the concrete g-function describing the transient concrete response and Rpipe [K∙m/W] is the thermal resistance of the pipes. The temperatures and thermal resistance arrangement are shown in Figure 3-2.
a)
b)
Figure 3-2: a) Temperature definitions in the energy pile cross section;
b) fundamentals of thermal resistances in the energy pile cross section.
G-functions are dimensionless response factors that describe the change in temperature in the ground around a heat exchanger with time as a result of an applied thermal load q [100]. Usually, both temperature change and time are normalised. In this study, the normalised temperature changes Φ and time Fo are defined as:
Ф =2πλs∆T
q (2)
Fo =αst
rb2 (3)
where ΔT [K] is the temperature change between the undisturbed soil temperature T0
[°C] and the average pile wall temperature Tb [°C], αs [m2/s] is the thermal diffusivity, i.e., the ratio between the thermal conductivity λs [W/m/K] and the volumetric heat capacity of the soil ρcps [J/m3/K], t [s] is the time and rb [m] is the pile equivalent radius. The pile radius is the radius that provides an equivalent circumference to the square perimeter.
For a single pile, the pile wall temperature depends on time and its aspect ratio AR (L/2rb), and it can be determined as:
Tb= T0+ q
2πλs∙ G(Fo, L
2rb) (4)
G-functions can be obtained by analytical, numerical and empirical methods. Figure 3-3 shows different types:
Figure 3-3: The infinite line [137] and infinite hollow cylinder [138] source solutions together with semi-empirical pile G-functions reported in [94].
The concrete G-function Gc, as defined by [94,139], describes the transient thermal resistance of the pile heat exchangers. It depends on the shape of the pile cross section, the position of the pipes and the thermal conductivity of the concrete λc. That is, it defines the thermal resistance of the concrete part. To incorporate the transient response of the pile concrete into the overall temperature response function (Equation 1), the proportion of the steady state thermal resistance that has been achieved at a given value of time Fo needs to be determined as:
Rc=Tp− Tb
q (5)
where Tp [°C] is the average temperature on the outer wall of the pipe
Finally, the pipe thermal resistance Rpipe [K∙m/W] is defined in Equation 6 as the sum of the pipe convective (first term on right hand side) and conductive (second term on right hand side) resistances:
Rpipe= 1
2nπrihi+ln(ro⁄ri)
2nπλpipe (6)
where n is the number of pipes in the pile heat exchanger cross section, ri [m] is the inner radius of the pipe, ro [m] is the outer radius of the pipe, hi [W/m2/K] is the heat transfer coefficient and λpipe [W/m/K] is the thermal conductivity of the pipe material.
hi can be calculated using the Gnielinski correlation as described in [140,141].
3.2.2. EXPERIMENTAL DATA
The experimental work related to the thermal aspects of the energy piles has been carried out in two test sites in Denmark (Figure 3-4). The work mainly consists of thermal response testing (TRT) of energy piles and laboratory measurements of the thermal properties of the soil at both test sites and of the concrete used in the piles.
The TRT is a field method of ground heat exchangers for estimation of soil thermal conductivity λs, ground heat exchanger thermal resistance (hereon concrete thermal resistance Rc) and undisturbed ground temperature T0 [122].
During the TRT, the heat carrier fluid (water) is circulated in the ground heat exchanger while being continuously heated at a specified rate. Heat dissipates to the ground heat exchanger and subsequently to the ground. The test records fluid inlet-and outlet temperatures inlet-and the fluid flowrate inlet-and logs them in 10-min intervals for at least 48h. Figure 3-5 shows the test setup and an example for the measurements.
Figure 3-4: Location of test sites, in Denmark.
Several tests are carried out in energy piles with different depths and pipe configurations (single-U and W-shape). During one of the tests, soil temperatures at a distance from the energy pile were also recorded at given depths (Figure 3-5).
Detailed information regarding the fieldwork is provided in [142], Appendix III.
a) b)
Figure 3-5: a) Thermal response test setup, after [143]; b) TRT field data of pile heat exchanger and weighted soil temperatures at 0.9 m from pile, after [144].
Independent measurements of the thermal properties of the soil and the concrete have been carried out by means of the Hot Disk apparatus [145]. The Hot Disk equipment (Figure 3-6) relies on the transient plane source method [146], which yields estimations on the thermal conductivity and volumetric heat capacity. An 18-m bore is drilled in each test site, where soil samples are collected every 0.5 m. Water content
and bulk and dry density measurements are also given for each sample. Details on the equipment, sample treatment and measurement procedures are provided in [147], Appendix IV.
Figure 3-6: Hot Disk sensor in between halved sample and room temperature sample holder. Courtesy of Hot Disk ® [145].
3.2.3. ANALYSIS METHODS
In practice, the TRT data is analysed with analytical models in order to estimate the soil thermal conductivity and the concrete thermal resistance. In this study, the TRTs of energy piles are interpreted with analytical, semi-empirical and numerical models by means of non-linear regression. First, the soil thermal conductivity is estimated by inverse 3D finite element modelling (Figure 3-7) of the TRT data and then compared to corresponding, independent laboratory measurements.
A fully 3D based TRT interpretation is not feasible for routine practical applications due to the computational cost of solving the inverse problem. Consequently, the study also explores the applicability of simpler analytical and semi-empirical models for interpretation of the TRT data. The tested models are summarised in Table 3-2.
The parameter estimation is performed with PEST Model-Independent Parameter Estimation software [148]. PEST employs the Gauss-Marquardt-Levenberg algorithm for minimising the weighted, squared difference between computed and observed fluid temperatures. PEST calculates linear confidence intervals for estimated parameter following the non-linear regression procedure. This algorithm is applied to all the models.
Figure 3-7: Description of the 3D finite element model: a) Schematic of the W-shape pile heat exchanger; b) Schematic of the Single-U pile heat exchanger; c)
Simulated meshed domains; d) Top view of a quarter of domain, after [35].
Table 3-2: Summary of models selected to evaluate the pile heat exchanger TRT data, after [35].
Model description
Analytical approaches
Infinite line source [137].
Infinite cylinder source [138].
Infinite solid cylinder source [95]
Finite solid cylinder source [95].
Semi-empirical approach G-function for pile heat exchangers [94]. The finite length of the pile is considered.
Numerical approach
Equivalent pipe model [149]. It neglects the finite length of the pile.
2D horizontal cross section finite element model.