Infinite Line Source Research Articles

Towards Improved Field Application of Using Distributed Temperature Sensing for Soil Moisture Estimation: A Laboratory Experiment.

The “dual probe heat pulse” (DPHP) method using actively heated fiber optic (AHFO) cables combined with distributed temperate sensing (DTS) technology has been developed for monitoring thermal properties and soil water content at the field scale. Field scale application, however, requires the use of robust and thicker fiber optic cables, corroborating the assumption of an infinite thin heat source in the evaluation process. We therefore included a semi-analytical solution of the heat transport equation into the evaluation procedure in order to consider the finite thermal properties of the heating cable without a calibration procedure to estimate effective thermal properties of the soil. To test this new evaluation procedure, we conducted a laboratory experiment and tested different heating scenarios to infer soil moisture from volumetric heat capacity. Estimates were made by analyzing the shift of the temperature amplitude at the sensing cable and the characteristics of the response heating curve. The results were compared with results from the calibrated infinite line source solution and in situ water content point measurements and showed a good approximation of thermal properties for strong and short heat pulses. Volumetric water content estimates are similarly accurate to the results of the calibrated infinite line source solution. Problems arose with the cable spacing and the resettlement process after burying the cable.

Shifted thermal extraction rates in large Borehole Heat Exchanger array – A numerical experiment

In large scale Ground Source Heat Pump (GSHP) systems, multiple Borehole Heat Exchangers (BHEs) are often connected with the pipe network array to extract shallow geothermal energy. In this study a comprehensive numerical model was developed. The heat transport within and around the BHEs and the pipe network is explicitly quantified in a coupled manner. The model allows a dynamic heat extraction calculation on the individual BHE that is determined by the hydro-thermal processes in the pipe network. The model is thus capable of capturing the long-term thermal interference among BHEs. The model was verified against analytical solution with respect to its hydraulic and thermal balances. Based on it, a series of numerical experiments have been performed to quantitatively investigate the amount of shifted thermal extraction rate in large BHEs array. It is found that, the heat extraction rate on the central BHEs was gradually shifted towards those located at the edge in the long-term operation. Over different seasons, the strongest shifting phenomenon was observed in the month with the lowest thermal load. The shift becomes significant with the increasing number of BHEs installed. The result of numerical study suggests that traditional super-positioned based infinite line source approach with a constant heat flux is not accurate enough for long-term prognosis since it does not fully consider the thermal recharge and the thermal interference effects.

Scattering from a Buried PEMC Cylinder due to a Line Source Excitation above a Planar Interface Between Two Isorefractive Half Spaces

A solution to the problem of scattering from a perfect electromagnetic conducting (PEMC) circular cylinder buried inside a half-space and excited by an infinite electric line source is provided. The line source is parallel to the cylinder axis, and is located in the other half-space. The two half spaces are isorefractive to each other. The source fields when incident at the planar interface separating the two half spaces, generate fields that are transmitted into the half-space where the cylinder is. These fields then become the known basic incident fields for the buried PEMC cylinder. Scattering of these incidents fields by the cylinder will consequently generate fields at the interface that get reflected back into the same half-space and transmitted frontward into the source half-space, all of which are unknown. Imposing appropriate boundary conditions at the surface of the buried cylinder and at a specified point on the interface, enables the evaluation of these unknown fields. The refection coefficient at the specified point is then computed for cylinders of different sizes, to demonstrate how it varies with the PEMC admittance of the buried cylinder, the intrinsic impedance ratio of the two isorefractive half-spaces, and the burial depth of the cylinder.

Investigation of groundwater flow and saturated multi-layer soil on the ground temperature response of an energy pile through thermal response and recovery testing

Thermal response testing (TRT) and thermal recovery testing were conducted on a full-scale precast energy pile installed in saturated multi-layer soils to investigate the thermal response on the pile and soils. The thermal and physical properties of the soil were assessed through laboratory testing and compared to the estimated properties from the thermal response and thermal recovery tests through use of the finite cylinder source (FCS) method and the moving infinite line source (MILS) method. The MILS method was used to assess Darcy’s velocity through thermal recovery testing at 13 m depth within the pile, eliminating the influence of ambient temperature and potential heat input from the TRT Unit. The ground temperature response was monitored parallel and perpendicular to groundwater flow in three locations during heating input and subsequent recovery, while the pile temperature distribution was monitored within the embedded U-loops. The results showed that thermal conductivity estimates were consistently higher from TRT results when compared to laboratory results, and application of the MILS method recorded values greater than the FCS method, particularly in thermal recovery. The MILS was shown to accurately predict Darcy’s velocity through thermal recovery testing, recording a value within the range estimated through laboratory testing. Furthermore, the ground temperature response displayed significant influence of the various soil layers through a temperature curvature with depth developing over time.

A note on the two-dimensional electrostatic field produced by a line charge near a dielectric wedge

Summary We shall consider the problem of determining the correct electrostatic field produced when an infinite two-dimensional line source is influenced by an adjacent infinite dielectric wedge. This result corrects a number of previous attempts at this problem, which are shown to be in error. The method avoids using the Mellin transform which has lead to some of these earlier errors. The method is used to solve a more general problem of the electrostatic field produced by an arbitrary number of line sources located in an arbitrary number of contiguous dielectric wedges.

A thermal-dissipation correction method for in-situ soil thermal response test: Experiment and simulation under multi-operation conditions

The in-situ thermal response test (TRT) is a key process to measure the soil thermal properties and has been widely applied for precise design and optimal operation of borehole heat exchanger (BHE). However, it is difficult to accurately calculate and simulate thermal dissipation from fluids to ambient air in test processes. This paper experimentally investigated geotechnical indoor testing and TRTs at 11 sites under multi-operation conditions. Furthermore, a thermal-dissipation correction method (TDCM) is introduced and established in TRNSYS to simulate the thermal dissipation. Infinite line source model (ILSM) and infinite cylindrical source model (ICSM) are adopted to comprehensively evaluate experimental and simulation results.The simulation results of TDCM are well verified and parallelly compared under multi-operation conditions. TDCM works well with ICSM on the correction of thermal resistance with improved accuracy over 10% and also improves the accuracy of average fluid temperature, soil thermal conductivities and heat flux per unit length of BHE. Besides, TDCM is more efficient to correct results processed with ICSM than that with ILSM and is valid for simulations with heating powers of 4–8 kW. This study proposes an efficient model to estimate thermal dissipation of fluids in TRTs and select appropriate operating parameters of BHEs.

A Novel Reciprocity Based Theory for Extraction of Scattered Field under Plane Wave Incidence from that of Near-field Illumination

ABSTRACT A novel theory based on the reciprocity theorem is presented for computing the plane wave scattered field of an arbitrary target, from its scattered field due to near-field illumination. The target is supposed to be illuminated by a transmitter probe on a cylindrical surface and the scattered near-field data is recorded in the near-field. The presented theory constructs a bridge between the scattered field due to the near-field illumination by a transmitter probe and that of plane wave illumination. Thus, referring to the definition of Radar Cross-Section (RCS) that assumes the incident field to be a plane wave, the RCS of the target can be analytically evaluated. For the verification of the proposed method, it is applied to an infinite conducting circular cylinder, which has an analytic solution for the infinite line source as a near-field source. Using the proposed theory the scattered field of such a target under plane wave incidence is evaluated, which is exactly identical to the same analytical scattered field under plane wave illumination.

A design and simulation tool for ground source heat pump system using energy piles with large diameter

Calculation of the underground temperature resulting from heat injection/extraction into/from ground heat exchangers (GHEXs) with hourly variation is one of the most noteworthy challenges to address when simulating and designing a ground source heat pump (GSHP). In order to overcome this challenge, the authors introduce a method to calculate the underground temperature, by considering heat injection/extraction into/from GHEXs with hourly variation. The method applies the superposition of the infinite cylindrical source (ICS) solution and the infinite line source (ILS) solution to calculate the temperature change due to heat injection/extraction into/from the considered GHEX and other neighboring GHEXs, respectively. The calculation method also considers heat injection/extraction from GHEXs with different heat injection/extraction rates and is able to accommodate GHEXs with large diameters such as energy piles. The calculation method was evaluated by applying it to calculate the temperature variation of the heat carrier fluid in a GSHP system with energy piles.

Improving thermal response tests with wireline temperature logs to evaluate ground thermal conductivity profiles and groundwater fluxes

A field method was developed to assess subsurface thermal conductivity profiles and groundwater fluxes from manual temperature logs using a wired probe lowered into a U-pipe during the recovery period of a thermal response test (TRT). Temperature and depth were recorded with a wired temperature and pressure data logger, which triggers a water level rise into a U-pipe. Depth correction methods were introduced and validated using subsurface temperature at equilibrium state measured into U-pipe. Wired temperature logs from recovery period after drilling operation were used to evaluate undisturbed subsurface temperature and during a conventional TRT to assess a thermal conductivity profile with approximately 1 m vertical spatial resolution. TRT analysis was improved by combining the infinite line source equation with the temporal superposition principle and slope method. The results reveal zones of higher apparent thermal conductivity identified as fractured zones in which Darcy’s flux has been quantified using the Peclet number analysis. The average subsurface thermal conductivity inferred with this method was 1.79 W m−1 K−1, similar to 1.75 W m−1 K−1 obtained using conventional TRT analysis. The estimated Darcy’s flux in the fracture zones is 3 × 10−9 to 1 × 10−8 m s−1. This method, based on wired temperature profiling along the borehole, provides a new approach using simple equipment and available analytical solutions to obtain more information from conventional TRT analysis.

Prediction of ground thermal diffusivity from thermal response tests

Ground-coupled heat pump (GCHP) systems are being increasingly utilized in recent years. There are several important parameters in the design of GCHP systems. As ground thermal conductivity is one of the most crucial parameters, it should be measured by conducting in-situ thermal response tests (TRTs). This paper presents the experimental results and analysis of thermal response tests to estimate the ground thermal conductivity and ground thermal diffusivity for three types of ground heat exchangers (GHEs), including U, 2U, and W-type ground heat exchangers (GHEs). Three different types of GHEs were installed in a dredged soil deposit, and continuous TRTs were conducted for 48 h. This study suggests a method to predict ground thermal diffusivity using the infinite line source model from the TRTs. Furthermore, soil samples were collected from different ground layers, and ground thermal properties were measured using laboratory tests. The equivalent ground thermal conductivity and ground thermal diffusivity measured through laboratory tests were compared with the in-situ TRT results that were newly derived in this paper. In addition, ground thermal diffusivity and ground thermal conductivity values that were obtained through research were used as input parameters in numerical analysis. The in-situ TRT results were numerically modeled using the finite element method.

A laboratory apparatus to study Thermal Response Test in the presence of groundwater flow

The standard approach to Thermal Response Test, based on conduction heat transfer in the ground, turns out to be unsuccessful under significant groundwater flow. The applicability of the Moving Infinite Line Source model to interpret the TRT in this case still needs to be proved. In order to study the TRT in the presence of a groundwater flow, an original laboratory apparatus has been developed. The Sand Box design is based on a heat transfer similitude between the real scale TRT problem and the laboratory scale one. The Sand Box sizes (1,2 m x 0.6 m x 1.0 m) are then set in order to keep the boundaries unaffected by the heat source during the TRT. The U-pipe heat exchanger is reproduced through a twocables electrical resistance 1 m long. A hydraulic loop with a peristaltic pump allows to obtain a Darcy velocity across the sandy soil up to 6,7510-5 m/s. The measurement system consists in several thermocouples in the porous medium and in a flow meter. The TRT results at null groundwater velocity allow to derive a reference thermal conductivity. The first tests with groundwater flow show the suitability of the apparatus and allow to derive some preliminary considerations.

Higher-order temporal scheme for ground heat exchanger analytical models

Several models have been proposed in the literature to simulate and design earth heat exchangers, mostly used in the context of ground-source heat pump systems. Some models were based on numerical approaches and other on analytical solutions. The analytic models are mostly based on a convolution of response factors, with a stepwise function assuming a constant heat load during each time step. In this work, Duhamel’s theorem is used to derive the analytical expression of the infinite line source, the infinite cylindrical source and the finite line source, and to obtain analytical expressions integrating high-order temporal schemes allowing the load to vary linearly between successive time steps. It is shown that when the load varies with time, and long-time steps are required, a high-order temporal scheme can be advantageous. The proposed method can be applied to the models where the temporal superposition of pulses was used previously.

Time-Domain Green’s Function of a 2-D Rectangular Room

Time-domain Green’s function of a 2-D rectangular room is presented using the characteristic Green’s function (CGF) and Cagniard-de Hoop methods. The electric fields are expressed as the convolution of the exciting current with this Green’s function. The frequency-domain Green’s function is first developed in an integral form by the CGF method. The resulting spectral-domain integral is converted to a Fourier-like integral by the Cagniard-de Hoop method, and time-domain Green’s function is obtained. In contrast to the conventional frequency synthesis approach, the Cagniard-de Hoop method is proved to be computationally more efficient and numerically more stable. In addition to the accurate time-domain result, an approximate solution is presented, which is much faster with an acceptable accuracy for propagation problems. Numerical results are obtained to provide modeling of a Gaussian pulse excitation of an infinite line source in a rectangular room.

Distributed Thermal Response Tests Using a Heating Cable and Fiber Optic Temperature Sensing

Thermal response tests are used to assess the subsurface thermal conductivity to design ground-coupled heat pump systems. Conventional tests are cumbersome and require a source of high power to heat water circulating in a pilot ground heat exchanger. An alternative test method using heating cable was verified in the field as an option to conduct this heat injection experiment with a low power source and a compact equipment. Two thermal response tests using heating cable sections and a continuous heating cable were performed in two experimental heat exchangers on different sites in Canada and France. The temperature evolution during the tests was monitored using submersible sensors and fiber optic distributed temperature sensing. Free convection that can occur in the pipe of the heat exchanger was evaluated using the Rayleigh number stability criterion. The finite and infinite line source equations were used to reproduce temperature variations along the heating cable sections and continuous heating cable, respectively. The thermal conductivity profile of each site was inferred and the uncertainly of the test was evaluated. A mean thermal conductivity 15% higher than that revealed with the conventional test was estimated with heating cable sections. The thermal conductivity evaluated using the continuous heating cable corresponds to the value estimated during the conventional test. The average uncertainly associated with the heating cable section test was 15.18%, while an uncertainty of 2.14% was estimated for the test with the continuous heating cable. According to the Rayleigh number stability criterion, significant free convection can occur during the heat injection period when heating cable sections are used. The continuous heating cable with a low power source is a promising method to perform thermal response tests and further tests could be carried out in deep boreholes to verify its applicability.

Exact solution for the acoustical impulse response of a line source above an absorbing plane.

The solution of the two-dimensional (2D) wave equation excited by a Dirac delta function above an infinite plane is the 2D impulse response of the half-space. Equivalently, it can be considered as the sound field generated by an infinite line source in three-dimensional (3D) space. For a point source in 3D, it is possible to express the resulting impulse function in closed form if the infinite plane is characterized by a masslike or pure absorbing behavior. Based on the 3D impulse response above an absorbing plane, a closed form solution for a line source over absorbing ground is found. This solution consists of the direct pressure contribution plus a reflected signal. The analytical expression for the reflection factor R is the main result of the work. R is a simple algebraic function of impedance, time, and spatial coordinates. The acoustical field generated by a line source with a general time dependence can be constructed by convolution. For a Gaussian signal, the resulting pressure is computed numerically. For a rectangular pulse, an analytical solution in 3D and 2D is derived. The deviations from the numerically obtained results are very small.

New perspectives in thermal performance test: Cost-effective apparatus and extended data analysis

Two different experimental methods, namely thermal response test (TRT) and thermal performance test (TPT), have been used in the field of ground-source heat pumps (GSHPs) for different purposes. TRT is a well-established method for estimating the design parameters of ground heat exchangers (GHEs), whereas TPT has recently been adopted to examine the thermal performance of newly developed GHEs. Although both methods provide important information, they are rarely performed together because they require different experimental apparatus. To overcome this limitation, we developed a cost-effective TPT apparatus by adding a general proportional-integral-derivative (PID) controller and a solid-state relay to an existing TRT apparatus without a hot water tank. The apparatus showed sufficiently fast and accurate controllability: when the setpoint was 25 °C, the rise time was ∼7 min from the initial temperature of 16.9 °C, and the steady-state error was within ∼±0.1 °C. Two TPTs were conducted using the developed apparatus and a 50 m-long borehole heat exchanger with two different setpoints of 30 °C and 40 °C. We applied a Bayesian inference technique using the infinite line source model as a forward model to extend the TPT data to the estimation of the GSHP design parameters. Thus, the information usually obtained from independent TPT and TRT can be obtained using a single TPT. Moreover, a new index, namely, the unit heat exchange rate, was defined to facilitate a comparison among TPT results obtained under different TPT setpoints, GHE configurations, and ground conditions.

Estimation of soil and grout thermal properties for ground-coupled heat pump systems: Development and application

Ground thermal properties, including the thermal conductivity and diffusivity of both soil and grout, are significant considerations for the design of a ground-coupled heat pump (GCHP) system. However, as a result of the limitations inherent in available response models, few in-situ thermal response tests (TRTs) can identify grout thermal conductivity and diffusivity. This paper proposes a new method to estimate the thermal conductivity and diffusivity of both soil and grout simultaneously using the recently developed infinite composite-medium line source (ICMLS) model. Firstly, a linear dependence analysis is performed on the aforementioned four parameters to ensure the feasibility of the proposed method, leading to an estimation of the minimum TRT duration. Secondly, uncertainty analysis is carried out to analyze the influence of U-pipe shank spacing, as it is considered a sensitive parameter in modeling the heat transfer of ground heat exchangers (GHEs). Thirdly, a genetic algorithm is used to identify these four parameters using the data collected from a TRT. The proposed method is verified using a well-designed sandbox experiment. Finally, its application is demonstrated and evaluated by applying it to the design of a GCHP system for an office building at Hunan University.

Bayesian inference of structural error in inverse models of thermal response tests

For the design of ground-source heat pumps (GSHPs), two design parameters, namely the ground thermal conductivity and borehole thermal resistance are estimated by interpreting thermal response test (TRT) data using a physical model. In most cases, the parameters are fitted to the measured data assuming that the chosen model can fully reproduce the actual physical response. However, two significant sources of error make the estimation uncertain: random error from experiments and structural bias error that describes the discrepancy between the model and actual physical phenomena. Generally, these two error sources are not evaluated separately. As a result, the suitability of selected models to correctly infer parameters from TRTs are not well understood. In this study, the Bayesian calibration framework proposed by Kennedy and O’Hagan is employed to estimate the GSHP design parameters and quantify the random and structural errors in the inference. The calibration framework enables us to examine structural errors in the commonly used infinite line source model arising due to the conditions in which the TRT takes place. Two in situ TRT datasets were used: TRT1, influenced by contextual disturbances from the outdoor environment, and TRT2, influenced by a strong groundwater flow caused by heavy rainfall. We show that the Bayesian calibration framework is able to quantify the structural errors in the TRT interpretation and therefore can yield more accurate estimates of design parameters with full quantification of uncertainties.

On the applicability of the moving line source theory to thermal response test under groundwater flow: considerations from real case studies

The classical methodology to perform and analyze thermal response test (TRT) is unsuccessful when advection contributes to heat transfer in the ground, due to the presence of a groundwater flow. In this study, the applicability, the advantages, and the limitations of the moving line source model to interpret TRT data are discussed. Two real TRT case studies from the Italian Alpine area are reported and analyzed, with both the standard infinite line source approach and the moving line source one. It is shown that the inverse heat transfer problem is ill-posed, leading to multiple solutions. However, besides minimization of the error between measurements and modeling, physical considerations help to discriminate among solutions the most plausible ones. In this regard, the MLS approach proves to be effective in the advection-dominated case. The original time criterion proposed here to disregard initial data from the fitting, based on a resistance–capacitance model of the borehole embedded in a groundwater flow, is validated in terms of convergence of the solution. In turn, in the case when advection and conduction are competitive, the MLS approach results more sensitive to ground thermal conductivity than to Darcy velocity. Although in this case a limited impact of the uncertainty in the groundwater velocity on the boreholes sizing is expected, future studies should focus on the development of a successful TRT methodology for this condition.

Simulation of Circular Cylindrical Metasurfaces Using GSTC-MoM

Modeling of circular cylindrical metasurfaces using the method of moments (MoM) based on generalized sheet transition conditions (GSTCs) is presented. GSTCs are used to link the integral equations for fields on the inner and outer contour of the circular cylindrical metasurface. The GSTC-MoM is validated by two examples: 1) anisotropic and gyrotropic metasurface capable of two field transformations; and 2) nongyrotropic metasurface capable of transforming the field generated by an infinite electric line source at origin to the field generated by a displaced electric line source. The formulations presented here can be used as a platform for deriving GSTC-MoM for 3-D spherical and conformal metasurfaces.