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(a) Geometry, temperatures, heat fluxes and thermal properties of the borehole. (b) The thermal network for the radial heat flow process for a borehole in the Laplace domain.
Source publication
Knowledge of borehole exit fluid temperature is required to optimize the design and performance of ground source heat pump systems. The borehole exit fluid temperature depends upon the prescribed heat injection and extraction rates. This paper presents a method to determine the fluid temperature of a single or a multiple borehole heat exchanger for...
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Context 1
... solution models the two legs of the U-tube as a single equivalent-diameter pipe and uses a single average value to represent the fluid temperatures entering and exiting the U-tube. The resulting radial heat transfer problem is shown in Figure 1. The heat flux q 0 is injected into the circulating fluid with temperature T f (t). ...
Similar publications
Knowledge of borehole exit fluid temperature is required to optimize the design and performance of ground source heat pump systems. The borehole exit fluid temperature depends upon the prescribed heat injection and extraction rates. This paper presents a method to determine the fluid temperature of a single or a multiple borehole heat exchanger for...
Citations
... While the point source solution cannot be applied directly for geothermal calculations, it is the fundamental building block to build other relevant solutions for heat transfer in solids. In particular, this allows us to extend the "non-history dependent" method to responses to a finite line source which is the critical solution for practical applications in borehole field simulations [3], [14], [15]. ...
... Again, effectively, computing Equation (31) is performing the two integrals in z, z ′ from Equation (15). Finally, we have ...
Simulations of the operation of fields of borehole heat exchangers involve a wide spectrum of time scales, and hourly simulations for decades are required for the evaluation of the heat transfer in the subsurface due to these systems. Most current models rely on time and space superposition of fundamental analytical solutions of the heat equation to build the solution for complex borehole fields configurations and loading conditions. These procedures are robust and accurate but do not have favorable scaling properties, and the problems can become quickly computationally intractable as the size increases. In this context, acceleration algorithms for temporal superposition are key to overcome this limitation. This paper presents developments on the so-called "non-history dependent" acceleration scheme and its application to the point and line source solutions, which are commonly used as building blocks in borehole field simulations. The results obtained show promising properties as the computational complexity of the proposed algorithm is linear in the number of time steps, and near double precision accuracy can be achieved by refining the discretizations used to compute the integrals arising from the scheme.
... Type of double U-tube BHE, defined in Table 1 2U102 1. 6 Type of double U-tube BHE, defined in Table 1 2U85 1.0 Type of double U-tube BHE, defined in Table 1 2U85 1. 6 Type of double U-tube BHE, defined in Table 1 Dimensionless × matrix, defined in Eq. (28) Dimensionless × matrix, defined in Eq. (40) Dimensionless heat load per unit length, =̇̇0 ⁄ ̅ Dimensionless heat load per unit length averaged along a BHE Distance between BHEs (m) ...
... Type of double U-tube BHE, defined in Table 1 2U102 1. 6 Type of double U-tube BHE, defined in Table 1 2U85 1.0 Type of double U-tube BHE, defined in Table 1 2U85 1. 6 Type of double U-tube BHE, defined in Table 1 Dimensionless × matrix, defined in Eq. (28) Dimensionless × matrix, defined in Eq. (40) Dimensionless heat load per unit length, =̇̇0 ⁄ ̅ Dimensionless heat load per unit length averaged along a BHE Distance between BHEs (m) ...
... Dimensionless vector, defined in Eqs. (29) and (41) Heaviside unit-step function U94 1.0 Type of single U-tube BHE, defined in Table 1 U94 1. 6 Type of single U-tube BHE, defined in Table 1 U54 1.0 Type of single U-tube BHE, defined in Table 1 U54 1. 6 Type of single U-tube BHEs (see Table 1) ...
The design and the simulation of a borehole-heat-exchanger (BHE) field is usually performed by simplified methods that yield either an overestimation or an underestimation of the thermal response. The methods employing the assumption of a uniform heat rate per unit BHE length overestimate the thermal response, while those employing the assumption of a uniform temperature of the external surface of the BHEs underestimate it. An accurate semi-analytical method to determine the g-function of a bore field with the real condition of BHEs fed in parallel with equal inlet temperature was developed by Cimmino (Int J Heat Mass Tran 91, 2015). An alternative semi-analytical method that yields the same results is presented in this paper. The method is implemented in a C++ program, available at the open-source online data repository of the University of Bologna. Thanks to several optimizations, the program yields a very accurate thermal response of bore fields of any shape with an extremely short computation time. The program is employed to analyze the inaccuracies caused by the assumptions of uniform heat rate and uniform surface temperature of the BHEs. It is also used to illustrate the low performance of the central BHEs in large and compact bore fields, and to show how the bore field can be optimized for a given plot of land and a fixed total length of the field.
... Several methods have been presented in the literature for the design of geothermal boreholes (BHEs), methods which are currently used in the main commercial calculation software adopted in the sector [9]. All are based on different basic solutions for the thermal response of the ground to the presence of the borehole field, known as Temperature Response Factors (TRF) or g-functions [10][11][12][13][14][15][16][17]. Available software (e.g. ...
... A similar model is applied by Agarwal et al. [35] to approach and solve problems in applications typical of the petroleum industry. Since then, various models and TRF have been developed to account for the geometrical complexity of real BHE fields [10][11][12][13][14]. The 1D models by Javed and Claesson [36], Lamarche [37] and Beier and Smith [38], following an approach similar to the ILS and ICS models, replace the multiple pipes in the borehole with a single equivalent pipe account for the thermal storage of the circulating fluid. ...
... However, for BHEs with different depths, Fossa and Rolando [59] proposed a correction to the evaluation of the Fourier number, which is used in the calculation of the E 1 function for Eq. (13). ...
... Taking into account the 40 2 = 1600 borehole g-functions adds up to a total number of 3776 g-functions considered in the model. The g-functions for the boreholes are calculated with the finite line source as presented by Claesson and Javed (2011). While the undisturbed ground temperature at the BHEs is assumed to be constant as T 0,b = 11 • C, the undisturbed ground temperatures at the horizontal pipes vary throughout the year and are calculated according to Phetteplace et al. (2013): ...
... Lamarche Lamarche (2019) has presented the general solution for the horizontal finite line source in a half space to calculate segment-to-segment temperature responses. Similar to the work in Claesson and Javed (2011);Lamarche (2011), the method of images is used to account for the ground surface boundary condition (Fig. 4). The temperature change in the ground T s at the outer wall of a single segment pipe (the horizontal line source) can be calculated as with where q is the length related heat load, the thermal conductivity of the soil and H is the length of the line source. ...
The heat transfer along horizontal connection pipes in geothermal bore fields can have significant effects and should not be neglected. As practical and design-related applications require simple and efficient models, we investigate suitability of different models for the first time within this context. Three ground and three pipe models of different complexity are studied. All model combinations are coupled with a fixed ground load boundary condition on one side and a borehole heat exchanger (BHE) model on the other side. Models are tested under a variety of realistic conditions to evaluate performance. The investigations show that all investigated pipe models are equally suitable for the application. For the ground models, the horizontal finite line source model and the numerical 2D model produce identical results for homogeneous ground properties. The soil resistance model neglects the temperature accumulation in the ground and thus leads to considerable deviations and should be avoided. Based on the findings, we propose a computationally efficient approach using a novel combination of established simple steady-state models for the BHE and connection pipes. In the selected example scenario, the consideration of a 30 m connection pipe attached to the BHE leads to an increase in the BHE load by 40% for the heating case and a reduction in the BHE load by 5% for the cooling case.
... He calculated the g-functions for different borehole fields by means of a finite difference scheme assuming uniform temperature along the borehole wall. Claesson and Eskilson [8], Zeng et al. [9], Lamarche and Beauchamp [10], and Claesson and Javed [11] proposed analytical models based on the FLS model to evaluate the g-function by imposing uniform heat extraction along the borehole wall. Cimmino [12] proposed an analytical solution based on the Stacked Finite Line Source (SFLS) model to calculate the borehole thermal response assuming uniform temperature along the borehole wallsame boundary condition proposed by Eskilson. ...
... We would like to thank Patrick Meisner for the language check and help with the development of equation (11). This project is supported by the Swedish Energy Agency under grant P43647-3. ...
Changes in the ground surface temperature, as it can occur in built-up areas or due to climate change, affect the temperatures of geothermal boreholes. Analytical models for the thermal simulation of boreholes and considering this factor have been proposed. However, they all impose a uniform heat extraction boundary condition along the borehole walls. This boundary condition overestimates the temperature change in the underground caused by the borehole heat extraction and underestimates it in case of rejection. More accurate results are most often obtained by imposing a uniform temperature boundary condition.
In this paper, we propose a new model to calculate the boreholes wall temperature taking into account both the heat extractions/rejections from all the boreholes in the area and the change in ground surface temperature. The model is tailored for areas with independent ground source heat pumps and imposes a uniform temperature boundary condition along the borehole walls, overcoming the limitation of the existing models.
We apply the new model to a real Swedish neighbourhood and show that existing systems may already be significantly affected by the increased ground surface temperature due to urbanization.
We also compare our new model with an existing similar model and show that while the two models provide similar results for smaller areas, their difference tends to be relevant for bigger areas –including the real Swedish neighbourhood analysed - thus making the application of our model important for neighbourhood- and city-scale studies.
... A similar idea is also found in other references e.g. [57]. An example of a square 3 3 borehole field is shown in Figure 6. ...
In order to progress towards more energy efficient buildings, vertical ground coupled heat pumps are a promising solution. Optimisation of both the design and operation of boreholes heat exchangers is a key factor to reduce energy consumption of such systems. This requires a fast evaluation of the thermal response factor of the ground heat exchanger, particularly if it contains numerous boreholes and operates for multiple years. To overcome this challenge, this article reports a new global model combining the finite line source (FLS) model, the two-dimensional (2D) heat conduction equation, and a newly developed three-points method. The borehole field is sorted in increasing distance categories, each being simulated with varying timesteps. The 2D heat conduction equation is used to determine: 1) when the detailed calculation needs to be performed; and 2) the growth of the timestep. A three-points method avoiding double integration of the temperature profiles is proposed to evaluate the borehole wall temperature. The global model calculation time and accuracy were evaluated. The thermal response factor calculation for a square field of 26 × 26 boreholes for one simulated year took 4 seconds, showing a calculation time reduction factor of around 1 000 000, and relative errors smaller than 2 % compared to the original FLS model with superposition principle. For 20 simulated years, the proposed model took only 1 minute. It is appropriate for various boreholes configurations. Its features such as accuracy, speed and load-independency are essential for its integration into building energy simulation tools.
... Eskilson [7] proposed the analytical finite line source (FLS) to estimate the g-function of a single borehole. The FLS was later superimposed on space to evaluate g-functions of fields of GHEs [8][9][10]. For larger fields, the FLS can lead to important errors in the g-functions due to a mismatch of the boundary conditions at the borehole wall between the FLS (uniform and equal heat extraction rates) and the g-functions (uniform and equal temperatures). ...
... Brussieux and Bernier [22] coupled a finite volume method for the interior of a GHE with the CHS solution for the exterior of a GHE, resulting in the definition of g * -functions for short-term effects. Claesson and Javed [8] coupled an equivalent geometry model for the interior of the GHE [23] and used the FLS solution to study thermal interactions between the GHEs. Li et al. [24] used the analytical infinite composite-medium line source (ICMLS) [25] to model the insides of GHEs and the g-functions to account for thermal interactions among GHEs. ...
... Prieto and Cimmino [36] introduced a coefficient updating scheme to consider the variations in fluid temperatures as step-wise variations at each time step n for every t (n−1) < t ≤ t n (where t n = t (n−1) + ∆t). The initial temperatures in Equations (3f) and (8) are replaced by the temperature at the end of the latest time step as follows: ...
The study of heat transfer in ground heat exchangers (GHEs) considering the fluid advection inside the pipes; the heat transfer between the fluid and the ground through the grout material; and the thermal interaction between GHEs is a challenging task. The present paper presents a new semi-analytical method that takes into account the aforementioned effects to consider both the short- to long-term effects of GHEs. The heat transfer between the fluid and grout was studied by a transient multipole expansion considering time-dependent fluid temperatures and an advection model for the pipes obtained from an energy balance on the heat carrier fluid. Thermal interactions were analyzed using an equivalent borehole method while penalizing the transient multipole expansion to include thermal interaction effects. Validation of the short-term predictions was performed by comparing the proposed model to experimental data found in the literature and to an FEA model. The proposed model was then compared with a FEA model in long-term simulations of a geothermal field comprised of 24 GHEs for multi-annual simulation. The method resulted in substantial reduction in computational time while preserving good accuracy when compared with the FEA model.
... He calculated the g-functions for different borehole fields by means of a finite difference scheme assuming uniform temperature along the borehole wall. Claesson and Eskilson [8], Zeng et al. [9], Lamarche and Beauchamp [10], and Claesson and Javed [11] proposed analytical models based on the FLS model to evaluate the g-function by imposing uniform heat extraction along the borehole wall. Cimmino [12] proposed an analytical solution based on the Stacked Finite Line Source (SFLS) model to calculate the borehole thermal response assuming uniform temperature along the borehole wall -same boundary condition proposed by Eskilson. ...
New model to calculate the boreholes wall temperature in an area with independent ground source heat pumps. This model takes into account the changes in the ground surface temperature, e.g. climate change, built environment and the non-uniform heat extraction profile along the boreholes.
... where [ • ℎ ] is the array of thermal response factors for a borehole positioned on the -th node on a borehole positioned on the -th node, multiplied by a constant. The ℎ factors are evaluated using the FLS model as proposed by Claesson & Javed (2011), with representing the radial distance between the -th and the -th borehole (with = ), the buried depth of the boreholes, and the ground thermal diffusivity. The factor is added to ensure a minimal spacing between the boreholes in the solution. ...
