A transient quasi-3D entire time scale line source model for the fluid and ground temperature prediction of vertical ground heat exchangers (GHEs)

doi:10.1016/j.apenergy.2016.02.099

Applied Energy

Volume 170, 15 May 2016, Pages 65-75

https://doi.org/10.1016/j.apenergy.2016.02.099 Get rights and content

Highlights

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The proposed model can accurately predict fluid and ground temperature.
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The relative error of the fluid temperature prediction is less than 5%.
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The maximum relative error of the ground temperature prediction is 3.85%.
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The heat flux profile is important for borehole distance determination.

Abstract

The energy performance of Ground-Coupled Heat Pump systems (GCHPs) depends heavily on the fluid temperature over the entire time scale inside the ground heat exchangers (GHEs) and the ground temperature profile outside the borehole. These temperatures define not only the coefficient of performance (COP) of GCHPs but also the ground receptivity of the energy demand of buildings. Although GCHPs have been studied for many years, it is still a great challenge to develop a model that can accurately predict both short-term and long-term responses due to the complex borehole configuration and the thermal capacity of grout. A new transient quasi-3D entire time scale line source model is proposed in this paper, which introduces the concept of transient borehole thermal resistance and considers the heat flux profile along the U-pipe as a variable. The proposed model is firstly compared with several existent models, including several traditional line source models and a full scale response model, using the data collected from a reported Sandbox experiment. The comparison study shows that the proposed model is able to predict the temperature with a relative error less than 5%. Then, the outside ground temperature profile that defines the borehole distance is analyzed and compared with the Sandbox experiment result, which shows that the proposed model leads to a maximum relative error being less than 3.85%. Finally, the impact of the heat flux profile along the U-pipe on the ground temperature profile prediction is investigated, which shows that when the heat flux profile along the U-pipe is considered as a variable, the determination of borehole distance will be much more accurate. Therefore, the transient quasi-3D entire time scale line source model is an effective method for the fluid and ground temperature prediction and may offer the theoretical basis for the system control and the borehole distance determination.

Introduction

Ground-Couple Heat Pump systems (GCHPs) have attracted global attention for their high energy efficiency and low greenhouse gas emissions. According to the report of the 2010 World Geothermal Congress (WGC 2010) [1], the installed capacity of GCHP units grew 2.15 times compared with 2005. The number of countries that began to implement GCHPs increased from 33 in 2005 to 43 in 2010. In China, the installed GCHP capacity increased from 383 MWt in 2004 to 5210 MWt in 2009. By the end of 2012, the number of GCHP units installed in China have been reported to be over 23,000 [2].

Generally, the performance of GCHP systems depends on the outlet temperature of the ground heat exchangers (GHEs). During the last two decades, many studies were proposed to predict the outlet temperature of the GHEs [3], [4], [5], [6]. The most common approach is the so called two-region model, which was well reviewed by Yang [7]. According to these studies, the heat transfer process for the whole region is analyzed by two separated regions. One is the ground region outside the borehole, which can be modeled by a line source model or a cylinder source model [8]. Another is the region inside the borehole, including the grout, the U-pipe and the circulating fluid, which can be modeled by a quasi-three-dimensional model [9] or a vertical temperature profile model [10].

However, these models are not valid for the first several hours of GCHPs’ operation due to the thermal capacity of the grout, U-pipe and fluid. For a more detailed energy analysis, especially for the system control simulation, the thermal performance of GCHP systems and the building load are required in a large time scale from sub-hours to years. Due to complex borehole configurations, it is still a great challenge to develop a transient heat transfer model.

Numerical method, which takes account of all the regions including grout, circulating fluid and U-pipe, was certified to be an efficient way to model the transient heat transfer process by lots of researchers from 1990s [11], [12], [13], [14], [15], [16], [17], [18], [19]. However, the calculation in the numerical method is very complicated and lack of flexibility for various applications. Another way is to develop analytical models [20], in which the pipe wall temperature is used as the reference temperature instead of the borehole wall temperature. To simplify the borehole geometric arrangements, the geometry of actual borehole is converted to be a completely cylindrical composite model by assuming equivalent diameter for the U-shaped pipe as shown in Gu and O’Neal [21], Beier and Smith [22], [23], Javed and Claesson [24], [25] and Lamarche and Beauchamp [26]. The equivalent diameter can be improved with the measured borehole thermal resistance by Beier [27]. These cylindrical composite models were modified by Li and his group [20], [28], [29], [30], [31], in which they distinguished the two legs of the U-shaped pipe and developed a new analytical model, and it was called infinite composite-medium line source (ICMLS) model based on Jaeger’s instantaneous line-source solution for a cylindrical composite medium [32]. This model was validated using Beier’s Sandbox experiment [33]. The relative error for both the inlet temperature and the outlet temperature was less than 10% [28]. To cover the entire simulation period, the ICMLS model was improved by combining the ICMLS model, the finite line source (FLS) model and the infinite line source (ILS) model together, and it is defined as a full scale response model [31].

Although the transient analytical model developed by Li and his team [20], [28], [29], [30], [31] offers relatively accurate results, there are still three issues that are important for enhancing the model accuracy but have not been addressed. Firstly, the heat flux along the U-pipe varies with length, but it is considered as a constant in current studies. Secondly, there is a thermal interaction between two legs of the U-shaped pipe, but current studies do not consider it and the heat flux for the inlet and outlet pipes is assumed to be the same. Thirdly, the current studies can only work effectively under a strict borehole geometric configuration. With the uncertainty in estimating ground and grout thermal conductivity and diffusivity, the absolute error of the inlet and outlet temperature for the Sandbox experiment could reach to 2–3 °C [28]. Poor accuracy is also observed when water is served as grout in Europe countries. Thus, an accurate transient model for both fluid temperature and ground temperature profile is still lacking.

This paper aimed to address the above issues by developing a transient quasi-3D entire time scale line source model. The heat transfer of the inlet pipe and the outlet pipe will be considered separately, and a set of energy balance equations for them will be developed as well. Meanwhile, the outside ground temperature profiles will be calculated by distinguishing the case of constant heat flux and the heat flux profile along the U-pipe. Besides, the proposed model will be calibrated using the measured borehole thermal resistance; and a correction coefficient for the borehole thermal resistance and the transient borehole thermal resistance is introduced and investigated. Finally, the reported Sandbox experiment will be used to evaluate the proposed model.

Section snippets

Transient quasi-3D entire time scale line source model

In principle, the heat transfer around the GHEs is a transient 3-D conduction and convection process. The schematic diagram of the heat transfer process around the GHEs is shown in Fig. 1. Similar to the electrical science, the whole region of the ground around the GHEs can be considered as an electric circuit, and the temperature profile around the GHEs can be written as a product of the heat flux and thermal resistance, as shown in Eq. (1) $T (z, θ, r) - T_{0} = q (z, θ, r) R (z, θ, r)$

Based on Eq. (1), the

Description of the model validation experiment

To demonstrate the accuracy and efficiency of the proposed transient quasi-3D entire time scale line source model, the reported Sandbox experiment is adopted and briefly introduced below. With a controlled environment, the well prepared Sandbox experiment for identifying ground thermal properties was built by Beier et al. [33]. The sandbox consisted of an 18.3 m U-pipe, an aluminum pipe (severed as the borehole wall) and a wooden box full of sand. Following the study of Yang and Li [17], the

Thermal resistances calculation

The proposed transient quasi-3D entire time scale line source model was compared with several conventional analytical models for both ground thermal resistance and borehole thermal resistance. Fig. 3 illustrated the ground thermal resistances calculated using different models. It can be seen that the ground thermal resistances given by the proposed model and the ICMLS model were similar to those given by the conventional line source models during the first few hours as the borehole wall

Conclusion

In this paper, a new transient quasi-3D entire time scale line source model has been proposed for the inside fluid and outside ground temperature profile predictions. Compared to current existent models, an improved transient borehole thermal resistance and the heat flux profile along the U-pipe were taken into account in the proposed model. Case studies show that:

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The proposed model can improve the accuracy of borehole thermal resistance when compared with the transient borehole thermal

Acknowledgements

This work was supported by the Project on the Integration of Industry, Education and Research of Guangdong Province and Ministry of Education (Grant No. 2010B090400301), International S&T Cooperation Program of China (ISTCP) (2015DFA61170) and Interdisciplinary Program of Hunan University in 2014.

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Citation Excerpt :
The boundary conditions and initial conditions are discussed below. Validation with experimental results is applied as a common practice to justify the proposed numerical models of BHEs [46–48]. A few laboratory-scale experimental system have been intentionally designed and built to provide reference data for model verification, and to examine the thermal behavior of model BHE as well.
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A transient quasi-3D entire time scale line source model for the fluid and ground temperature prediction of vertical ground heat exchangers (GHEs)

Highlights

Abstract

Introduction

Section snippets

Transient quasi-3D entire time scale line source model

Description of the model validation experiment

Thermal resistances calculation

Conclusion

Acknowledgements

Geothermics

Energy Build

Appl Energy

Appl Energy

Renew Energy

Geothermics

Energy Build

Appl Energy

Renew Energy

Int J Therm Sci

Geothermics

Energy Convers Manage

Energy

Int J Heat Mass Transf

Appl Therm Eng

Appl Energy

Appl Energy

Appl Energy

Geothermics

Renew Energy

Int J Heat Mass Transf

Geothermics