A Fast Simulation Approach to the Thermal Recovery Characteristics of Deep Borehole Heat Exchanger after Heat Extraction
Round 1
Reviewer 1 Report
This is a highly original idea on the future of deep borehole heat exchanger optimal design with climate change and urban densification on the horizon. It is original because it treats this challenge in scientific (thermodynamics) terms, simply, in ways that are reproducible. The treatment is itself original because it is based on an ingenious thermodynamic model of a deep borehole heat exchanger. This kind of heat exchanger can be very usefull also within urban greenhouses where the urban greenhouse space is enclosed by a double skin envelope. I have not seen this model before. The paper is a clear contribution to the field. In sum, this paper is an ideal example of how the topic of sustainability must be approached in future studies regarding heat exchangers in general. Just I propose to the authors in the introduction to cite that an interisting application of this heat exchanger is also the implementation on urban greenhouses. On this note, it may help the readers if this additional reference is included in this paper:
Constructal Macroscale Thermodynamic Model of Spherical Urban Greenhouse Form with Double Thermal Envelope within Heat Currents, Sustainability 2019, 11(14), 3897; https://doi.org/10.3390/su11143897.
Author Response
Thanks for your valuable comment. I have included an interesting implementation of borehole heat exchanger on urban greenhouses in the introduction part of the paper. As for the deep borehole heat exchanger (DBHE), it is still a new concept, perhaps many applications of DBHE would be utilized on urban greenhouses in the future.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
mesh independency test required.
Why are the relevant parameters selected? parameters studies will be useful.
205d - 205 days should be used.
Author Response
Point 1:. mesh independency test required.
Response 1:
Mesh independence test
Temporal and spatial step sizes selection
The temporal and spatial size determines the accuracy of results and computational efficiency. The impact of step size on the computational stability was analyzed in this part. In our simulation, the extended finite line source model is employed for fast spatial and temporal superposition of the heat impulses to evaluate their corresponding thermal response contributions under the intermittent condition. Considering that the thermal response function in finite line source model is characterized for the cross scale merit with respect to simulation time, therefore, large time step could be safely chosen. The temporal step size was set as 0.5h, 1h, 3h,12h. With different time steps, the simulated results of borehole bottom temperature during the intermittent period for 205 days were compared in Figure 1, which shows that during the first 40 days unsteady stage, there exist larger deviations among the borehole temperatures of different time steps compared with the detailed solution and then the deviations gradually decrease and stabilize during the steady stage from 40 days to 205 days. With the temporal step size less than or equal to 3h, the simulated temperatures have the maximum relative error of 1.92%. Therefore, 3h is selected as the time step for calculation.
The effect of spatial step in z direction Δz was also analyzed. The step was set from 0.1 to 5m (the time step is 3h) for simulation of borehole bottom temperature at the 40th day of intermittence. The comparison results were shown in Figure 2. The simulated borehole bottom temperature keeps constant, when theΔz is less than or equal to 1m. Accordingly, 1m is considered as the step size in the simulation. In this scenario, the number of nodes is 2600.
Figure 1. Comparison of borehole bottom temperatures of deep borehole heat exchanger for 205 days intermittence with different time steps.
Figure 2. Comparison of borehole bottom temperatures of deep borehole heat exchanger for 205 days intermittence with different step size in z direction along depth.
Point 2: Why are the relevant parameters selected? parameters studies will be useful.
Response 2:
As a matter of fact, it has been pointed in part 4 of the paper that the deep borehole heat exchanger (DBHE) in study comes from a pilot demonstration project in Qingdao, located in Shandong Province, China. All the relevant parameters come from in-situ field tests for the project. In view of the paragraph length limit, this paper focus on the simulation results for its thermal recovery after heat extraction, and details of the in-situ field test as well as parameter analysis will be introduced in our future work.
Point 3: 205d - 205 days should be used.
Response 3:
Thanks for your valuable comment. After a careful check, I have replaced all the 205d for 205 days in the paper.
Author Response File: Author Response.pdf
Reviewer 3 Report
The motivation for this research is well justified and nicely presented. My main problem with this paper is related to the style of presentation of mathematical concepts. After integrating a Green function (1) with respect to time \tau’ and space position z’ , the Authors obtained the overall temperature response \theta (x,y,z,\tau). What about x’ and y’?
We do not need either an appendix to introduce the Green function used in the paper; it is sufficient to say that for a thermal conduction problem in Cartesian geometry the Green function has a following form (and present the appropriate equation). I’m not convinced that the reference provided is the best reference to describe the Green function (perhaps Methods of Theoretical Physics, pts. I and II. By Philip M. Morse and Herman Feshbach. McGraw-Hill, New York-London, 1953. xxx + 1978 pp. is more appropriate). When introducing the subject, it is useful to mention what is the form of the Green function in other coordinate systems including a cylindrical system which is quite relevant for this study.
Please rewrite the mathematical part. I would also expect that the formal derivation of eq. (10) is provided; it is not sufficient to say: “Therefore, flow temperature varies along the depth....”. By the way, it is not clear what is the “flow temperature”?
The current version is not precise, and in some parts, it is not correct.
Author Response
The motivation for this research is well justified and nicely presented. My main problem with this paper is related to the style of presentation of mathematical concepts.
Point 1: After integrating a Green function (1) with respect to time \tau’ and space position z’ , the Authors obtained the overall temperature response \theta (x,y,z,\tau). What about x’ and y’?
Response 1:
Thanks for your valuable comment. As a matter of fact, according to the finite line source model depicted in Figure 4 of the paragraph as shown below (Figure 3), the deep borehole heat exchanger is treated as a finite line heat source, it is one dimensional in z direction along depth. Therefore, the overall temperature response in any point of rock zone is obtained by integration of the Green function with respect to time \tau’ and space position z’ along depth of the finite line.
Figure 3. Finite line source model.
Point 2: We do not need either an appendix to introduce the Green function used in the paper; it is sufficient to say that for a thermal conduction problem in Cartesian geometry the Green function has a following form (and present the appropriate equation). I’m not convinced that the reference provided is the best reference to describe the Green function (perhaps Methods of Theoretical Physics, pts. I and II. By Philip M. Morse and Herman Feshbach. McGraw-Hill, New York-London, 1953. xxx + 1978 pp. is more appropriate).
Response 2:
Thanks for your valuable comment. Actually, Appendix A in the paper focuses on the derivation of finite line source model based on Green function. This derivation is necessary to enhance the classical finite line source model which is adaptable only for shallow borehole heat exchanger without consideration of multi-geological layers effect as well as heat flux density distribution along depth in the scenario of deep borehole heat exchanger. Additionally, the author was acknowledged for the reviewer’s suggestion for the classical reference about Green function, and I have already included the recommended reference of Methods of Theoretical Physics, pts. I and II in the revised manuscript now.
Point 3: When introducing the subject, it is useful to mention what is the form of the Green function in other coordinate systems including a cylindrical system which is quite relevant for this study.
Response 3:
Thanks for your valuable comment. As pointed by the reviewer, if we study heat transfer of a single deep borehole with coaxial tubes, it is useful to mention what is the form of the Green function in other coordinate systems including a cylindrical system. However, for future study of heat transfer of multiple DBHEs, thermal interference among the deep boreholes could not be negligible, therefore, cylindrical system would not be so convenient because the circumferential homogeneous effect (which is the case for a single DBHE) could never be ignored. Due to this consideration, the author did not present the form of the Green function in a cylindrical system. Of course, the Green function in a cylindrical system could be easily obtained with a minor modification of Equation.(1) in the paper.
Replace , here, and are the position of the rock zone and the line source in the radial direction.
We have the Green function in a cylindrical system as:
Accordingly, one can obtain the overall temperature response located at point and at time by integrating Green function with respect to time and space position as:
Point 4: Please rewrite the mathematical part. I would also expect that the formal derivation of eq. (10) is provided; it is not sufficient to say: “Therefore, flow temperature varies along the depth....”. By the way, it is not clear what is the “flow temperature”? The current version is not precise, and in some parts, it is not correct.
Response 4:
Quasi-steady state modeling of DBHE inside borehole
In the quasi-steady state model, the fluid temperature and borehole wall vary in the axial direction, and the inlet and outlet temperature of the circulating fluid changes with time due to the heat flux distribution along the borehole wall determined by the temperature difference and thermal resistance. The model is able to evaluate the influence of short-circulating among the branch pipes (Diao et al. 2006; Yang et al. 2010).
During thermal recovery of DBHE, return water from the heat pump units flows into the inner pipe (circular domain) and circulates out from the outer pipe (annular domain). On the basis of the energy equilibrium equations for upward and downward flow of the circulating fluid in DBHE shown in Fig.4 are formulated to model the dynamic heat transfer process. According to the quasi-steady state heat transfer analysis inside the borehole, it is assumed that convection overweighs conduction for the circulating fluid in pipes along the axial direction and the temperature and velocity distribution of the fluid in pipes at any cross section are uniform, turbulence influence on heat transfer could be incorporated into the convection coefficient by lumped parameter method. Therefore, the temperature of the fluid flowing upward in the outer tube (named as pipe 1) and downward in the inner tube (named as pipe 2) varies along the depth as:
Where, and are the local heat flow between the flow pipes and the borehole wall respectively.
Figure 4. Quasi-steady state heat transfer analysis based on energy equilibrium inside the borehole.
TRCM model for thermal resistances within the borehole
Thermal resistances within the borehole can be implemented into the quasi-steady state heat transfer model by applying the methods in analogy to the electric networks. A geometrical simplification is made such that the different parts of the borehole are represented by single nodes (as depicted in Fig.5).
Figure 5. Cross section of annular pipe borehole and the corresponding thermal circuit
Numerical models based on this methodology have earlier been referred to as thermal resistance and capacity model (TRCM), TRCM model for coaxial BHEs have been published by Refs.(Bauer et al. 2011; Michele et al. 2010; Johan et al. 2011; Beier et al. 2011,2014). A thermal circuit would be used to describe the local heat flow between the flow pipes and the borehole wall and heat flow as functions of the temperature difference, and derive the corresponding thermal resistances of outer pipe to the borehole wall (also ) and inner pipe to the outer pipe :
Consistently, the local heat flow between the flow pipes and the borehole wall in the quasi steady state model of DBHE inside borehole could be formulated as:
Therefore, we can obtain the quasi-steady state heat transfer equation for fluid flow inside the borehole:
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
All my major concerns have been taken into account. The text reads quite well and the paper will provide a useful reference for future research. I would like to congratulate the authors for carrying out this interesting study.
Author Response
Thank you for your valuable comment and effort for my paper revision
