Closed-loop Geothermal Systems Research Articles

Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise

Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise

Novel hybrid coaxial-U-shaped closed-loop geothermal system for enhanced energy extraction: a numerical analysis

ABSTRACT Insulated U-shaped closed-loop geothermal system (UCLGS) is an efficient and environmentally friendly method of harnessing geothermal energy. However, heat available in the rock surrounding the insulated production pipe has not been extracted yet. Therefore, this study proposed a novel hybrid coaxial-U-shaped CLGS (HCLGS) to enhance the overall thermal performance of CLGS. The HCLGS comprises an insulated UCLGS for harnessing energy from the reservoir rock, alongside a coaxial double pipe system designed to extract additional thermal energy from the rock enveloping the insulated production pipe of the UCLGS. Further, the study explored the thermal feasibility of the proposed HCLGS in various rock formations for geothermal energy extraction at different depths. The results showed that the HCLGS installed in the pyroxenite rock produced primary hot fluid with a maximum temperature of 94°C and secondary hot fluid at a temperature of 38.5°C for the base case, achieving a COP of 1.15. The proposed HCLGS can extract up to 18% higher thermal power than the insulated UCLGS, with a COP ranging from 1.11 to 1.16. Finally, the proposed hybrid system significantly harnessed the additional heat available in the rock encompassing the insulated production pipe of UCLGS.

Study Outlines Challenges of Repurposing Oil and Gas Wells for Geothermal Applications

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 215751, “Challenges for Repurposing Oil and Gas Wells for Geothermal Applications,” by Livio Santos, SPE, and Arash Dahi Taleghani, SPE, The Pennsylvania State University. The paper has not been peer reviewed. _ This work provides insights into the technical and economic viability of repurposing oil and gas wells for geothermal energy production for direct heating. The potential benefits—especially for areas that are highly dependent on fossil fuels for heating—and challenges of this approach are assessed, while technical, market, and policy barriers for future development are identified. Repurposing oil and gas wells for geothermal energy requires a multidisciplinary approach that involves expertise in geology, engineering, and regulatory affairs. Introduction Repurposing abandoned wells for power generation in geothermal energy projects can present challenges because of the low temperatures associated with these wells. The low temperature gradients and thermal energy available in such wells have been found to limit overall power-generation potential. The same low-temperature geothermal resources can still be used effectively, however, for community heating applications. Community heating systems have lower temperature requirements compared with those necessary for power generation and are compatible with available heat from low-temperature geothermal sources. In particular, closed-loop geothermal systems, specifically designed for community heating, offer several advantageous features. The closed-loop design ensures that the fluid remains contained within the system. These systems are environmentally compatible because they do not extract or discharge water from natural sources. Additionally, with better control over the working fluid and heat-exchange process, closed-loop systems deliver optimized system performance, energy efficiency, and cost savings. They also require less maintenance and feature lower operational risks because of the closed circuit. Closed-loop systems offer fluid-selection flexibility and streamlined regulatory compliance. Not all abandoned wells are suitable for repurposing, however. Thorough feasibility assessments and well-integrity evaluations are necessary to ensure that the abandoned well is structurally sound, has the appropriate depth and geological characteristics, and can extract geothermal heat effectively. The condition of the well casing, cementing, and any potential wellbore damage also must be evaluated. An economic analysis is performed to assess the financial viability and cost-effectiveness of the geothermal community heating system. Finally, an environmental impact assessment evaluates the environmental consequences of the geothermal community heating system.

Numerical modelling of flow and heat transport in closed mines. Case study Walsum drainage province in the Ruhr coal-mining area

High geothermal potential and multiple mine-water-based geothermal installations in Germany and other countries improve the relevance of detailed studies and modeling of promising sites. In this context, we developed a numerical model of water flow and heat transport in the Walsum mine drainage province in the west of the Ruhr coalmining area using the available data on geology, mining, water levels, pumping, and the temperatures of deep rocks and mine water. The model was validated by varying the parameters of groundwater recharge and hydraulic conductivity to achieve sufficient consistency with measured inflows and pumping rates from the central pumping facility located in the Walsum 2 shaft. The calculated mine water temperature of 30.3 ºC is close to the average of the measured temperature varied within the range of 29 – 33 ºC during the last years of mine maintenance. Using the numerical model, we evaluated the expected thermal capacity of a hypothetical open-loop circulation system and two closed-loop geothermal systems within the study area. The installation and operation of these systems would enable the generation of a thermal capacity from a few dozen kW to 1 MW sufficient for small-size to mid-size heat consumers with insignificant impact on the high thermal energy potential of the Walsum mine drainage province.

Numerical simulation of the thermal response of seabed sediments to geothermal cycles in Suvilahti, Finland

Renewable thermal energy from seabed sediment is used for heating and cooling houses in Suvilahti, Vaasa, Finland. Innovative coaxial polyethylene pipes (Refla) filled with heat collection fluid are laid horizontally in the sediment layer. This study aims to evaluate the adequacy and possible overuse of this shallow geothermal energy. This entailed a numerical analysis of the heat relations of a “coaxial closed-loop geothermal system (CCGS)" to investigate the thermal behavior of the sediment. The numerical model was developed in Midas GTS NX software with a thermal analysis module. The objective was to understand temperature fluctuation in the sediment, which is influenced not only by geothermal energy exploitation, but also by seasonal weather. The numerical model, due to its design, provided information mainly about changes in sediment temperature due to the geothermal energy exploitation during the different seasons. The results show that in the first third of the total length of the Refla pipes, the sediment environment is significantly affected by energy exploitation's temperature loading. It is advisable to exclude the first third from an analysis of total geothermal energy reserves. The remaining two-thirds of the length shows potential to provide sustainable, long-term geothermal energy (GE) exploitation at the current rate.

Numerical investigation of closed-loop geothermal systems in deep geothermal reservoirs

Closed-loop geothermal systems (CLGSs) rely on circulation of a heat transfer fluid in a closed-loop design without penetrating the reservoir to extract subsurface heat and bring it to the surface. We developed and applied numerical models to study u-shaped and coaxial CLGSs in hot-dry-rock over a more comprehensive parameter space than has been studied before, including water and supercritical CO2 (sCO2) as working fluids. An economic analysis of each realization was performed to evaluate the levelized cost of heat (LCOH) for direct heating application and levelized cost of electricity (LCOE) for electrical power generation. The results of the parameter study, composed of 2.5 million simulations, combined with a plant and economic model comprise the backbone of a publicly accessible web application that can be used to query, analyze, and plot outlet states, thermal and mechanical power output, and LCOH/LCOE, thereby facilitating feasibility studies led by potential developers, geothermal scientists, or the general public (https://gdr.openei.org/submissions/1473). Our results indicate competitive LCOH can be achieved; however, competitive LCOE cannot be achieved without significant reductions in drilling costs. We also present a site-based case study for multi-lateral systems and discuss how our comprehensive single-lateral analyses can be applied to approximate multi-lateral CLGSs. Looking beyond hot-dry-rock, we detail CLGS studies in permeable wet rock, albeit for a more limited parameter space, indicating that reservoir permeability of greater than 250 mD is necessary to significantly improve CLGS power production, and that reservoir temperatures greater than 200 °C, achieved by going to greater depths (∼3–4 km), may significantly enhance power production.

Heat extraction through conductive proppants

The absence of direct contact between the working fluid and the geothermal formation in closed-loop geothermal system (CLGS) limits its application to direct heating. Enhancing the CLGS heat extraction performance could lead to an effective utilization of CLGS for power generation purposes. Optimizing CLGS performance is still an active area of research. In this paper, we propose a fractured closed-loop geothermal system (FCLGS) with a conductive plug installed downhole and a fracture propped with conductive proppants to improve heat exchange performance of CLGS. Using an integrated numerical approach, we first explore the potential to improve the thermal conductivity of common types of proppants by applying metallic coatings. Applying a thick layer of copper coating to Aluminum Nitride proppants yields a thermal conductivity of 44.91 W/(mK), 24 times larger than that of uncoated sand. The proppant/coating with largest thermal conductivity is used for field-scale simulations. Heat extraction performance of FCLGS is evaluated at the field-scale under different parameter combinations. Numerical results show that incorporating a conductive plug into a fractured CLGS improves net power output by 30.60 %. Conductive plug shape and size play an important role in heat extraction in FCGLS. In addition, a circulation velocity of 2 m/s is observed to yield the highest increase in net power output.

Analysis of an enhanced closed-loop geothermal system

Closed-loop geothermal systems are a novel heat extraction technique not requiring continuous fluid supply. Heat extraction efficiency of closed-loop systems is often limited due to the small heat exchange area between the circulating fluid in the wellbore and surrounding rock. An enhanced closed-loop geothermal system (ECLGS) is considered here to improve the heat extraction efficiency of closed-loop systems. The objective of ECLGS is to increase heat transfer from surrounding hot rocks to the wellbore by artificial fractures propped with high-thermal-conductivity proppants. To explore the mechanism controlling heat extraction through ECLGS, a systematic study was conducted. First, a three-dimensional hydrothermal coupled model was developed and validated, and the effects of several parameters were analyzed. It was found that highly-conductive fractures could enhance heat extraction through closed-loop systems considerably. As the flow velocity of the circulating fluid increases, the net power starts to decrease after reaching a peak value. An increase in fluid heat capacity would enhance heat extraction but without requiring more pumping power. The four-wing fracture configuration causes higher heat extraction improvement than the branched fracture. A dimensionless analysis was performed for ECLGS, and three dimensionless numbers integrating different involved parameters were proposed. The proposed dimensionless numbers have been verified numerically. The effect that each dimensionless number has on production temperature has been investigated. Via dimensionless numbers, it becomes feasible to comprehensively assess the impacts of involved parameters, thus making designing closed-loop systems more convenient. The possible ways to apply proposed dimensionless numbers to building a closed-loop system have been illustrated.

Performance analysis of phase change slurries for closed-loop geothermal system

Geothermal energy offers a route to clean power generation with baseload reliability that is unique among renewable sources. Current geothermal energy systems employ water, brine, or CO2 as a working fluid and rely on sensible heating to extract energy. In this work, latent heat based phase change slurries (PCSs) are assessed as geo-fluids for use in closed-loop geothermal systems. The PCS property data and its thermal, physical, chemical, and rheological behaviour at reservoir-relevant high-pressure and high-temperature conditions are studied. The experimental data is further used to model the viability of PCS-based closed-loop geothermal. The thermal output performance of PCSs is assessed and optimized in comparison to water. The resulting strategy achieves a 12.5% enhancement in thermal output compared to the water base case. The additional thermal power gain achieved with the PCS motivates further advancement in geo-fluids for geothermal energy systems.

Model for economic evaluation of closed-loop geothermal systems based on net present value

Closed-loop geothermal systems (CGSs) represent an effective means of extracting low-grade thermal energy from hot dry rock (HDR). Thus far, there has been much debate on whether they can be profitable because no economic evaluation model has been available. This study highlights the profitability of geothermal projects by developing a techno-economic coupling model for economic evaluation of CGSs based on net present value (NPV) for the first time. The proposed model was applied and analyzed in the case of the Gonghe Basin, Qinghai, China. The results indicate the following conclusions. (1) The technical model was extremely accurate, and the results were consistent with experimental results and the results of calculations using other classical models (the relative errors were all less than 9.8 %). (2) In terms of the heat extraction effect, the heat extraction power of U-shaped closed-loop geothermal systems (UCGSs) is ∼ 1.3 times that of coaxial closed-loop geothermal systems (CCGSs). The heat extraction power for direct heating is ∼ 1.1 times that for electricity generation. (3) Increasing the rate of the one-off government subsidy to 30 %, increasing the heat price by 30 %, and reducing the drilling cost by 30 % can make a CGS profitable within a production time of 50 years. This study will serve as a theoretical foundation for policymakers and engineers to develop HDR resources using CGSs.

Numerical simulation study of closed-loop geothermal system well pattern optimisation and production potential

Previous numerical simulation studies on U-shaped closed-loop geothermal systems (UCLGS) have been based on single horizontal well, but the reservoir temperature influence range of single well is limited and cannot effectively utilize deep geothermal resources. There are few studies on the well pattern design of UCLGS, especially concerning the shape, number, and arrangement of well. Therefore, we proposed a grid-shaped horizontal well heat exchange system of multi-layer and multi-branch to achieve optimal heat transfer and fluid circulation effect. The effects of the well layout location, injection-production ratio, number of grids, and well length on the heat extraction performance were investigated by numerical simulations after establishing a three-dimensional geothermal mining model. We explored the temperature response characteristics of the reservoir. Finally, the actual production characteristics and potential of the grid-shaped horizontal well heat extraction system were characterised using production indicators, such as production temperature, heat production rate, accumulated thermal energy, and generated energy. The results showed that the diagonal well layout is more effective for heat extraction. Multiple injection wells and one production well mode is the optimal design scheme under the conditions of this study. In addition, when using the diagonal well layout, the system’s heat extraction capacity is positively correlated with the number of grids. However, when using the horizontal well layout, it is no longer proportional to the number of grids once the injection rate exceeds 3 kg/s. Increasing the well length can improve heat production efficiency, but it is limited by the total thermal energy storage capacity of the reservoir. When a single-layer well layout cannot meet the heat extraction requirements, a multi-layer well layout can be used to enhance the heat extraction capacity and reduce the interference of the well network system with the reservoir. Therefore, this well design promotes the efficient development of deep geothermal resources.

Hybrid geothermal energy and hydrocarbon resources production by repurposing horizontal wells in shale gas reservoirs in Horn River Basin, British Columbia, Canada

The production of geothermal energy from the depleted oil and gas reservoirs could represent one of the practical pathways of the clean energy transition for the hydrocarbon energy industry. In this study, a new strategy for repurposing oil and gas wells into geothermal energy production system is proposed to co-produce hydrocarbon and geothermal resources during the transition. At a certain stage in a hydrocarbon production system of fractured horizontal wellbores, one vertical injection well is added to connect to the end of the horizontal wellbores to form a U-loop system with open fractures in the lateral part. Cold water is injected through the injection well, transported along the U-loop system, and geothermal energy is produced along with natural gas for utilization. Natural gas production continues as long as the pressure of the horizontal well remains lower than the reservoir pressure. An example application case from a typical shale gas production system in the Horn River Basin of British Columbia, Canada has been examined in this study. The results show that reduction in cumulative gas production is limited if the injection well is added at a middle or later stage. The outlet temperature from this system behaves like those in a closed-loop geothermal system and could be stable for several decades of operation. Although the time of conversion is sensitive to gas production, it does not affect the bottom hole temperature much. The water flow rate does not affect the cumulative gas production but considerably affects the formation temperature. The horizontal length of the fractured wellbore will significantly affect the gas production performance and geothermal energy output. Fracture length and permeability have critical impact on gas production but less influence on geothermal energy recovery.

Impact of microannulus on the efficiency of heat transfer in the bottomhole

To ensure feasible power generation from closed-loop geothermal wells, deeper wells are required to reach higher temperature zones. However, weak bonding between cement and casing or cement and formation may allow formation of a small gap (known as microannulus), which could have a negative effect on the heat extraction rate and consequently compromises the entire investment. Previous projects have reported that the output temperatures were significantly lower than the expected values, and the cause is believed to be cement debonding. This study aims to develop a reliable simulation model to demonstrate the impact of microannulus in closed-loop geothermal systems. Multi-physics finite element analysis is used to construct models with and without microannulus. The microannulus is modeled based on real cement evaluation logs, with gaps varying between a few micrometers to few millimeters. In extreme cases, the presence of microannulus is found to decrease the geothermal power by more than 35%. Furthermore, the possibility of heat loss containment is investigated by a sensitivity study of wellbore parameters. These sensitivity analyses demonstrate that cement and geothermal fluids with higher thermal conductivity can improve but cannot compensate the presence of microannuli. The results also highlight the importance of proper cementing design to ensure wellbore integrity and avoid geothermal power loss.

Comparative thermo-economic analysis of co-axial closed-loop geothermal systems using CO2 and water as working fluids

Co-axial closed-loop geothermal heating systems coupling abandoned oil/gas wells face more grievous corrosion issues than the shallow ground source heat pumps due to the higher temperature and pressure conditions. Instead of water, using CO2 as a working fluid would extract more power and eliminate the corrosion issue; however, these advantages might be offset by the energy and capital cost of high operating pressure. To better guide the working fluid selection for the co-axial closed-loop geothermal heating systems, the effects of using CO2 as a working fluid on the system thermo-economic performance as well as the effects of geothermal gradients and tube diameters are evaluated. We first established a 2D thermodynamic model to simulate the heat extraction from a vertical co-axial closed loop geothermal well and the heat conduction in host rocks and then applied the key parameters into an economic model for the calculation of thermo-economic indicators. It is found that the wellbore set with smaller tube diameters can profit the thermo-economic performance of the co-axial closed-loop geothermal heating system by substantially lowering the total annual cost, promoting the energy efficiency, and slowing down the growth of cost and the descent of energy efficiency within the lifetime of the system. Besides, CO2 will be a better working fluid for small-scale co-axial closed-loop geothermal heating systems leading to lower cost per net power compared to water, and it also can slow down the descent of energy efficiency year by year.

Expertise in Complex-Well Construction Leveraged for Geothermal Wells

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 204097, “Constructing Deep Closed-Loop Geothermal Wells for Globally Scalable Energy Production by Leveraging Oil and Gas Extended-Reach Drilling and High-Pressure/High-Temperature Well-Construction Expertise,” by Eric van Oort, SPE, Dongmei Chen, SPE, and Pradeepkumar Ashok, SPE, The University of Texas at Austin, et al. The paper has not been peer reviewed. _ In the complete paper, deep closed-loop geothermal systems (DCLGS) are introduced as an alternative to traditional enhanced geothermal systems (EGS) for green energy production that is globally scalable and dispatchable. The authors demonstrate that DCLGS wells can generate power on a scale comparable to that of EGS. They also highlight technology gaps and needs that still exist for economically drilling DCLGS wells, writing that it is possible to extend oil and gas technology, expertise, and experience in extended-reach drilling (ERD) and high-pressure/high-temperature (HP/HT) drilling to construct complex DCLGS wells. Introduction CLGS is considered a subset of EGS, but the authors write that it is a distinct entity. EGS mostly involves well designs that rely on fractures for heat extraction. Such systems are different from CLGS wells in that the latter use closed conduits for thermal fluid circulation and heating. CLGS relies on fluids pumped through a closed loop. The authors treat CLGS systems as being different from EGS systems, with the understanding that drilling technologies discussed in the paper as enablers for CLGS wells apply equally to EGS wells. In the geothermal (GT) domain, the majority of attention and funding currently is assigned to EGS projects. A case is made in the complete paper to continue to develop DCLGS technology because of its favorable risk profile compared with EGS. Part I of the complete paper introduces a hydraulic model coupled with a thermal model suitable for calculating the power generation of DCLGS wells. This synopsis concentrates instead on Part II of the complete paper, in which technology gaps and needs of DCLGS drilling and well construction are highlighted and opportunities identified where oil and gas experience and technology can be directly applied and leveraged. Similarities and Differences of Deep GT and Oil and Gas HP/HT Wells - GT wells generally use larger production hole sizes than typical land wells. - Casing-cement annuli typically are cemented back to surface. - GT wells can be drilled in more-forgiving pore-pressure fracture gradient (PPFG) environments with wider drilling margins than geopressured HP/HT wells in hydrocarbon systems. - Severe lost circulation appears to be a universal problem in deep GT wells. - Drilling costs can account for 50% or more of the total capital costs for a GT energy project. - Data sets on GT wells are much smaller than those for oil and gas wells.

Comparative Analysis of Heating Efficiency of a Single-Well Geothermal System in the Cold Region of Northeast China

Geothermal energy is a type of renewable energy that has rich reserves, is clean, environmentally friendly and has been widely used in the heating industry. The single-well closed-loop geothermal system is a technology with the characteristics of “taking heat without taking water” and is mainly used for geothermal energy heating. Although the heating requirements in the cold region of Northeast China are urgent, the traditional heating mode not only has high economic costs but also causes serious damage to the environment. Therefore, it is of important practical significance to change the heating structure and develop and utilize geothermal energy for heating according to local conditions. In this study, the actual operating single-well geothermal system in the Songyuan area of Jilin Province is used as a case study, and a numerical model is established based on the T2WELL simulation program. The flow production temperature and heat extraction response law of the single-well system in the M1 and M2 wells are contrasted and analyzed under the three key factors of geothermal gradient and injection temperature and flow rate. Based on the simulation results, an optimized development and utilization plan for the M1 and M2 wells is proposed. These results provide a theoretical reference and heating potential evaluation for the promotion of single-well geothermal systems in Northeast China. Taking the geothermal gradient of 4.2 ° C/hm as an example, after 30 years of operation, the heat extraction of the M1 well is 406 kW, and that of the M2 well is 589 kW. Compared with the M1 well, although the M2 well has higher heat extraction, the radial variation in reservoir temperature is more than 50 m under long-term operation, which is not conducive to long-term development and utilization.

CPINNs: A coupled physics-informed neural networks for the closed-loop geothermal system

CPINNs: A coupled physics-informed neural networks for the closed-loop geothermal system

Technical Performance Comparison between U-Shaped and Deep Borehole Heat Exchangers

The geothermal industry is fronted by a fundamental decade to grow and become an energy supplier in transitioning to a sustainable energy system. The introduction of Closed-Loop Geothermal energy systems (CLG) can overcome the negative social response and increase the attractiveness of geothermal developments. The present work aims to investigate and compare the performance of CLG systems. For the comparison, the case study of Campi Flegrei was chosen. The maximum depth was fixed at 2000 m, and the two configurations were set up to analyse the performance and evaluate the best operational configuration. Both CLG configurations showed decay in the output temperature of the working fluid during the production time. For a U-shaped design, it is possible to find a working condition that allows constant thermal power over time. The DBHE specific power was always more significant, up to 350 kW/m, compared to the U-shaped, which attained a maximum of 300 W/m (15%). The comparison with Beckers et al. analysis highlights the similarity of our results with their base case. The consideration of the CLG system’s length is related to the heat exchange and investment costs. For longer exchangers, there are higher investments and lower specific power.

Well-posedness and finite element approximation for the steady-state closed-loop geothermal system

In this article, we study a multi-physics model for the closed-loop geothermal system consisting of some pipelines and a reservoir. The mathematical model for simulating this system mainly focuses on heat transfer between porous media in the reservoir and fluid flow in the pipelines. Besides, to describe the fluid and heat transfer in two regions, the governing equations of the mathematical model are denoted by the steady Navier-Stokes/Darcy equations coupled with the convection-diffusion equations. Next, we design a perturbation system to prove the well-posedness of the governing equations. Moreover, in variational formulation, a Nitsche's penalty term is added to overcome the difficulty of instability caused by the interface condition of heat flux. Furthermore, a decoupled iterative algorithm is proposed to solve the considered equations based on the finite element method, and the stability and convergence of the presented algorithm are shown. Finally, some numerical tests are performed to verify the accuracy and efficiency of the proposed algorithm.

Factors affecting the efficiency of closed-loop geothermal wells

Geothermal energy has a promising prospect of becoming a feasible source of clean energy in the future. However, conventional open-loop geothermal systems need continuous water circulation, which limits their application in dry areas. On the other hand, long-term fluid injection into the reservoir may cause issues such as seismicity. To overcome these shortcomings, closed-loop geothermal systems are not requiring water withdrawal or water injection. However, the mechanism controlling the efficiency of closed-loop systems has not been well analyzed. To resolve the issue, a systematic study was conducted on factors affecting production efficiency of a closed-loop geothermal well. First, a coupled three-dimensional model was established, which was then validated against experimental results available in the literature. Using this model, the effects of several critical design factors were investigated quantitatively. These factors include cement thermal conductivity, cement thickness, fluid circulation rate, injected fluid temperature, heat capacity of circulating fluid, heat exchanger height, and tubing outer diameter. Our simulation results indicate that increasing the cement thermal conductivity enhances both produced fluid temperature and thermal power. The effect of cement thickness is correlated with the value of its thermal conductivity. As the circulation rate increases, the temperature of produced fluid decreases while thermal power increases asymptotically. Using a circulating fluid with high heat capacity would result in higher thermal power. A large heat exchanger height would boost both the production temperature and thermal power significantly. Finally, a dimensionless analysis was conducted, through which two dimensionless numbers were derived to integrate the effects of different factors. By comparison analysis, the accuracy of the dimensionless numbers was validated. Also, the impact of the dimensionless numbers on dimensionless produced fluid temperature was determined. With the dimensionless numbers, the effects of different individual factors can be evaluated comprehensively, which brings convenience to parameter design. The potential applications of the derived dimensionless numbers in parameter design of closed-loop systems were discussed. Results and conclusions obtained in this paper may provide a reference for optimally designing closed-loop systems.