Joule-Thomson Effect Research Articles

Investigations on a 1 K hybrid cryocooler composed of a four-stage Stirling-type pulse tube cryocooler and a Joule-Thomson cooler. Part B: Experimental verifications

In this part, a hybrid cryocooler composed of a four-stage Stirling-type pulse tube cryocooler (SPTC) and a Joule-Thomson cryocooler (JTC) is worked out to verify the theoretical analyses carried out in Part A. The heat loss on each precooling stage is calculated to evaluate the feasibility of thermal coupling between the SPTC and the JTC. The JT compressors are tested and then the compression capacity is studied. A bypass-accelerated-cooling approach is adopted to speed up the cool-down process. The experimental results show that with He-4 in the SPTC and He-3 in the JTC, a no-load temperature of 1.36 K is reached when the three precooling temperatures are kept at 71.2 K, 40.5 K and 8.12 K, respectively. The cooling capacity of 13.9 mW at 1.8 K is measured with the JT loop high pressure and the last stage precooling temperature being at 0.5 MPa and 8.1 K, respectively. Satisfactory agreements are observed between theoretical and experimental studies.

Dimensionless Scaling and Prediction of Vortex Tube Temperature Separation

Ranque–Hilsch vortex tubes split an incoming fluid stream into two outgoing streams: one with a higher total temperature than the incoming fluid and the other with a lower total temperature. This 90-year-old device accomplishes its temperature separation with no moving parts and no external power sources, and despite a diversity of extant theories regarding the underlying mechanism, it is clear that a rapidly swirling flow inside the tube is key to its operation. While many parametric studies have characterized the degree of temperature separation as a function of a host of variables (e.g., inlet pressure, inlet temperature, working gas, tube geometry), no previous effort has identified the governing nondimensional parameters that would simplify further characterization. In the present work, we nondimensionalize the energy equation as it applies to Ranque–Hilsch vortex tubes and identify the governing nondimensional parameters and nondimensional dependent variables. The governing nondimensional parameters are shown to be the Reynolds number, the Péclet number, and a lesser-known nondimensionalization of the Joule–Thomson coefficient, . To discern the dependence of temperature separation on these nondimensional parameters, experimental data were analyzed. We show that a unique dimensionless form of the product of the total temperature, density, and specific heat collapses across a wide range of the aforementioned governing nondimensional parameters, although some dependence remains evident, particularly in the case of , for which an extremely wide range of values were obtained through comparison of carbon dioxide with air. A method is demonstrated to predict temperature separation for new cooling scenarios, reducing the experimental burden for new applications.

The effect of nonlinear electrodynamics on Joule-Thomson expansion of a 5-dimensional charged AdS black hole in Einstein-Gauss-Bonnet gravity

We have studied in this paper the Joule-Thomson expansion of a new charged AdS black hole with a nonlinear electrodynamics in framework of Einstein-Gauss-Bonnet gravity in AdS space. We investigated effects of mass (m), electric charge (q), GB coupling constant (α) and nonlinear electrodynamics parameter (k) on Joule-Thomson expansion by depicting different graphs. The fact that inversion temperature tends to decrease by increasing k, is in contrast to the effect of electric charge. The divergent point as well as the zero point of Joule-Thomson coefficient are also discussed. Results show that, this black hole exhibits a phase transition similar to that of van der Waals system. Furthermore, the isonthalpic curve is obtained and an interesting dependence of these curves on charge and nonlinear electrodynamics parameter is revealed. In T−P graphs, the cooling region shrinks with charge, while this region expands both with mass and with nonlinear electrodynamics parameter. Our study shows that nonlinear electrodynamics parameter plays an important role in Joule Thomson expansion.

Sustainable offshore natural gas processing with thermodynamic gas-hydrate inhibitor reclamation: Supersonic separation affords carbon capture

Dew-point adjustments and reclamation of thermodynamic gas-hydrate inhibitors injected in subsea flowlines are common operations in offshore natural gas rigs characterized by high costs, power consumption and carbon emissions. Typically, offshore plants employ triethylene-glycol absorption for water dew-point adjustment and Joule–Thomson expansion for hydrocarbon dew-point adjustment, while hydrate inhibitor reclamation is applied only on aqueous-inhibitor streams for re-concentration. To implement economically sustained post-combustion carbon capture on offshore rigs, a more efficient gas processing is necessary. This work contemplates a new process – supersonic separator inhibitor-reclamation – which recovers inhibitors from saturated raw-gas via supersonic separation with liquid-water injection additionally yielding exportable liquefied-petroleum-gas and natural gas with adjusted dew-points. This process dramatically improves profitability for thermodynamic inhibitors methanol, ethanol and monoethylene-glycol, creating an economic leverage that affords a post-combustion capture plant avoiding about 43% of carbon emissions. Even including post-combustion capture penalties, the supersonic separator inhibitor-reclamation process achieves a higher net value than the conventional gas processing without carbon capture. A multi-criteria sustainability assessment, based on quantitative metrics and qualitative aspects over all sustainability dimensions, evinces the supersonic separator inhibitor-reclamation process with carbon capture as more sustainable than the conventional counterpart for all inhibitors, whereas among the three inhibitors, the gas processing with monoethylene-glycol was ranked as the more sustainable for offshore rigs.

Modeling and characterizing the thermal and kinetic behavior of methane hydrate dissociation in sandy porous media

Methane hydrates (MHs) are deemed to be a potential energy source to meet the overwhelmingly growing demand for natural gas. The production of MHs involves many controlling processes such as heat transfer, fluid dynamics and hydrate reaction kinetics. Local thermal equilibrium (LTE) between solid and fluid phases was typically assumed when modeling the hydrate production, however, in many cases the local thermal non-equilibrium (LTNE) caused by the complex heat transfer processes at the pore scale cannot be neglected. In this work, a LTNE model was adopted to reflect the multiple heat transfer forms associated with hydrate dissociation and gas-liquid flow, particularly the solid-fluid interfacial convective heat transfer and the Joule-Thomson effect. The LTNE phenomenon and Joule-Thomson effect were hinted by a hydrate dissociation experiment carried out for model verification. The results show that the average estimation errors of temperature, cumulative gas and water production profiles between experiment and LTNE model were 7.70%, 6.63% and 4.02%, respectively, which are considerably better than those of the traditional LTE model (16.44%, 11.74% and 10.89%, respectively). A tenfold increase of interfacial convective heat transfer coefficient could reduce the local solid-fluid temperature difference by about 78.5%. A special focus was given to the Joule-Thomson effect which seemed to have less impact at the experimental condition in this work but become more significant when the gas saturation exceeds 40%. This work could add further insights into the thermal performance during hydrate dissociation in porous media, help to reveal the thermo-kinetic relationship at the pore scale and provide a more accurate estimation of the hydrate dissociation dynamics. It could be potentially employed to optimize the methane hydrate production strategies, minimize the risk of hydrate re-formation and maximize the overall energy recovery efficiency.

Performance Comparison of H2O and CO2 as the Working Fluid in Coupled Wellbore/Reservoir Systems for Geothermal Heat Extraction

CO2 is considered as a novel heat-transmission fluid for extracting geothermal energy from enhanced geothermal systems (EGS), attributed to its high compressibility, expansivity and low viscosity in comparison to water. In order to compare the performance of CO2 and H2O as the working fluid in EGS, a classical five-spot model based on the geologic and geothermal conditions at the Songliao Basin, China, was constructed. Results obtained from the coupled wellbore/reservoir model revealed that the net heat extraction and flow rate are greater for CO2 than for H2O at a fixed operation pressure difference between the injection and production wellheads. However, the wellhead temperature is far lower for CO2 than for H2O due to the strong Joule–Thomson effect of CO2 in the wellbore. Moreover, a stronger pressure change in the wellbore is observed by using CO2, attributed to the gravity and high flow velocity of CO2; this pressure change induces a drop in the frictional pressure. For CO2, the enthalpy change in the wellbore is mainly contributed by the gravitational potential, while for H2O, it is contributed by the gravitational potential and lateral heat exchange. The heat extraction performance depends on the operation pressure difference and injection temperature for H2O-based EGS, while it depends on the wellhead pressures of both the injection and production wells as well as the injection temperature for CO2-based EGS. A high operation pressure is favorable for improving the heat extraction performance (especially the production temperature) for CO2. With the temperature drop limitation at the downhole of the production well, the heat extraction performance is better by using H2O than that by using CO2 as the working fluid. However, the low-power consumption for maintaining fluid circulation demonstrates the application potential of CO2-based EGS.

Tuning the natural gas dew point by Joule-Thomson and Mechanical Refrigeration processes: A comparative energy and exergy analysis

In this study, to adjust the dew point of natural gas, two methods of Joule-Thomson and mechanical refrigeration have been analyzed in terms of technical, energy and exergy. The basis of design was a natural gas feed with an intensity of 1743 kgmole/h. The initial dew point of the feed before processing was equal to 9.65 °C. Comparison of the results showed that the mechanical refrigeration method had much better performance and the energy efficiency in this system was 99.84%, while this value was 97.8% for the Joule-Thomson method. Furthermore, the intensity of gas condensate production, as one of the products of dew point adjustment unit, for mechanical refrigeration that uses propane refrigerant for natural gas refrigeration was estimated as 1842 barrels/day. This value exhibited an increase of 185% compared to Joule-Thomson method. The dew points of produced gas through Joule-Thomson and Mechanical refrigeration processes were −8.4 and −24.91 °C, respectively. It implies that the quality of hydrocarbons dew point in the produced gas was much lower for Joule-Thomson unit compared to mechanical refrigeration. However, in terms of exergy analysis, the mechanical refrigeration method had an efficiency of 88.69%, while the exergy destruction rate for this method was 15.78% less than the Joule-Thomson process.

Interpretation of temperature transient data from coupled reservoir and wellbore model for single phase fluids

This paper presents a non-isothermal, transient coupled reservoir/wellbore model that accounts for the Joule-Thomson (J-T), isentropic expansion, conduction and convection effects for predicting the transient temperature behavior and computing the wellbore temperature at different gauge depths. In this study, single phase fluid flow of oil or geothermal brine from a fully penetrating vertical or inclined well in an infinite-acting homogeneous single layer reservoir is modeled. The coupled simulator solves mass, momentum and energy conservation equations simultaneously for both reservoir and wellbore. We improve solutions by the functional iteration procedure that updates fluid properties based on available correlations as a function of pressure and temperature at a given time step. A comparison of the developed model with a commercial simulator is provided. To understand and identify diagnostic characteristics of temperature transients at gauge locations at the sandface and above the sandface that may arise during a well test, we examine the sensitivity of the model parameters appearing in the coupled non-isothermal reservoir/wellbore model through a synthetically generated test data sets and history matched field application. The drawdown and buildup sandface transient temperature data are obtained from the coupled model and used to interpret and analyze temperature transients. In addition to the J-T coefficient of fluid, history matching transient temperature data provides estimates for the skin zone radius and permeability when analyzed jointly with the conventional pressure test analysis (PTA). An investigation on the effect of gauge location on temperature data shows that the early-time response is influenced by the wellbore phenomena while the J-T effects are clearly identified at later times at typical gauge locations up to 100 m above the top of the producing horizon. Logarithmic derivative of temperature transients are found as a useful diagnostic tool to differentiate the wellbore phenomena from the reservoir response. It is also shown that the temperature transient is more reflective of the properties of the near wellbore region (e.g., skin zone) than the pressure transient. For this reason, analyzing temperature transients together with the pressure transients could add more value to the analysis to better examine near wellbore characteristics. • A fully implicit transient single phase coupled wellbore/reservoir non-isothermal simulator is presented. • New diagnostic plots for temperature transient analysis are provided. • For typical gauge locations (usually up to 100 m above the producing horizon), reservoir response is acquired through drawdown wellbore temperature data. • An effective well radius is introduced for history matching due to the placement of many tools in wellbore during temperature transient analysis. • Reservoir and skin zone parameters are estimated reliably by history matching pressure and temperature field data together.

Performance characteristics of a divergent vortex tube

In this paper performance characteristics for a Ranque-Hilsch vortex tube are presented. The vortex tube was of divergent design and optimised for both size and performance, to be used to supply cold air to turbomachinery experiments with feed air in the supply pressure range 3.0–7.0 bar, and with overall feed mass flow rates in the range 0.05–0.14 kg/s. Optimum fixed parameters selected from the literature were: the ratio of cold-exit diameter to tube inlet internal diameter (0.50); the ratio of combined inlet-nozzle throat area to the tube inlet internal cross-sectional area (0.22); the vortex tube half-angle (3°); the number of inlet nozzles (6); the half-angle of the hot exhaust conical splitter (45°) and the ratio of tube length to tube inlet internal diameter (20). Free parameters that were investigated were: the overall total-to-static pressure ratio between inlet and cold-exit conditions (3.0, 5.0, and 7.0) and the cold-gas mass fraction (0.0–1.0). Comprehensive experiments were performed to characterise the vortex tube performance in terms of the following parameters: non-dimensional cold and hot-outlet temperatures; non-dimensional cold and hot-gas energy separation; isentropic efficiency referenced to cold-gas expansion; and coefficient of performance (COP). A transient heat flux correction technique was implemented to correct for unsteady heat flux terms from the working gas to the vortex chamber walls. Joule-Thomson effects were accounted for by introducing a corrected inlet total temperature. Cold-exit and hot-exit temperature separations were found to increase with pressure ratio. Non-dimensional cold-exit temperature had a minimum at cold-gas mass fraction of approximately 0.40. Non-dimensional hot-exit temperature increased approximately linearly with cold-gas mass fraction. COP decreased with increasing pressure ratio and had a peak value at cold-gas mass fraction approximately 0.60. Isentropic efficiency was relatively insensitive to pressure ratio and had a peak value at cold-gas mass fraction approximately 0.23. In the range investigated the lowest non-dimensional cold-exit temperature (normalised by inlet temperature) was 0.91, the highest coefficient of performance was 0.097, and the highest isentropic efficiency was 0.23. Full details of the design are given, allowing implementation by other researchers.

Thermodynamic geometry and Joule-Thomson expansion of black holes in modified theories of gravity

The implication of Ruppeiner thermodynamic geometry theory to spherically symmetric black holes in anti--de Sitter (AdS) spacetime produces an unavoidable singularity in the line element of thermodynamic geometry. The singularity occurs due to the nonindependence of volume and entropy of black holes (BH) in AdS spacetime. In this work, we present an efficient and easy approach to deal the thermodynamic geometry of AdS BH in Einstein-Maxwell-scalars theory. We show that enthalpy instead of internal energy is central thermodynamic quantity. We develop the particular forms of the line element of thermodynamic geometry in different phase spaces. We find out that the curvatures in different phase spaces are identical and positive which leads to the repulsive interaction information between black hole molecules. We also analyze the thermal stability of AdS BH in Einstein-Maxwell-scalars theory and regularized Lovelock theory in the presence of thermal fluctuations. We observed that momentum relaxation parameter and coupling constants of Lovelock theory increase the thermal stability of the BHs. Finally, we also study the Joule-Thomson expansion for black hole in regularized Lovelock theory.

Stall Warning Strategy Based on Fast Wavelet Analysis in a Multistage Axial Flow Compressor

Abstract As a reliable stall warning strategy, the fast wavelet method was introduced to successfully predict the aerodynamic instability of a multistage axial flow compressor. One single sensor installed at each stage is proved to be sufficient to predict the stability status in a three-stage axial flow compressor. The whole prediction strategy includes the dynamic pressure signal capture, disturbance extraction using decomposition and reconstruction via fast wavelet transform, and stall warning index calculation based on statistical probability distribution. On this premise, the first occurrence of the stall in this three-stage axial flow compressor is predicted to be within the first stage, which is consistent with the stall route captured by the eight transducers around the casing wall. Thereafter, the stall warning index is used to monitor the stability status during the continuous throttling process. Furthermore, the validation using tip air injection and inlet radial distortion indicated that the stall warning index decreases as the compressor's stability improves. Conversely, the deterioration of stability causes the increase of the stall warning index. Thus, experimental results demonstrate that the stall warning method based on fast wavelet analysis can predict the aerodynamic instability in actual application.

EXPERIMENTAL INVESTIGATION OF THERMOHYDRODYNAMIC PROCESSES DURING FILTRATION OF WATER-OIL EMULSIONS

Temperature measurements in downhole conditions (well thermometry) are widely used in monitoring the development of oil and gas reservoirs. With the use of thermometry many oilfield tasks are solved: determination of the working intervals, behind-the-casing flows, casing leakage etc. Variation of temperature in formation is caused by thermodynamic effects: Joule-Thomson and adiabatic. One of the main effects in determining the working intervals by downhole thermometry methods is the Joule-Thomson effect, which is associated with the Joule-Thomson coefficient, characterizing the composition of the fluid flowing from the perforated formation into the well. Therefore, it is necessary to know the value of Joule-Thomson coefficient for different fluids, especially it is important for quantitative interpretation of thermometry data. Currently, the Joule-Thomson coefficient is well studied for various pure fluids and gases, but practically not investigated for emulsions of different types. Meanwhile, the inflow of oil and water from the formation often leads to the formation of emulsion in the bottomhole zone of the formation. Therefore, experimental determination of Joule-Thomson coefficient for such systems and study of thermohydrodynamic processes during emulsion filtration is relevant. The paper presents the results of experimental studies of thermohydrodynamic processes during filtration of emulsion through a choke cell. The description of the experimental setup and the experimental procedure are given. The main design features of the throttle cell, approaches to reduce heat transfer during the movement of fluid through the throttle cell are discussed in detail. The data of experimental studies of temperature and pressure changes during filtration of type “oil in water” — oil dispersion in water and reverse emulsions of type “water in oil” — water dispersion in oil on the throttling cell are given. The paper also obtained estimates of the Joule-Thomson effect for liquids like water, oil and oil-water emulsions.

ALGORITHM FOR SOLVING THE PROBLEM OF METHANE HYDRATE DECOMPOSITION IN A CLOSED HYDRATE-CONTAINING REGION OF A POROUS MEDIUM

Russia has a law on limiting greenhouse gas emissions, signed by the President of the Russian Federation in July 2021. The list of greenhouse gasses includes methane, and its reserves in the form of gas hydrates largely prevail over the reserves of free methane. The formation of gas hydrates occurs at sufficiently low temperatures and high pressures. A significant part of the “sleeping giants”, as journalists dubbed the methane-containing deposits, are in the Arctic region, and they are very sensitive to warming. The release of this greenhouse potential would have very negative consequences for the Earth’s climate. Thus, it is necessary to study the process of gas hydrate decomposition under thermal impact on a hydrate-containing porous medium. It is worth noting the importance of theoretical research to solve this problem, which includes the construction of a mathematical model of the process under study, algorithmization, software implementation and computational experiments. The paper presents a two-dimensional approximation of the formulated problem of heating at the upper boundary of the porous medium’s closed region containing methane and its hydrate initially. A mathematical model is presented, which is based on the mass conservation equations for methane, water and gas hydrate; Darcy’s law for the motion of the gas and liquid phases; the equation of a real gas state, the energy conservation equation considering thermal conductivity, convection, adiabatic cooling, the Joule-Thomson effect and absorption of latent heat during hydrate formation. An algorithm for the numerical implementation of the mathematical model is constructed and a computer code is developed to calculate main parameters of the process in the work, the computer program is updated to calculate the main parameters of the process researched in the article.

Cavitation characteristics and suppression research of refrigerant in capillary tube

Cavitation is easy to form and develop due to the large pressure drop during throttling in the refrigeration system. To effectively suppress the throttling cavitation phenomenon, an aerating method through bypass capillary is proposed. Based on the Homogeneous Equilibrium Mixture (HEM) model, the Schnerr-Sauer cavitation model and the Realizable k-ε turbulence model were adopted, and the throttling process in the capillary tube and transition tube was simulated by Fluent, and the pressure distribution and vapor volume fraction distribution in the capillary tube and transition tube were obtained for both original model and aerating model. Results show that after adopting the aerating technology, the vapor volume fraction at the capillary outlet is almost unchanged, while the vapor volume fraction in the transition tube is obviously increased, which indicates that the bubble collapsing phenomenon in the transition tube is weakened, and then realized the purpose of suppressing the cavitation effect.

Mathematical modeling of non-stationary gas flow modes along a linear section of a gas transmission system

Article demonstrates the applicability of modeling non-stationary non-isothermal gas flow along a linear section of a gas transmission system by means of using various numerically simulated models and sophisticated numerical techniques. There are described several models of non-stationary non-isothermal regimes of gas flow along the pipeline section. They are included in the considered general model and their comparative analysis is carried out by the virtue of numerical simulation. The finite difference algorithm is used to solve the simultaneous equations of the numerically simulated model for the pipeline section. The results of calculating the gas flow parameters using various models are presented: both with and without taking into account kinetic energy, as well as both with and without taking into account the Joule–Thompson effect. The matter of choosing the appropriate model is discussed. The obtained results can be used at the stage of transfer pipeline system operation in order to develop scientifically well-founded recommendations for improving the safety and efficiency of the pipeline transportation system.

Performance Prediction of a Distributed Joule-Thomson Effect Cooler with Pillars

Performance Prediction of a Distributed Joule-Thomson Effect Cooler with Pillars

Joule–Thomson expansion in charged AdS black hole surrounded by a cosmological fluid in Rainbow Gravity

In this work, we analyze the Joule–Thomson expansion in an AdS Reissner–Nordström black hole, surrounded by an average cosmological fluid, in Rainbow Gravity. We plot the graphs corresponding to the inversion temperature curves and numerically calculate the ratio between the minimum of the inversion temperature and the critical temperature, with the aim of investigating how Rainbow Gravity alters such behaviors compared to General Relativity.

Investigation on the choked mass-flow characteristics of the helium fluid during the Joule-Thomson process in micro-orifice under different high pressures

The micro-orifices during the Joule-Thomson process are used in a variety of energy applications such as microfluidic systems and large-scale cryogenic helium systems. The structure of the micro-orifice impacts the cooling capacity by adapting the mass flow rate and pressure drop in the refrigeration system. The pressure drops and flow characteristics in micro-orifices with diameters of 20–40 μm and thicknesses of 50 μm were numerically and experimentally investigated. Helium was used as the working fluid, and the simulations using CFD (Computational Fluid Dynamics) method were conducted with upstream pressure (pu) of 0.5–2.0 MPa (Mega-Pascal) and downstream pressure (pd) of 0.1–1.5 MPa at 293 K (Kelvin). The simulations found that the completely choked pressure ratio of each micro-orifice is smaller than the theoretical critical pressure ratio λ*=0.487 and revealed the flow field characteristics in micro-orifices flow under different working conditions. Experiments were carried out with four micro-orifices with the thickness of 50 μm and the effective diameter of 23.88 μm, 26.53 μm, 33.30 μm, 40.69 μm under different pressure conditions (pu 0.5–2.0 MPa, pd 0.1–1.5 MPa), respectively. The errors between the numerical and experimental mass flow rates in different micro-orifices corresponding to the pressures are within 10 %. The W. B. Brower model was used to predict the mass flow rates that are consistent with the experiments and simulations. The semi-empirical model is developed using the data set for the modifications to W. B. Brower model and with an error range of ± 0.08 mg/s in the calculated flow rate compared to the experimental value.

Joule–Thomson expansion of Reissner–Nordström-Anti-de Sitter black holes with cloud of strings and quintessence

The Joule–Thomson expansion is studied for Reissner–Nordström-Anti-de Sitter black holes with cloud of strings and quintessence, as well as its thermodynamics. The cosmological constant is treated as thermodynamic pressure, whose conjugate variable is considered as the volume. The characteristics of the Joule–Thomson expansion are studied in four main aspects with the case of ω=−1 and ω=−23, including the Joule–Thomson coefficient, the inversion curves, the isenthalpic curves and the ratio between the minimum inversion temperature Timin and the critical temperature Tc. The sign of the Joule–Thomson coefficient is possible for determining the occurrence of heating or cooling. The divergent point of the Joule–Thomson coefficient corresponds to the zero point of the Hawking temperature. Unlike the van der Waals fluids, the inversion curve of the black hole is not closed. The inversion curve is the dividing line between heating and cooling regions, above which the slope of the isenthalpic curve is positive and cooling occurs, and the cooling–heating critical point is more sensitive to Q. Concerning the ratio TiminTc, we calculate it separately in the cases where only the cloud of strings, only quintessence and both are present.

Joule-Thomson expansion of lower-dimensional black holes

The Joule-Thomson expansion is extended to the lower-dimensional regime by considering the rotating BTZ metric in the (2+1)-dimensional space-time. Specifically, the properties of three important aspects of the Joule-Thomson expansion, namely the Joule-Thomson coefficient, the inversion curve and the isenthalpic curve are focused on. The divergence point of the Joule-Thomson coefficient and the zero point of the Hawking temperature are studied. The inversion temperature curves and isenthalpic curves in the $T-P$ plane are obtained and the cooling-heating regions are determined. Furthermore, the minimum inversion temperature is found to be zero, and the black hole becomes an extremal black hole. The ratio between the minimum inversion temperature and the critical temperature for the BTZ black hole doesn't exist, since the BTZ black hole does not have the critical behavior in $P_c$, $T_c$ and $V_c$.