Cold End Temperature Research Articles

Autonomous Cooling Design and Temperature Control Mechanism of Hypersonic Vehicle

Aiming at the thermal control problem of the key parts in hypersonic vehicle, an adsorption autonomous cooling device based on porous media was designed. The transient behavior of two-phase flow and heat transfer inside porous media was studied numerically and experimentally in this paper. The two-phase mixing model was used in the numerical method and verified by the visualization experiment. The results show that the heat transfer performance of adsorption autonomous cooling device was more significant under the hot-end exhaust design. It shows that the temperature of cold end was still within 100 °C at 1500 s. Moreover, the motion direction of the vapor phase was opposite to gravity, and the flow direction of the liquid phase was squeezed by the vapor phase and related to the outlet position. The liquid working fluid was more strongly bound by the pore structure. Furthermore, the liquid working medium was further vaporized near the interface after the formation of the two-phase mixing zone.

The reasonableness and feasibility of replacing standing wave vibration with air piston mode vibration in thermoacoustic refrigerator

On the basis of analyzing the longitudinal forced vibration of the gas column in the resonator of a half-wavelength standing wave thermoacoustic refrigerator, the idea of replacing the standing wave vibration with the first-order piston mode vibration of the gas column in the thermoacoustic refrigerator was proposed, so as to increase the vibration amplitude of the gas in the resonator, realize the large alternating flow of gas, and improve the working efficiency of the thermoacoustic refrigerator. At the same time, its rationality and feasibility are discussed from the aspects of modal vibration, pressure distribution of gas column piston, and temperature variation of gas microcluster vibration in equilibrium position. Combined with the existing research results, the superiority of the application of the first-order piston mode vibration of the gas column in the thermoacoustic refrigerator is further explained. The results provide a new research idea and direction for further reducing the cold end temperature of thermoacoustic refrigerator.

Research on temperature performance prediction of vortex tubes based on artificial neural networks

Abstract This study constructs a hybrid neural network model by integrating the physical constraints of the Bernoulli equation and Nikolaev's formula. The model is designed to explore and predict the variation pattern of the cold end temperature in a vortex tube. The input parameters include inlet pressure, inlet temperature, and cold mass fraction, with the cold end temperature as the output parameter. The network employs a multilayer feedforward model and the Levenberg-Marquardt learning algorithm, using a hyperbolic tangent function as the activation function. To evaluate the statistical validity of the developed model, the coefficient of determination (R²) and root mean square error (RMSE) are utilized, along with an analysis of the model's uncertainty and robustness. The hybrid model achieves an R² of 0.9936 and an RMSE of 0.3392, demonstrating strong performance in terms of uncertainty and robustness. These results indicate that the model accurately predicts the cold end temperature variation in the vortex tube. Furthermore, the findings reveal an optimal pressure range (0.49 MPa to 0.76 MPa) and cold mass fraction range (0.1 to 0.2) that minimize the cold end temperature.

Optimization of a Biomass-Based Power and Fresh Water-Generation System by Machine Learning Using Thermoeconomic Assessment

Biomass is a viable and accessible source of energy that can help address the problem of energy shortages in rural and remote areas. Another important issue for societies today is the lack of clean water, especially in places with high populations and low rainfall. To address both of these concerns, a sustainable biomass-fueled power cycle integrated with a double-stage reverse osmosis water-desalination unit has been designed. The double-stage reverse osmosis system is provided by the 20% of generated power from the bottoming cycles and this allocation can be altered based on the needs for freshwater or power. This system is assessed from energy, exergy, thermoeconomic, and environmental perspectives, and two distinct multi-objective optimization scenarios are applied featuring various objective functions. The considered parameters for this assessment are gas turbine inlet temperature, compressor’s pressure ratio, and cold end temperature differences in heat exchangers 2 and 3. In the first optimization scenario, considering the pollution index, the total unit cost of exergy products, and exergy efficiency as objective functions, the optimal values are, respectively, identified as 0.7644 kg/kWh, 32.7 USD/GJ, and 44%. Conversely, in the second optimization scenario, featuring the emission index, total unit cost of exergy products, and output net power as objective functions, the optimal values are 0.7684 kg/kWh, 27.82 USD/GJ, and 2615.9 kW.

Experimental study on a novel split thermoelectric cooler of big temperature difference for combined cooling and heating supply

The improvement in maximum temperature difference is very important for thermoelectric cooler potential applications under severe and complex conditions. Currently, the maximum temperature difference of commercial single-stage thermoelectric modules represented is usually below 70 K and the energy efficiency is not high. In this paper, 12 kinds of split thermoelectric modules are developed according to different semiconductor and connector structures, and their performance is compared by experiments. The results show that the temperature difference of novel split thermoelectric cooler can exceed 200 K, which is more than 180 % higher than the currently commercial module (70 K), and it also has good performance under high ambient temperature. The new thermoelectric module can achieve the application of cold and hot ends energy on long-distance different occasions and has a module coefficient of performances of 2.68 and 1.29 under temperature differences of 57.43 K and 197.73 K, respectively. Meanwhile, a new asymmetric design method for independent control of hot and cold end temperatures is presented. It can achieve independent heat flux density design for cold and hot ends, and its performance can be further enhanced by regulating the temperature of connector.

Multi-criteria decision analysis and experimental study on heat pipe thermoelectric generator for waste heat recovery

In the realm of heat conversion and utilization, gravity heat pipes have garnered significant attention due to their exceptional heat conduction efficiency. As a strategy to effectively enhance the efficiency of thermoelectric generator (TEG) systems through shallow geothermal heat recovery, optimizing the performance of gravity heat pipes holds significant implications for their widespread application in the field of sustainable energy. Experimental evaluations comprehensively assessed the performance of a 6 m gravity heat pipe within a temperature difference power generation system, employing multi-criteria decision-making to test thermal performance, efficiency, reliability, stability, and working fluid studies. The study demonstrates that the heat transfer efficiency of gravity heat pipes is highest under film boiling conditions at 250 °C to 350 °C. By adjusting the inclination angle to 60° to 90° and the filling ratio to 30 %–60 %, the power generation efficiency can be improved to 9.71 %. With an increase in wind speed to 8 m/s, the cold end temperature decreases by 30 %, resulting in a peak power generation efficiency of up to 11.3 %. Additionally, CHIME connection methods reduce energy losses to 8.1 %, providing crucial scientific foundations for the design and optimization of temperature difference power generation systems. These findings not only elucidate the heat transfer mechanisms of gravity heat pipes but also offer technical guidance for efficient energy conversion in thermoelectric generator systems.

Performance investigation of a novel free piston Stirling pump for reverse osmosis desalination systems

Renewable energy (RE)-powered desalination technology is considered a cleaner and promising alternative for alleviating energy exhaustion and environmental pollution resulting from traditional desalination. Currently available solar thermal reverse osmosis (RO) desalination systems are based on the organic Rankine cycle. With the aim to expand the driving mode of reverse osmosis desalination as well as simplify the transmission chain and reduce mechanical energy losses, a novel hydraulic free piston Stirling pump (FPSP) is developed and a simple model is designed and tested for conceptual validation. A free piston Stirling pump - powered reverse osmosis desalination system is presented, and a thermodynamic–dynamic coupling model is established. A parametric analysis is performed with an emphasis on the effects of temperatures, charge pressure, spring stiffnesses, damping coefficients, and masses on the system performance, including the frequency, power output, water yield, and efficiency. Results showed that the frequency in the pumping process was higher than that in the suction process because of the different loads. Within the range investigated, the performance improved as the hot-end temperature, charge pressure, spring stiffnesses, and damping coefficient of the power piston increased, while it deteriorated with a decrease in the damping coefficient of the displacer and the piston masses. Specifically, the power output and water yield reached the optimum value as the cold-end temperature was maintained around 310 K. Under representative parameter conditions, the power output could reach 12.15 kW and the water yield was approximately 6.07 × 10−4 m3/s. In addition, the temperatures and charge pressure influence the system efficiency most.

Effect of thermodynamic-dynamic parameters on the oscillation and performance of a free piston Stirling engine

This paper presents an analysis of the oscillation and parametric performance of a gamma-type free piston Stirling engine using a thermodynamic-dynamic nonlinear model. Firstly, the concept of oscillation and impact cylinder points is defined and a method for determining the stable oscillation zone of the free piston Stirling engine is proposed. Besides, the effects of various thermodynamic parameters on the system’s performance, such as charging pressure, hot-end and cold-end temperatures, and dynamic parameters (such as the masses and spring stiffness of the displacer and piston), are investigated. The oscillation zone takes on a “V” shape when the piston and displacer masses or stiffness, and hot-end temperature and displacer stiffness, are varied. Finally, the output power and operating frequency of the impact cylinder line are determined. The results show that maximum output power and operating frequency of 75.36 W at 58.21 Hz or 76.73 W at 58.61 Hz are achieved at the “V” bottom when either the piston and displacer masses are adjusted or their stiffness properties are varied. Additionally, consistent output power and operating frequency of 125.35 W and 75.08 Hz, and 104.63 W and 68.59 Hz, are obtained regardless of changes in both displacer mass and stiffness.

Thermoelectric system investigation with the combination of solar concentration, greenhouse and radiative cooling for all-day power generation

Thermoelectric generator (TEG) can utilize solar heating to generate electricity without any fossil fuel consumption. However, conventional solar driven TEG fails to achieve high efficiency power generation for 24-h, due to the losing of solar concentration at the hot end and additional cooling capability at the cold end. Therefore, a novel TEG system with the combination of solar concentration, greenhouse and radiative cooling is proposed. With the aim to significantly increase the temperature of hot end, a dish-type concentrator is introduced to concentrate solar radiation and a greenhouse seals up heat. Radiative cooling panel is used to decrease the temperature of cold end, which can realize temperature differences of TEG at night. The eight-day outdoor experimental test indicates that the thermoelectric system achieves a maximum temperature difference of 47.5 °C and a voltage output of 1293.8 mV. The system attains the average power outputs of 3.6 W/m2 on sunny and 0.16 W/m2 on cloudy. Moreover, even at the nights of high humidity and low temperature, the system also achieves a maximum power output of 0.08 W/m2, which can enable continuous power generation throughout the day. This innovative TEG system presents a viable strategy for powering small-scale devices in remote areas.

Enhanced performance of thermoelectric generation system by increasing temperature difference using spray cooling

Thermoelectric generation system (TEGS) can directly convert the waste heat to electricity in practice. The power generation is highly impacted by the temperature difference between the hot and cold ends. To significantly improve the temperature difference, various cooling technologies have been applied to TEGS. However, there is still room to further increase the power generation efficiency of TEGS by enhancing the cooling performance. Herein, spray cooling was introduced to the TEGS and influencing factors were investigated. The experimental study demonstrates that, upon the spray cooling, the cold-end temperature can be dramatically reduced up to 27 °C. Further, the output performance of the TEGS was substantially increased by optimizing spray heights, flow rates and heat source temperatures, etc. For example, for the spray height of 3 cm, the TEGS achieves a maximum output voltage because the thermoelectric generation module (TGM) is completely covered by the spray area in this study. Moreover, the temperature difference between cold/hot ends and voltage of the TEGS are further increased to 55 °C and 1.39 V at a flow rate of 100 L/min. On the other hand, increasing the heat source temperature (hot end temperature) can also strongly improve the output performance of the TEGS. Particularly, compared to passive air cooling, spray cooling can enable the TEGS to significantly increase its output voltage and current by 17 and 27 times, respectively. Finally, this study would potentially provide a new cooling strategy to strongly enhance the output performance of TEGS.

Integrated micro thermoelectric devices with self-power supply and temperature monitoring: Design and application in power grid early warning

The Internet of Things (IoT) and wearable technology have led to the development of wireless sensors at an unprecedented pace. Within the IoT framework, smart grid systems require real-time temperature monitoring of various switch contacts. Especially in the process of power transformation (China: ultra-high voltage → 10 kV → 380 / 220 V), poor switch contact will lead to overheating, resulting in equipment damage and major safety accidents. Micro-thermoelectric devices (micro-TEDs) exhibit a significantly greater thermoelectromotive force than metal-based thermocouples. By establishing two bivariate first-order equations for hot-end and cold-end temperatures, accurate detection of both temperatures and simultaneous self-power supply can be achieved. In this work, we observe that the Seebeck coefficient of hot extruded n-type Bi2Te2.7Se0.3 and p-type Bi0.5Sb1.5Te3 thermoelectric couple exhibits slight fluctuations around 435 μV/K, while its resistivity follows an almost linear trend with temperature from room temperature to 120 °C. By utilizing these two bivariate first-order equations, the temperatures at both the hot and cold ends can be determined. Based on this, we have fabricated a micro-TED (size: 14 × 6 × 2.5 mm3, TE leg: 0.8 × 0.8 × 1.6 mm3) with a temperature measurement accuracy of up to ± 0.1 °C and a power density of 10.12 mW/cm2 under a temperature difference of 30 °C. Furthermore, we used thirteen micro-TEDs and one circuit board to build a self-power supply temperature monitor. When this TE sensor device was deployed on the plum blossom connector surface and conducted online assessments for the 10 kV substation over a span of approximately 200 h, it continuously outputs a temperature signal and the power supply maintained normalcy, exhibiting no aberrations. This temperature monitoring system can be extended to other fields that require all-weather wireless temperature monitoring, which provides a new direction for the development of micro-TEDs.

Numerical Study of Pore Water Pressure in Frozen Soils during Moisture Migration

Frost heaving in soils is a primary cause of engineering failures in cold regions. Although extensive experimental and numerical research has focused on the deformation caused by frost heaving, there is a notable lack of numerical investigations into the critical underlying factor: pore water pressure. This study aimed to experimentally determine changes in soil water content over time at various depths during unidirectional freezing and to model this process using a coupled hydrothermal approach. The agreement between experimental water content outcomes and numerical predictions validates the numerical method’s applicability. Furthermore, by applying the Gibbs free energy equation, we derived a novel equation for calculating the pore water pressure in saturated frozen soil. Utilizing this equation, we developed a numerical model to simulate pore water pressure and water movement in frozen soil, accounting for scenarios with and without ice lens formation and quantifying unfrozen water migration from unfrozen to frozen zones over time. Our findings reveal that pore water pressure decreases as freezing depth increases, reaching near zero at the freezing front. Notably, the presence of an ice lens significantly amplifies pore water pressure—approximately tenfold—compared to scenarios without an ice lens, aligning with existing experimental data. The model also indicates that the cold-end temperature sets the maximum pore water pressure value in freezing soil, with superior performance to Konrad’s model at lower temperatures in the absence of ice lenses. Additionally, as freezing progresses, the rate of water flow from the unfrozen region to the freezing fringe exhibits a fluctuating decline. This study successfully establishes a numerical model for pore water pressure and water flow in frozen soil, confirms its validity through experimental comparison, and introduces an improved formula for pore water pressure calculation, offering a more accurate reflection of the real-world phenomena than previous formulations.

Numerical simulation and experimental validation of the heat transfer characteristics in a circuit gas gap heat switch for the dilution refrigerator

The gas gap heat switch (GGHS) used for controlling heat transfer between different stages can be an important component for the precooling process of some dilution refrigerators. In this paper, a novel circuit GGHS is a rotationally symmetric heat switch assembly with annular fin arrangements to strengthen the heat-transferring effect. A numerical model considering 4He actual gas properties is proposed to investigate the heat transfer characteristics in the circuit GGHS, in which the effects of charge pressure, cold end temperature, thickness and length of walls on the mean thermal conductance (MTC) are studied. Simulation results show that the MTC increases with the growing cold end temperature, wall thickness, and length, respectively. Given the pressure of 10 kPa, cold end temperature of 4.2 K, wall thickness of 0.97 mm, and wall height of 66 mm, the theoretical MTC is 0.828 W/K. Experimental results indicate that the proposed simulation model is reasonable. Furthermore, the increment of the MTC decreases with the growing temperature. The GGHS used in experiments had a cold end temperature of 4 K, wall thickness of 0.65 mm, and wall height of 33 mm; the measured MTC was 0.219 W/K. Only when the temperature is above 10 K does the charge pressure have a pronounced effect on the MTC. This study provides helpful theoretical guidance for the design and optimization of the circuit GGHS.

Thermoelectric cooler with embedded teardrop-shaped milli-channel heat sink for electronics cooling

Thermoelectric coolers (TECs) offer a site-specific, on-demand, and solid-state thermal management solution for on-chip electronics cooling. To maximize their cooling performance, it is essential to integrate an effective heat exchanger, such as microchannels, on the TEC’s hot side. However, traditional microchannel heat sinks pose challenges, including the necessity for high-cost equipment and the manifestation of deleterious parasitic contact thermal resistance with the TEC. Recent studies have demonstrated the superior performance of teardrop-shaped microchannels compared to their conventional counterparts, yet their effectiveness at the milli-scale remains unexplored. In this work, through systematic numerical simulations across nine milli-channels with distinct variable cross-sections, we propose a teardrop-shaped milli-channel heat sink designed for fabrication through cost-effective mechanical machining. The optimized teardrop-shaped milli-channel exhibits a superior overall thermo-hydraulic performance evaluation factor (PEC) of approximately 1.8 with respect to its rectangular cross-sectional counterpart. Furthermore, we seamlessly embed the designed milli-channel heat sink into the substrate of the TEC to mitigate the parasitic contact thermal resistance. Measurements on the TEC embedded with the teardrop-shaped milli-channel heat sink show that the cold end temperature reaches a minimum value of −18.6 °C at an optimal input electrical current of 3.5 A and an inlet coolant flow velocity of 0.2 m/s. This temperature represents an improvement of approximately 3.5 °C and 4.4 °C compared to the TEC with an attached teardrop-shaped milli-channel and a conventional water block heat sink, respectively, under identical operating conditions. Additionally, a prototype TEC cooling system with the embedded teardrop-shaped milli-channel for high-power field-effect transistor (FET) cooling is demonstrated. Operating at an input electrical current of 3.5 A and an inlet flow velocity of 0.6 m/s, our proposed TEC cooling system achieves a steady-state working temperature of 51.3 °C for the FET at a high heat flux of 75 kW/m2, outperforming the TEC with an attached teardrop-shaped milli-channel and a conventional water block heat sink by approximately 1.4 °C and 2 °C, respectively.

Generation of Current in Thermoelectric Generator by Absorption of Heat Using ANSYS Thermal

Thermoelectric power generation is a renewable energy conversion process that directly converts heat to electricity. This research proposes a novel fluid-thermal-electric multi-physics numerical model for predicting the performance of a thermoelectric-producing system. On the ANSYS platform, numerical simulations are done along with the exhaust temperature and mass flow rate. The range of hot end temperatures is from 100 to 450 degrees Celsius. Similarly, the cold end temperature can reach a maximum of 25 degrees Celsius and a minimum of 10 degrees Celsius. When the temperature at the generator's hot end rises, it means the temperature at the generator's cold end must fall. It means both have an inverse relation to each other. The maximum heat absorbed at the hot junction is 32.762 W and the minimum heat absorbed at the hot junction is 4.6305 W. Hence the maximum and minimum current values produced in this paper by the thermoelectric generator are 72.156 A & 10.816 A respectively. The hot side heat exchanger's position of the thermoelectric modules has a significant impact on output. Combining the benefits of many models are advised to construct an inclusive thermoelectric generator system for use in real-world applications.

Theoretical and experimental study of free piston Stirling generator for high cold end temperatures

The cold end temperature of the free piston Stirling generator (FPSG) should be increased to reduce the heat sink radiator area for space application. A comprehensive model coupled with thermodynamics, dynamics, and electromagnetism was built and solved numerically to investigate the intrinsic relationship between cold end temperature and FPSG performance. It shows that, with fixed heating power, as the cold end temperature increases, the magnetic field strength of alternator air gap decreases. The output power and efficiency show a steadily declining trend. A 100W prototype was tested for model verification. The relative error between experiment and model prediction of the amplitude of piston and displacer is 2.48 % and 11.97 % respectively. For phase difference between piston and displacer, thermal efficiency, and working frequency, the relative error is 54.32 %, 27.23 %, and 2.39 % respectively. When the cold end temperature rises from 320.5K to 394K, the output power and efficiency relative error is 26.06 %∼37.85 % and 27.67 %∼34.24 % respectively. The predicted parameter effect trend is well consistent with experiment, showing the important guiding role for generator optimization operating at high temperatures. The exergy analysis shows that increasing cold end temperature causes input exergy decline and aggravates exergy loss, which is main cause of generator performance degradation.

Rapid prediction of regenerator performance for regenerative cryogenics cryocooler based on convolutional neural network

The regenerator is the core component of the regenerative cryogenic refrigerator, for its structure sizes, operating parameters and phase characteristics at the cold and hot ends co-determine the power and efficiency of the refrigerator, and the design parameters of other coupled components. Efficiently predicting the regenerator performance can reduce the design period of cryogenic refrigerators. Addressing the long computational time constraints in the traditional numerical simulation methods, a novel approach based on a one-dimensional convolutional neural network (1D-CNN) was proposed. Initially, a program capable of multi-threading and automatically running the specialized regenerator calculation software REGEN 3.3 was developed. The performance of the regenerators with various parameter combinations at the cold end temperature of 60–120 K were calculated and 181,440 pieces of data were obtained. Subsequently, the architecture and hyperparameters of the model were determined. The trained model exhibits an average relative error of 3.83% for predicting regenerator power, 0.13% for predicting pressure ratio at the hot end, and 1.55% for predicting the coefficient of performance (COP). The model's generalization ability was confirmed by generating data points beyond the original dataset. Additionally, the model allows for the simultaneous calculation of multiple sets of irregular regenerator parameters, and reduces the calculation time from 2500 min for 1000 pieces using REGEN 3.3 software to just 130 ms, representing a decrease by nearly six orders of magnitude. This approach effectively resolves the long computation time associated with traditional numerical simulation methods, and will present a new solution for the rapid and precise design of regenerators.

An approach to improve the efficiency of cooling enhancement of a thermoelectric refrigerator through the use of phase‐change materials

AbstractIn this study, a technique is suggested to enhance the temperature difference for cooling and energy efficiency of thermoelectric cooler (TEC) used in cooled detectors when subjected to high current conditions. Embedding a structure for storing heat during phase change is the basis of this method at the heat sink. A simulation model was created for a common two‐stage series TEC with an asymmetrical design, which is coupled with a structure for storing heat through phase change. The research examined how phase transition heat storage impacts the coefficient of performance (COP) and refrigeration temperature difference across various phase‐change substances, heat transfer coefficients at the hot end, currents, and the height of the phase‐change material (PCM). The findings suggest that the suggested approach of combining PCM with TEC can efficiently lower the cold end temperature of TEC by a maximum of 20 K, enhance the temperature gap by a maximum of 16 K, and preserve the consistency of the optimized quantity across various hot end heat transfer coefficients. During the phase‐change process of PCMs, the COP of TECs integrated with PCM is found to increase by an average of 2%–3% compared to TECs without PCM integration. Under the maximum current operating condition, the cryogenic temperature can be optimized to a minimum of 238 K. In summary, the proposed method of integrating phase‐change heat storage with TECs provides a promising solution for improving their cooling performance and energy efficiency in cooled detectors under high current conditions. Additional investigation can be conducted to explore the practical application of this approach and enhance the design parameters for various uses.

Analysis and optimization of a novel high cooling flux stacked T-shaped thermoelectric cooler

To meet the cooling demands of high heat flow density hotspots in scenarios such as electronic chips, a novel three-dimensional stacked T-shaped thermoelectric cooler (STTEC) is designed in this study. Under steady-state conditions, a finite element method with coupled thermal–electrical–mechanical physical fields is utilized, and the temperature dependence of thermoelectric (TE) materials is considered. First, the cooling flux, coefficient of performance (COP), and minimum cooling temperature of STTEC under different input-current and thermal boundary conditions are investigated and compared to the traditional π-shaped thermoelectric cooler (π-TEC). Second, the effects of geometrical parameter variations under optimal currents on the cooling performance and reliability of STTEC are studied. Finally, the structural parameters are optimized. The results show that the STTEC altered the path of TE conversion and transfer, which significantly improved the optimal current. The STTEC has a remarkable advantage in cooling performance under low temperature differences or high cooling loads. Compared to the π-TEC, STTEC enhances cooling flux by 101.6%, rises COP by 358.5%, and lowers the cold-end temperature by 46.6 K. At optimal current conditions, by optimizing the thickness of the T-shaped copper slice and the height difference between the TE leg and the T-shaped copper slice, the thermal stress decreased by 18.4%. The STTEC’s novel design could inspire the manufacturing and commercialization of high-performance thermoelectric coolers.

Optimization and comparison of two-stage thermoelectric generators considering the influence of temperature variation on materials for waste heat utilization

Thermoelectricity technology, as a kind of cost-effective and pollution-free power generation solution, is often used for waste heat recovery and utilization. In this paper, the temperature distribution of a Two-stage Thermoelectric Generator (TTEG) under constant temperature conditions has been studied using a one-dimensional heat conduction model. Moreover, by combining the obtained temperature distribution with the three-dimensional size of TTEG, a calculation formula of resistance and voltage was developed based on the calculus method. When the sum of cross-sectional areas of all the PN-type thermoelectric arms respectively in high- and low-temperature layers is constant, the optimal ratio between cross-sectional areas of a single PN-thermoelectric arm respectively in high- and low-temperature layers can be calculated using the proposed formula in this study to achieve the maximum output power. Results also showed the relationship between the heights of PN-type thermoelectric arms and the temperature distributions in high- and low-temperature layers. Using PbTe as the medium temperature thermoelectric material and Bi2Te3 as the low temperature thermoelectric material, a case study was conducted on the PN-type thermoelectrics with the same total height and the same total cross-sectional area. The theoretical calculation results showed that the bigger of maximum output power between the two-stage thermoelectric generator and that of the Segmented Thermoelectric Generator (STEG) is related to the hot and cold end temperature.