Transient Supercooling Research Articles

Numerical study of transient supercooling performance and thermal stress analysis of segmented annular thermoelectric cooler

A multiphysics-coupled numerical simulation of a segmental annular thermoelectric cooler (SATEC) was performed using a three-dimensional finite element method. A coupling model of the temperature, stress, and electric fields was established based on the three-dimensional transient states. Moreover, the effects of different pulse width (τ), pulse current amplitude (Pa), pulse current shape parameter (m), heat transfer coefficient (h), angle (θ), and length (H) of the thermoelectric leg on the SATEC transient properties and von Mises stress were studied. The results show that τ and Pa are the key influencing factors of SATEC. The larger the h, the more optimized the heat transfer effect of the SATEC hot end. However, when h increased to a certain value, the changes in the transient performance were insignificant. The optimal pulse can be achieved when the input pulse current Iop = 6.47 A and pulse width τ = 20 s. When the length of the thermoelectric leg H4 was 3 mm and the thermoelectric leg angle θ2 was 7°, the performance of SATEC was optimal, with Tc,min being 249.57 K, Tc,max being 279.50 K, thold being 5.5 s, and the maximum von Mises stress being 37.48 MPa. The results show that different supercooling indices and von Mises stresses could be obtained for different pulse current shapes. Geometric optimization of SATEC can improve the cooling performance and reduce the von Mises stress, thereby providing a theoretical reference for practical engineering applications and thermoelectric cooling research.

A hybrid battery thermal management system coupling with PCM and optimized thermoelectric cooling for high-rate discharge condition

The lithium-ion battery is now abundantly available in the market due to its excellent performance. However, battery overheating is increasingly prominent when the battery discharges at high speed. In this study, an active-passive BTMS (Battery Thermal Management System) combining PCM (Phase Change Material) and TEC (Thermoelectric Cooler) was proposed. The effects of different TEC currents and delayed cooling at different PCM melting rates were analyzed under the 4C discharge condition to enhance the thermal performance. The results demonstrate that battery temperature gets effectively controlled as the TEC current increases, but the temperature difference and PCM utilization are poor. However, applying a delayed current after the PCM melting rate reaches 80% reduces energy consumption and provides better temperature uniformity. Considering the large battery heat under 4C discharge, combined with the transient supercooling effect of TEC, a continuous pulse current was used to provide additional cooling power. Results show that the battery with TEC pulsed current operates 57.8% longer at 40 °C with a temperature difference maintained at 2.5 K under 4C discharge conditions compared with the PCM model.

Supercooling behaviours of thermoelectric coolers with different continuous Double-pulse current waveforms

This study innovatively identifies techniques to improve the transient supercooling performance of thermoelectric coolers (TECs) through continuous double-pulse currents with different waveforms. Some applications need to operate at a specific temperature while obtaining the requirement for long periods of refrigeration, thus the new effective cold-holding time (teff-hold) for the defined effective cooling line and the variation curve of the cold-end temperature was used as the main index to measure the transient performance of the TEC. In addition, the cold-end minimum temperature (Tc,min), overshoot temperature (Tc,max), and the time for recovery to steady-state temperature (trec) of TEC were considered comprehensively. A three-dimensional (3D) multi-physics coupling transient model was constructed using numerical simulations, and its accuracy was verified by experimental data. Five typical pulse waveforms (-t−2, -t1/2, t0, t1/2, and t2) were used as the research objects. The results demonstrated that the waveform of the continuous double-pulse current plays an important role in improving the teff-hold of TECs under transient working conditions. In this simulation, the pulse widths (τ1 and τ2) of the first and second pulse currents were both 4 s when the pulse initial current was the optimal current (0.6A). When the pulse amplitudes (P1 and P2) were equal and small, the pulse current combination of t0 + t1/2 was applied, which could achieve relatively large teff-hold values. However, -t2 + t1/2 was more appropriate. Additionally, large teff-hold values for TECs could be achieved by reasonably choosing the pulse widths (τ1 and τ2) and amplitudes (P1 and P2) of continuous double-pulse waveforms and the initial current of the second pulse current (Iini,2). It is important to ensure that P1 is not too large, or it may lead to significant increases in Tc,min and Tc,max, a growth in trec, and a reduction in teff-hold. In this study, the optimal teff-hold was achieved (8.38 s) when the pulse current combination was -t2 + t1/2, P1 = 4, P2 = 3, τ1 = τ2 = 4 s, and Iini,2 = 0.55A.

Impact of intrinsic properties and interface contacts on thermoelectric transient supercooling

The thermoelectric transient supercooling is expected to significantly suppress temperature rise in electronic devices since Peltier cooling has the highest response rate among various refrigeration techniques. Conventional researches assume that the better transient supercooling performance is predicated on zT as high as possible, while the power factor is overlooked. And there is a lack of methods to quantify the interface effect on transient supercooling losses. Here, using a series of numerical simulations and experimental methods, we found that the overall transient benefit only depends on the final zT value of materials, but the higher power factor will lead to a shorter response time and lower overshoot temperature. And thermal contact resistance (TCR) and electrical contact resistance (ECR) negatively impact transient supercooling, but the ECR is worse. Finally, the impact of semiconductors-metal interface resistance on transient supercooling was precisely investigated by in-situ measurement and agreed well with the numerical results. The TCR and ECR interface losses lead to an overall transient benefit reduction of 11% and 43% at two times the optimum current and ten seconds pulse width. The present results have an excellent reference for quantifying the effect of intrinsic properties and interface contacts on transient supercooling.

Irreversibility of thermoelectric cooling model: Newtonian linear heat transfer and finite time thermodynamics

The irreversibility of the thermoelectric cooling (TEC) model could analyze and optimize refrigeration performance. In present work, a one-dimensional thermal conductivity model of the internal irreversible thermoelectric element was first established. Subsequently, two categories of evaluation indexes were further analyzed, i.e., cooling capacity Q c and coefficient of performance (COP). The thermodynamic model of TEC was then established jointly with finite time thermodynamics. The influences of current and the ratio of heat transfer areas between cold and hot sides on the steady-state characteristics of TEC were further discussed under Newtonian linear heat transfer. In addition, a transient thermodynamic model of TEC equipment was built to compare the effects of pulse width, pulse gradient, and pulse current on the cold side temperature of the TEC module. Theoretical results demonstrated that the effective working range of the steady-state operating current is 2A ≤ I ≤ 25A, and the area ratio of heat transfer between cold and hot sides is 0.07 ≤ x ≤ 1. The cooling capacity (Q c) and coefficient of performance (COP) could reach up to about 50W and 2.0, respectively. In addition, the increase of external heat transfer capacity of the hot side could reduce the transient supercooling temperature and the peak overshoot temperature, simultaneously. Afterward, the cooling capacity was remarkably enhanced. Current research could provide a reliable support for the optimization design and performance improvement of the thermoelectric cooler. Highlights Thermal conductivity model of internal irreversible thermoelectric element was established; Effects of cold and hot ends on the TEC were discussed under Newtonian linear heat transfer; Effects of pulse width, gradient and current on TEC cold end temperature were investigated; Heat transfer capacity of TEC hot end could reduce supercooling and overshoot temperatures; The irreversibility of TEC model will enhance its potentials for low grade energy exploitation.

Transient supercooling analysis of an I-type thermoelectric cooling element by the experiment

Structure improvement and optimal operating parameters of the thermoelectric cooling component will favor obtaining a lower transient supercooling temperature. The contribution proposed a new I-type thermoelectric cooling structure by connecting the P-type arm with the N-type arm by the sandwich arrangement structure. A one-dimensional transient governing equation is constructed based on the physical model, and the finite volume method provides a numerical solving solution. Moreover, an experimental system is built to analyze the supercooling performance of the I-type cooling layout under different operating conditions. The experimental results validated that the proposed numerical simulation method is feasible and accurate for the component modeling and cooling performance analysis. Moreover, we analyzed the influences of single pulse current amplitude and reference-pulse current groups on the cold end temperature of the I-type thermoelectric cooling structure. The cold end temperature will reach the minimum when the pulse current amplitude is 20 A. Meanwhile, the combined reference-pulse current operation effectively reduces the transient subcooling temperature at the cold end. The experiments show that the minimum cold-end temperature reduces by 31.8 K when the input reference-pulse current waveform is 5 A-20A. In summary, the I-type thermoelectric cooling structure achieves better cooling performance under combined reference-pulse current operating conditions.

Transient thermal behavior of a segmented thermoelectric cooler with variable cross‐sectional areas

In this work, a simulation model based on finite element analysis for a one-dimensional p-type segmented semiconductor element with different cross-sectional areas is analyzed to predict transient thermal behavior for thermoelectric coolers (TECs). The proposed model considers all thermoelectric effects, including the Peltier effect, Thomson effect, Joule heating, and Fourier's heat conduction. The supercooling occurs when a pulse current higher than the steady-state optimum current is applied to a TEC. Dynamic characteristics, such as minimum cold side temperature and holding time of transient state, which arise in transient supercooling are very important to be considered for the design and operation of TECs. In this study, we present a new approach of transient performance based on leg geometry shape, considering variable cross-sectional areas, as well as individual element lengths of two different thermoelectric materials using a segmentation model proposed. A variety of model designs are analyzed, considering rectangular, trapezoidal, and inverse trapezoidal legs, where also optimum pulse current is found as a function of the element lengths. It is demonstrated that an improvement of 4.75% in the cooling is possible when using trapezoidal legs compared with conventional rectangular systems. Our presented model is mainly a new alternative in characterizing Peltier supercooling as a function of both pulse current ratio as well as the leg shape.

Transient supercooling performance of thermoelectric coolers with a continuous double current pulse

In this work, the transient supercooling performance of thermoelectric coolers is numerically investigated via a three-dimensional, multiphysics, and transient model. The particular focus is placed on how to increase the holding time of supercooling state to meet some specific cooling requirements. A new concept of effective cooling temperature line is proposed to evaluate the supercooling performance. On the basis of this concept, the frequently used holding time of supercooling state is replaced by an effective holding time, teff-hold. As a reference baseline, the transient supercooling performance is first evaluated under a single current pulse and the largest teff-hold is determined by searching for the optimal pulse width and amplitude. Subsequently, the supercooling performance and the effective holding time are evaluated under a continuous double current pulse whose shape can be characterized by the pulse width (τ1) and amplitude (P1) of the first pulse, the pulse width (τ2) and amplitude (P2) of the second pulse, and the initial current (Iini,2) of the second pulse. The results show that among the five parameters Iini,2 is the most dominant parameter to achieve a larger teff-hold. Only when Iini,2 is higher than the initial current of the first pulse, employing a double pulse can yield a larger teff-hold than employing the optimal single pulse. The results also demonstrate that with a proper Iini,2 the value of teff-hold can be further prolonged by optimizing τ1, τ2, P1, and P2, and the largest teff-hold for the optimal double pulse is increased by 59.8%, as compared with that of the optimal single pulse. Finally, the principles for selecting parameters of a double current pulse are discussed.

Numerical study on transient supercooling performance of annular thermoelectric cooler

Transient supercooling can effectively improve the performance of thermoelectric coolers (TECs), especially seen from the studies on flat-plate TECs. But for an annular facility, annular thermoelectric coolers (ATECs) are a better choice because their annular structures increase contact areas. For further studying it, a 3D transient model coupling temperature field and electric potential field is established to study the effects of pulse current amplitude (P), pulse width (τ), annular thermoelectric angle (θ), leg length (r) of thermoelectric element, convective heat transfer coefficient (h) and pulse current shape on the transient performance of ATEC made of variable properties materials. The results show that P and τ are still key factors for the said ATEC, let the P is constant, there is an optimal pulse width minimizing the cold end temperature. If the annular thermoelectric angle θ increases, ATEC with a cold end cross-sectional area larger than that of the hot end makes the supercooling poorer. Let the leg lengths r of the ATEC are 0.8 mm and 0.7 mm respectively, the minimum cold end temperatures (Tc,min) are 236.41 K and 246.18 K as the cross-sectional area of the cold end gets smaller but larger than that of the hot end. The larger of the h is, the better the heat will be dissipated from the hot end of the ATEC, till a certain value of h, the transient performance no longer changes significantly for no much more heat could be dissipated. As the study shown, applying different pulse current shapes leads to the acquisition of different supercooling indicators.

Numerical analysis for transient supercooling effect of pulse current shapes on a two-stage thermoelectric cooler

Transient supercooling of TECs (Thermoelectric coolers) is a technology that can effectively strengthen cooling performance. It has been found that the pulse current shape has a significant impact on the enhancement of the transient supercooling. However, previous studies focused on a single-stage TEC or a two-stage TEC with the same square shape applied in the cold stage and hot stage. In this work, a new method is proposed to improve the transient supercooling performance by combining a two-stage TEC with pulse currents of various shapes (−t−2, −t1, −t1/2, t0, t1/2, t1, t2). A three-dimensional multi-physical field coupled transient model is used to verify the efficiency of the method. Pulse shapes, pulse amplitudes and pulse widths are analyzed in detail to reveal their effects on several important transient performance indexes of the two-stage TEC, including the minimum cold end temperature (Tc,min), the overshoot temperature (Tc,max), the holding time for supercooling state (thold), and the time to recover to steady temperature (trec). The results show that when the pulse current shape of one TEC stage is fixed, the Tc,min will become lower if the other stage is given a falling pulse current with larger index or a rising pulse current with smaller index. Moreover, distinct optimal combined shapes are obtained when pursuing different high supercooling performance indexes. The combined pulse current t0 + t0 can be adopted to achieve a lower Tc,min, the combination of −t1/2 + −t1/2 can provide a lower Tc,max and a shorter trec for the two-stage TEC, and the holding time thold can be prolonged by using the combination of t2 + t2.

Transient thermal stress analysis of a thermoelectric cooler under pulsed thermal loading

A three-dimensional, nonlinear, and dynamic finite element model coupling the thermal, electrical, and mechanical fields was established to investigate transient thermal stress of the integrated thermoelectric cooler (TEC) under the pulsed thermal load. The model considered the temperature-dependent material properties. Results showed that pulsed thermal loading reduced the cooling performance of the TEC and caused the dramatic variations of temperature and thermal stress in the device. With a pulse load of 30 W, the chip temperature fluctuation amplitude was about 170 °C, and the fluctuation amplitude of the thermal stress at the contact interface between the chip and TEC reached 686 MPa. The fluctuations amplitudes increased with pulse amplitude and time. Using the TEC′s transient supercooling characteristics to relieve temperature and thermal stress peaks appeared effective, which could be utilized to improve the robustness and functionality of TEC for the electronics cooling.

Experimental study on transient supercooling of two-stage thermoelectric cooler

In this work, an experimental study was conducted to evaluate the cooling performance of two-stage thermoelectric coolers (TECs) under current pulse operation so as to achieve a larger temperature drop. Measured data disclosed that the steady electric current varied from 0.1 to 4.1 A and the optimal current was found to be Iopt = 2 A. The implementation of transient supercooling was found to reduce the minimum temperature from 236.65 to 232.90 K, corresponding the temperature drop of 3.75 K. The results further indicated that the increment of pulse current leads to the reduction of the Tc,min. However, for τ = 5 s, further increment of the pulse current from 4 A to 4.5 A did not lead to the lower Tc,min, which can be attributed to the large magnitude of the current so that the heat sink cannot remove much heat and it would be fed back to the cold side. Thus, the least Tc,min was achieved for 4 A. Regarding the variation of pulse width, it was obtained that Tc,min is decreased tending to a particular value. Accordingly, the least Tc,min of 232.8 K was attained by applying a pulse current with τ = 6 s.

Deep supercooling enabled by surface impregnation with lipophilic substances explains the survival of overwintering buds at extreme freezing.

The frost survival mechanism of vegetative buds of angiosperms was suggested to be extracellular freezing causing dehydration, elevated osmotic potential to prevent freezing. However, extreme dehydration would be needed to avoid freezing at the temperatures down to −45°C encountered by many trees. Buds of Alnus alnobetula, in common with other frost hardy angiosperms, excrete a lipophilic substance, whose functional role remains unclear. Freezing of buds was studied by infrared thermography, psychrometry, and cryomicroscopy. Buds of A. alnobetula did not survive by extracellular ice tolerance but by deep supercooling, down to −45°C. An internal ice barrier prevented ice penetration from the frozen stem into the bud. Cryomicroscopy revealed a new freezing mechanism. Until now, supercooled buds lost water towards ice masses that form in the subtending stem and/or bud scales. In A. alnobetula, ice forms harmlessly inside the bud between the supercooled leaves. This would immediately trigger intracellular freezing and kill the supercooled bud in other species. In A. alnobetula, lipophilic substances (triterpenoids and flavonoid aglycones) impregnate the surface of bud leaves. These prevent extrinsic ice nucleation so allowing supercooling. This suggests a means to protect forestry and agricultural crops from extrinsic ice nucleation allowing transient supercooling during night frosts.

Transient supercooling behaviors of a novel two-stage Peltier cooler

A new design concept is proposed to enhance the transient supercooling performance, in which variable leg cross-sectional areas are introduced into two-stage TECs. Since the cross-sectional area ratios of hot end to cold end for the hot stage (γh) and the cold stage (γc) can be larger, smaller, or equals than/to 1, nine designs with various combinations of γh and γc are investigated. A three-dimensional multiphysics model is used to compare the transient supercooling performance of the nine designs at various pulse amplitudes and widths. The results show that as compared with the conventional TEC with a constant cross-sectional area, the new design significantly reduces the cold-end temperature (Tcc) and employing variable cross-sectional area in the cold stage is more effective for reducing Tcc than that in the hot stage. Design with γh > 1 and γc > 1 yields the lowest Tcc among the nine designs. The reduced Tcc can be attributed to the fact that the variable cross-sectional area makes the current density and thermal resistance asymmetrical such that more Joule heat transports to the hot end due to its larger cross-sectional area. The cooling performance of the design with γh > 1 and γc > 1 is additionally enhanced by increasing γh and γc.

Enhanced Peltier cooling of two-stage thermoelectric cooler via pulse currents

As compared with single-stage thermoelectric coolers (TECs), two-stage TECs can achieve a larger temperature drop when TECs operate at steady state conditions. In this work, pulse currents are respectively supplied to the hot and the cold stage of a two-stage cascaded TEC to seek further increase in temperature drop across the TEC. The results show that the two-stage TEC yields a lower cold end temperature and a longer holding time of supercooling state than the single-stage counterpart. The minimum cold end temperature, Tc,min, can be further improved by the matching between the pulse amplitude and width on the cold stage (Pc and τc) and the pulse amplitude and width on the hot stage (Ph and τh). When Pc and τc are specified, the same Ph (=Pc) and a shorter τh (<τc) are required to yield a lower Tc,min. When the current pulse applied to the hot stage is triggered earlier than that to the cold stage, a lower cold end temperature and a longer holding time are observed. The results obtained in this work should be useful for both design and use of transient supercooling of TECs.

Seasonal variation of freezing resistance mechanisms in north-temperate alpine plants

Alpine herbaceous plants employ two mechanisms to prevent freezing injury to leaves during the growing season, i.e., they avoid freezing by transient supercooling or tolerate the occurrence of extracellular ice. I examined seasonal changes in the two freezing resistance mechanisms in leaves of alpine plants from two temperate mountain regions, Niwot Ridge (3500 m) in the Front Range of the Rocky Mts. (USA) and Modre sedlo (1510 m) in the Giant Mts. (Czech Republic). Although plants from Niwot Ridge were on average ca. 2.6 K more resistant to freezing than plants from the Giant Mts., patterns of freezing resistance changes during the growing season were consistent between the two regions. Both freezing resistance mechanisms, i.e., avoidance by supercooling and tolerance of extracellular ice were encountered. Plants predominantly avoided freezing injury during early summer and tolerated extracellular freezing at the end of the growing season, and the seasonal change from avoidance to tolerance mechanisms was significant for both regions. The avoidance and tolerance mechanisms provided comparable freezing resistance to the plants during early summer but, unlike in avoidant plants, in tolerant plants resistance increased over the course of the season. The species formed three groups with regard to resistance mechanisms employed during the season: (1) species that avoided injury by supercooling during early summer and employed freezing tolerance towards autumn; (2) species that were freezing tolerant during the entire growing season; and (3) species with ephemeral shoots that solely avoided freezing by supercooling. Although freezing tolerance is a common feature in north-temperate alpine plants, avoidance by supercooling as a freezing resistance strategy might be selected for in species with particular life histories such as ephemeral hemiparasites.

Enhancement of maximum temperature drop across thermoelectric cooler through two-stage design and transient supercooling effect

In this work, a new design concept which combines two-stage design with transient supercooling effect is proposed to enhance the maximum temperature drop across thermoelectric coolers (TECs). A three-dimensional, multiphysics, and transient model is used to examine the design effectiveness. Step current pulses with various amplitudes (P) and widths (τ) are supplied to the two stages of a two-stage TEC in series. The results show that, as compared with the single-stage counterpart, a significant improvement in the maximum cold-end temperature drop (ΔTc,max) is observed for the two-stage TEC. Meanwhile, the new design also greatly reduces the temperature overshoot (Tc,max) and increases the holding time of supercooling state (Δthold). Subsequently, effects of the pulse amplitude, width, and shape are discussed and two important geometry parameters: the cross-sectional area ratio of p-type leg to n-type leg and the leg length ratio of cold stage to hot stage are investigated. These results confirm that ΔTc,max, Tc,max, and Δthold can be further improved by optimizing the pulse and the geometry parameters. This work provides a feasible cooling approach for some specific cooling targets, such as mid-infrared laser gas sensors or any other semiconductor devices which require a temporary but large temperature drop.

Improvement of transient supercooling of thermoelectric coolers through variable semiconductor cross-section

In this work, a new design of thermoelectric cooler (TEC) with variable semiconductor cross-sectional area is proposed to improve its transient supercooling characteristics. Four key evaluation indicators of transient supercooling for the conventional and new designs, including the minimum cold end temperature, maximum temperature overshoot, holding time of transient state, and recovery time ready for next steady-state, are examined and compared by a three-dimensional, transient, and multiphysics model. Two additional effects are observed in the TEC with variable semiconductor cross-sectional area. First, the variable cross-sectional area makes the thermal circuit asymmetric, so that Joule heat is preferentially conducted toward to the end with a larger cross-sectional area. Second, more Joule heat is produced close to the end with a smaller cross-sectional area. The present simulations find that these two effects can be utilized to achieve the desired evaluation indicators by changing the cross-sectional area ratio of hot end to cold end. When a lower cold end temperature, a smaller temperature overshoot, and/or a longer holding time are/is required, a larger cross-sectional area at the cold end is recommended. However, to achieve a shorter recovery time, a smaller cross-sectional area at the cold end is needed.

Optimal pulse current shape for transient supercooling of thermoelectric cooler

This work developed a three-dimensional, multiphysics, and transient model to investigate transient supercooling of TECs (thermoelectric coolers). The model coupled the heat and electricity conductions and considered all thermoelectric effects. The model was well validated by dynamic test data for a TEC (Thermoelectric cooler) start-up process and was compared with the previous heat conduction model. The comparison confirmed that the multiphysics model has a more superior performance for predicting the key evaluation parameters of the transient supercooling. Then the model was used to investigate various current pulses (t0, t1/2, t1, t2, t3, t4, and t5) to search for the optimal shape. The results showed that the optimal shape is only determined by the time to reach the minimum cold end temperature (tmin) and the pulse width (τ). For the pulses with tmin < τ, a higher power pulse provides a lower cold end temperature, for the pulses with tmin = τ, however, the trend is opposite. The present results reasonably explain the divergence for the optimal pulse shape reported by the previous studies.

Local and bulk melting of shocked columnar nanocrystalline Cu: Dynamics, anisotropy, premelting, superheating, supercooling, and re-crystallization

We perform large-scale molecular dynamics simulations to study shock-induced melting transition of idealized hexagonal columnar nanocrystalline Cu. The as-constructed nanocrystalline Cu consists of unrotated (reference) and rotated columnar crystals, relative to the columnar axis. Shock loading is applied along three principal directions of the columnar Cu: two transverse (zigzag and armchair) and one longitudinal directions. Dynamic local melting processes are highly anisotropic with respect to the shock directions. For the transverse directions, hotspot effect and disparate dynamic responses of grains with different orientations may lead to partial or complete premelting of the initially rotated grains, which in turn leads to transient supercooling and heterogeneous recrystallization, and thus, the formation of nanocrystalline solids with modified grain structures or solid-liquid mixtures, depending on the extent of supercooling. With increasing shock strengths, the reference grains melt heterogeneously at interfaces and homogeneously inside. Conversely, "bulk" premelting of the rotated grains is absent for the longitudinal direction, except for grain boundary melting. The progression of recrystallization or heterogenous melting diminishes and eventually eliminates the transient premelting or superheating of the system via latent heat and thermal diffusion. Premelting or superheating appears unlikely for bulk melting or well-defined Hugoniot states, if the thermal and mechanical equilibria are achieved, and the thermodynamic melting curve coincides with the partial melting Hugoniot states of a polycrystalline solid.