Derivation of generalized thermoelectric energy equations and the study of thermoelectric irreversible processes based on energy, exergy, and entransy analysis

Thermoelectric (TE) generation is becoming a valuable and promising research direction. Many researchers have carried out system analysis and performance optimization of thermoelectric technologies based on the generalized thermoelectric energy balance equations. However, it is assumed that TE legs have no heat exchange with the ambient except at the junctions of the hot and cold ends where heat flows in and out. Based on basic thermoelectric effects and fundamental theories of heat transfer, a detailed derivation of the revised generalized thermoelectric energy equations considering convective heat transfer between TE legs and the ambient has been carried out. Irreversible heat transfer processes have been analyzed by employing energy analysis based on the first law of thermodynamics and exergy analysis based on the second law of thermodynamics. The results show that convective heat transfer leads to a decrease in both energy and exergy efficiencies: the rate and magnitude of the decrease in exergy efficiency are greater than those of the decrease in energy efficiency. The exergy efficiency is relatively high despite the low energy efficiency in operation, revealing the features and advantages of thermoelectric generators (TEGs) in low‐grade energy utilization. For TEG efficient operation, the load resistance value should match the system's internal resistance, or at least be greater than that, to avoid a sharp drop in power output and efficiencies. In an attempt at theoretical analysis, the concept of entransy was first introduced into thermoelectric analysis, yielding two concise relational equations which reflect the intrinsic link between Carnot cycle efficiency, energy efficiency, exergy efficiency, and entransy flow transfer efficiency. The entransy analysis based on the index of entransy flow transfer efficiency, together with energy analysis and exergy analysis, may be a novel and valuable guideline for the operation and optimization of TEGs, which needs to be further investigated.


| INTRODUCTION
Thermoelectric (TE) power generation is an all-solidstate energy conversion technology that uses thermoelectric materials to convert thermal energy directly into electrical energy.The compact structure of the thermoelectric generator (TEG) and the absence of rotating parts make it quiet, reliable, and basically maintenance-free.With no piping and chemical fluids, the thermoelectric generator is very environmentally friendly. 1Because of these advantages and the current industrial background of low carbon and energy saving, thermoelectric power generation technology is becoming a valuable and promising research direction. 2owever, the widespread application of thermoelectric power generation technology is limited by the low thermoelectric conversion efficiency of existing thermoelectric materials. 3Bell 4 pointed out that there are two paths to promote the application of thermoelectric components.One is what materials scientists are working on-improving the performance of thermoelectric materials.The other is from a system perspective, that is, how to use, configure, or integrate thermoelectric components.Chen 5 presented a solution for improving system-level efficiency by integrating thermoelectric generators with combined heat and power (CHP) units.Since there is no breakthrough in thermoelectric materials for the time being, designing a system from the perspective of cascade utilization of multiple energy sources has proven to be a viable approach to current thermoelectric power generation technology.
Research on thermoelectric power generation technology is extensive, and in addition to engineering applications, system analysis, and optimization are also important aspects.Rowe 6 developed a program for evaluating the performance of thermoelectric modules (core components of TEG).Chen 7 used irreversible process thermodynamics to construct a model to study the performance of thermoelectric generators with external and internal irreversibility.In their analyses, they all viewed TEGs as devices composed of a single thermoelectric pair.However, the actual thermoelectric modules are composed of several thermoelectric pairs.In other words, the object of study should be a multielement TEG rather than a single-element device.Accordingly, Chen 8,9 and Pan 10 conducted system analysis and optimization studies on multielement TEGs.
Many researchers [11][12][13][14] have carried out system analysis and performance optimization of thermoelectric technologies based on the so-called generalized thermoelectric energy balance equations.Even in the study of the performance of thermoelectric refrigerators 15 and the structural optimization of thermoelectric modules, which are the core components of the cooling system of photovoltaic devices, 16 the generalized thermoelectric energy balance equations are still the basis of research.The above-generalized equations take into account the irreversibility inside the thermoelectric device due to the Joule heat and the heat conduction inside the TE couple leg.However, it is assumed that the TE couple leg has no heat exchange with the ambient, that is, they are thermally insulated, except at the junctions of the hot and cold ends where heat flows in and out. 12ince thermoelectric generators are composed of multiple pairs of TE legs, should the heat exchange between the TE legs and the environment be neglected as the size of the device increases (using an increasing number of TE modules)?This is a question that deserves to be studied and evaluated.In this paper, a detailed derivation of the thermoelectric energy equations will be presented, taking into account the convective heat transfer between the TE legs and the ambient.Furthermore, based on the newly derived energy equations and the analysis of the first and second laws of thermodynamics, the performance of a multielement TEG, especially its multiple irreversibility, has been investigated.
There have been numerous studies on the modeling and analysis of thermoelectric elements in terms of several aspects in recent years, represented in particular by a book edited by Christophe Goupil 17 which provides a solid foundation for thermoelectric element and module design in the technical development process.To have a more comprehensive and detailed understanding of the various thermoelectric fundamental effects, and on this basis, to introduce irreversible convective heat transfer and to analyze its impact on the performance of the TEG, it is necessary to give a detailed derivation of the thermoelectric energy equations and an introduction to the various thermoelectric fundamental effects.The main innovation of the work in this paper is that the convective heat transfer between the TE legs and the ambient, which is usually neglected, is taken into account, derived, and analyzed.More innovatively, the concept "entransy" proposed by Guo 18 for the analysis of heat transfer processes is first introduced into the analysis of thermoelectric transfer processes.
Entransy theory, a method used for the optimization of heat transfer processes, has received some developments and applications in recent years. 19This paper attempts to integrate the efficiency analysis of energy conservation (the first-law-based thermodynamic analysis), the exergy analysis of energy grade (the second-lawbased thermodynamic analysis), and the specially introduced "entransy" analysis which aims to evaluate the heat transfer capability and efficiency, to explore the intrinsic mechanism and unique characteristics of thermoelectric conversion compared with traditional energy conversion from a multidimensional perspective, and to establish a multidimensional evaluation system for thermoelectric conversion that can fully reflect the above connotations, so as to provide a guide for the evaluation and optimization of thermoelectric power systems.Since no literature has been published on the entransy analysis of thermoelectric systems, the work in this paper is exploratory and cutting-edge, and is a continuation of the authors' previous work. 20

| SYSTEM MODELING AND EQUATIONS DERIVATION
A typical thermoelectric generator, operating between a high-temperature heat reservoir at T 1 and a lowtemperature heat reservoir at T 2 , consists of multiple thermoelectric couples, as shown in Figure 1.Each thermoelectric couple consists of a P-type semiconductor leg and an N-type semiconductor leg.All thermoelectric couples are connected together in series and/or parallel through a circuit and packaged into a thermoelectric module, that is, a thermoelectric generator with a load resistor R L connected in series.
The Peltier effect is one of the thermoelectric effects, which occurs at the junctions of TE couples and manifests itself as heat absorption at the hot-end junction (Q ph ) and heat release at the cold-end junction (Q pc ).The Peltier heat can be expressed as follows: where α P and α N are the Seebeck coefficients of the P-and N-type thermoelectric materials, and α α α = − P N .I is the current in the circuit, which is the direct cause of the Peltier effect, and T h and T c indicate the temperature of the hot and cold sides of the TEG.
The individual thermoelectric couple is analyzed next, as shown in Figure 2. The heat flow in P-and N-type legs are represented by Q p and Q n , respectively.ρ p and ρ n are the electric resistivities, and k p and k n are the thermal conductivities.In the case where the operating temperature range is not too large, the physical parameters of the thermoelectric material can usually be considered temperature-independent.To simplify the model and match the actual production, the length of each TE leg, represented by L, and other geometric dimensions are considered to be identical.It should be noted in particular that the cross-sectional area of the TE leg is represented by S p and S n .
As mentioned in Section 1, the convective heat transfer between the TE legs and the ambient will be considered.A detailed discussion of the heat loss from the TE legs to the ambient can be found in the author's previous work. 21Relevant theoretical analysis 22 and experimental study 20 can also be referred to.
Consequently, based on the principle of energy conservation, the energy control equation within the infinitesimal cell (microelement with a length dx) of the thermoelectric leg (in the case of P-type) in steady state can be written as Then, Equation (3) can be changed to where h is the convective heat transfer coefficient, T 0 represents a fixed ambient temperature value, and P is the circumference of the TE leg.Q p and T p represent the conductive heat flow and the temperature value inside the leg varying with the coordinate x.The rightmost two terms of Equation ( 4) refer to the Joule heating rate and the convective heat transfer rate, respectively, which are equivalent to playing the effect of heat source and heat sink.The left-most term is the net increase in heat flow along the x-coordinate direction from position x to x + dx.
Integrating Equation (4) from 0 to x yields Here, it is assumed that T p (x) is distributed linearly.Although in practice, due to the reversible heat-to-work conversion and the irreversible Joule heat loss, the temperature within a TE leg cannot be linearly distributed.Considering the convenience of integral processing and that the error should be acceptable when the temperature difference is not large, there is assumed to be a linear temperature distribution.A comparison of the mathematical calculation results and the numerical simulation results will be made later to verify the reasonableness of the assumption.Thus, it is obtained that Bringing the boundary condition at position L into Equation ( 5), the following equations can then be derived, It is easy to see that the second term to the left of the equal sign of Equation ( 8) is equal to the total amount of Joule-generating heat in a TE leg, and the third term is equal to the total amount of convectively dissipated heat in a TE leg.Due to the assumption of a linear distribution of temperature within a TE leg, the third term can be equated to the total amount of convective heat transfer between the surface, with temperature everywhere equal to (T h + T c )/2, of a TE leg and the ambient with temperature T 0 .It is helpful to understand the total amount of convectively dissipated heat in a TE leg (the third term) in this way to simplify the integration process later on to arrive at a concise expression.However, it should be noted that the above "can be equated to" is only valid for the entire TE leg length (i.e., x = L), not at any other position (any x except L) of the TE leg; after all, the temperature at x is T p (x), not T c .Don't forget the expression of Fourier's law in this scenario reflecting the relationship between the conductive heat flow and the temperature gradient: Thus, Equation ( 7) can be replaced by Integrating the above equation yields Note the boundary condition at the cold (hot) side of the TEG, It is not difficult to obtain the following equations, It is easy to see that the second term to the right of the equal sign of Equation ( 14) is equal to half of the total amount of Joule-generating heat in a TE leg, and the third term should be equal to half of the total amount of convectively dissipated heat in a TE leg.
Then, Equation ( 14) should be rewritten by Comparing Equations ( 14) and ( 15), it can be concluded that Equation ( 17) is actually the result of the assumption mentioned above, that is, due to the assumption of a linear distribution of temperature within a TE leg, the total amount of convectively dissipated heat in a TE leg can be equated to the total amount of convective heat transfer between the surface, with temperature everywhere equal to (T h + T c )/2, of a TE leg and the ambient with temperature T 0 .
On the other hand, Equation ( 17) can be derived by giving a specific expression for the temperature distribution T p (x) assumed to be linearly distributed where the coefficients a and b are determined by the following boundary conditions: The above method involves only mathematical techniques and does not require extra-textual explanations.Still, the explanation, which seemed somewhat redundant, is necessary to make the physical meaning clearer.After all, this is about the description of a physical scenario, not simply a demonstration of a math trick.
Consequently, Equation ( 15) is rewritten by where K p and R p refer to the thermal conductance and the internal electric resistance, respectively, and The equation derived is obviously not concise enough, so Q pconv is employed to represent the convective heat transfer heat flow in place of the complicatedlooking term in Equation ( 21), Thus, a concise expression is derived, The heat flow at position L can be obtained from Equation (8) Based on the same assumptions and derivations, the expression for the heat flow Q n in the N-type TE leg can be obtained.Consequently, the equation for a TEG consisting of a single pair of thermocouple leg is where and Similarly, we define After integration of the Peltier effect expressed by Equations ( 1) and (2), Equation ( 27) can now be replaced by the generalized thermoelectric energy balance equation for a TEG consisting of a single pair of thermocouple legs, as follows: It is important to note that Equation (31) is the equation that takes into account the convective heat transfer between the TE legs and the ambient, which should be called the revised generalized thermoelectric energy balance equation.][13][14][15][16] It should be noted that the effect of convection has been considered in several studies and the topic has been analyzed already to a different extent. 23Apart from this, there are also recent works on this topic. 24Some works have been carried out in detail and depth, and are theoretically rigorous.The work in this paper only considers exploring the derivation of generalized formulas that are easy to compute and analyze, starting from the fundamental effects of thermoelectricity from a practical engineering point of view (employing a linear temperature distribution).Comparison of the formula calculation results and the numerical simulation results will be made later, it will be found that the above method is effective and the error is small under specific working conditions.Although the work in this paper is relatively rough, it is no less practical and operable, and is informative and useful for analyzing and calculating specific engineering problems.
In fact, in addition to the internal irreversible heat transfer process, there is also an external irreversible heat transfer process between the thermoelectric generator and the high-and low-temperature heat reservoirs.It is easy to write the expression for the total heat flow (Q H and Q C ) of a thermoelectric generator consisting of multiple pairs of thermoelectric legs, as follows: where n is the number of pairs of TE legs, k 1 and k 2 refer to heat transfer coefficients between the thermoelectric generator and heat reservoirs which included interface effects such as contact thermal resistance, and S 1 and S 2 represent the external heat transfer areas of the thermoelectric generator.

| Energy transfer processes and efficiency analysis
It is well known that thermoelectric power generation involves four macroscopic energy transfer processes, which are revealed by the energy balance equations derived above.Specifically, they are, respectively, the Joule effect, the Peltier effect, heat conduction, and convective heat transfer.
Peltier effect, as one of the basic thermoelectric effects, occurs at the junctions of TE couples and manifests itself as heat absorption at the hot-end junction, expressed by Equation (1), and heat release at the cold-end junction, expressed by Equation (2).
Joule effect, or Joule's law, as one of the basic theories of electricity, is macroscopically equivalent to an internal heat source effect occurring everywhere inside the thermoelectric generator, especially throughout the TE leg.From Equation (31) it seems to follow that half of the Joule heat flow generated by the TE leg is directed to the hot end and the other half is transferred to the cold end.In fact, as can be seen from the detailed derivation of the previous section, the factor 0.5 in the Joule term 0.5I 2 R is a product of integration, which is essentially a result of the assumption of homogeneous material properties rather than an approximation based on expediency.
Chen 12 hit the nail on the head by pointing out that the Joule heat generated in TE leg is nothing but heat conduction, which is transferred as a whole from the hot end to the cold end following Fourier's law, only the value of the heat flow differs from place to place.Similarly, the factor 0.5 in the convective dissipation term 0.5Q conv is also determined for the same reasons.Next, the effect of convective heat transfer is analyzed by describing the performance of the thermoelectric generator.
The power output of the thermoelectric generator consisting of multiple pairs of TE legs can be expressed as where T Δ is the temperature difference between the hot and cold ends, and T T T Δ = − h c .Power output can also be written in terms of voltage output as Consequently, according to the first law of thermodynamics, the energy efficiency of the TEG system is defined as The effect of convective heat transfer on TEG performance can be clearly understood from Equation (35).
On the one hand, convective heat transfer results in the direct dissipation of a portion of the heat that should have been transferred to the cold end and which could have been partly converted to electricity.On the other hand, the presence of the convective dissipation term 0.5Q conv in the denominator of Equation ( 35) directly leads to a reduction in the energy efficiency of the TEG system.
The convective heat transfer does not affect power output according to Equation (33) or (34).The reason stems from the difference in the boundary conditions assigned.Specifically, in this paper, for the convenience of computational analysis, the boundary condition of given temperature is assigned to both ends of the TEG.Once the temperature or temperature difference is determined, the thermoelectric effect is also determined.In reality, the convective heat transfer certainly affects the temperature distribution (and thus the temperature difference), regardless of degree, eventually leading to a reduction in power output.However, a given heat flux boundary condition rather than a given temperature boundary condition is required to represent the above coupling effect.For the boundary condition of given heat flux, the convective heat transfer need to be calculated iteratively to affect the temperature distribution and thus the power output.In a sense, it can also be argued that this is due to a nonlinear temperature profile.Qualitatively, if the temperature difference is kept constant, the heat flux is greater when convective heat loss is considered than when it is not considered; If the heat flux remains constant, the temperature difference is smaller when convective heat loss is considered than when it is not considered.As mentioned above, the work in this paper considers to explore the derivation of generalized formulas that are easy to compute and analyze, starting from an engineering practical point of view.That is, the effect of convective heat transfer on temperature (and thus power output) is already implicitly included in the given boundary temperature.
Based on the derived theoretical equations, it is now possible to study quantitatively the effect of each irreversible process, especially convective heat transfer, on the performance of the TEG system.The following geometrical dimensions and material properties obtained from typical commercial TE modules have been used in the study: number of TE couples n = 31, uniform TE leg dimensions: 2 × 2 × 3 mm, electric resistance R = 0.086 Ω, thermal conductance K = 3.84 × 10 −3 WK −1 and Seebeck coefficient α = 9.6774 × 10 −4 VK −1 .
As mentioned in the derivation of thermoelectric energy equations considering the convective heat transfer between TE legs and the ambient, it is necessary to compare the results of the mathematical calculation with the results of the numerical simulation to verify the reasonableness of the assumption of a linear distribution of temperature.A numerical simulation was carried out for a pair of TE legs, focusing on the convective heat transfer between the leg surface and the ambient, and the results were compared with the results calculated through Equation ( 30), as shown in Figure 3.Note that the results shown are cumulative values for a typical commercial TE module (containing 31 pairs of TE legs).As shown in Figure 3, the calculated convective heat transfer increases linearly with increasing hot-end temperature, which is a result that can be clearly seen from Equation (30).The simulated convective heat transfer, on the other hand, increases exponentially with the increase of the hot-end temperature, although this exponential increase is not obvious from the curve and is closer to a linear increase.In general, the simulated convective heat transfer is always slightly larger than the calculated one, and this difference gradually increases with increasing temperature.However, in the scenario of low-temperature or low-grade heat utilization, the difference between the simulated and calculated values is acceptable, specifically, in the calculation range of this paper, the difference is controlled within 5%.This suggests that a linear distribution of temperature is an acceptable and reasonable assumption for lowtemperature or low-grade heat utilization.The nonlinear increasing (although not obvious) simulated values are always larger than the calculated ones revealing that the temperature inside the TE leg is not strictly linearly distributed, but is slightly higher than the assumed linear distribution temperature everywhere.
As previously mentioned, the work in this paper has been carried out from an engineering practical point of view and is relatively rough.It must be emphasized the fact that the calculated and simulated temperature profiles coincide does not validate the linear assumption is valid with all thermoelectric materials as each thermoelectric material behaves differently. 25Temperature profile and therefore the error between calculated (based on a linear distribution) and simulated temperature profiles vary widely among different material classes for different temperature ranges and the one considered in this paper is only one case that happens to have less error.Typical values for bismuth telluride thermoelectric materials are used in this work, in other words, the work in this paper only verifies that a linear temperature distribution in typical bismuth telluride thermoelectric materials, which have good performance at room temperature, is valid (with less error) in the scenario of low-temperature or low-grade heat utilization.The properties of thermoelectric materials are very complex, and the linear assumption is certainly the simplest and most popular treatment for engineering calculations.Follow-up studies plan to examine the differences in the properties of different thermoelectric materials and their applicable temperature ranges, that is, the applicable conditions (material type and operating temperature) for the linear assumption (or other practical and operable treatments) need to be clearly defined.Such work will be of value for engineering applications.
A quantitative analysis of the impact of convective heat transfer on energy efficiency allows further evaluation of the share of irreversible heat transfer to prove the necessity of considering irreversible convective heat transfer.As shown in Table 1, when the coefficient of convective heat transfer is greater than 10 Wm −2 K −1 , the effect of irreversible heat transfer on energy efficiency becomes so large as to be non-negligible.In general, the effect of irreversible heat transfer may not be considered when the TEG is indoors or in a natural convection environment, but it is necessary to consider it when outdoors or in a forced convection environment.Special attention should be paid to the fact that only the scenario of a small TE module is considered here.For large TEG equipment, due to the larger volume and surface area of thermoelectric material, the impact of irreversible convective heat transfer will be even greater, and it is important to take care of thermal insulation.Table 1 also shows that the higher the temperature, the smaller the effect of irreversible heat transfer; when using TEG for low-temperature thermal energy utilization, particular attention should be paid to reducing the effect of convective heat transfer.

| Exergy analysis
Due to the limitation of thermoelectric materials, the efficiency of thermoelectric power generation technology is low compared with the efficiency of conventional heat engines.However, because of its unique performance, the thermoelectric generator has irreplaceable advantages in some special occasions, especially in utilizing low-grade thermal energy such as low-temperature waste heat.For this reason, second-law-based thermodynamics analysis (exergy analysis) is applied to make a more indepth analysis to reveal the unique advantages and operation laws of TEGs.
The first law of thermodynamics considers only the quantity of energy, while the second law of thermodynamics focuses on the difference in the quality of energy, which leads to the concept of exergy.Energy is classified into different grades based on its ability to be converted into other energy, and exergy is the concept used to measure how much of energy can be converted into other energy.For example, electrical energy is high-grade energy because it can be converted entirely into mechanical or thermal energy spontaneously without any cost, that is, electrical energy is all exergy.Thermal energy, on the other hand, cannot be converted completely into mechanical or electrical energy spontaneously.It is sub-grade energy, which contains the amount of exergy (the amount of thermal energy that can be converted to other energy) limited by the second law of thermodynamics, and this is the reason why there is an upper limit to the efficiency of heat engines.
The efficiency of an ideal heat engine cycle is expressed in terms of Carnot cycle efficiency in thermodynamics: where, T h refers to high-temperature heat source temperature, and T c represents low-temperature heat source temperature.Therefore, based on the above analysis, the share of exergy in thermal energy can then be measured by Carnot cycle efficiency.Naturally, the amount of exergy in thermal energy can be expressed by the following equation where Q H is the amount of thermal energy input from high-temperature heat source.Since energy efficiency η cannot reflect differences in the grade or convertibility of energy, consequently, the concept of exergy efficiency, which can quantitatively measure energy convertibility, is proposed, as expressed by ( ) To study the effect of irreversible convective heat dissipation on the performance of thermoelectric generators, the convective heat transfer coefficient h can be used as a variable to analyze the consequent changes in TEG performance, in a selected baseline state (R L = 2.7 Ω equal to the internal resistance, T 0 = 298 K, T c = 313 K).Although the power output P out seems to be unaffected by h as found from Equation (34), the energy efficiency η and exergy efficiency η x are affected by h and their differences can be analyzed by comparison.
As shown in Figure 4, convective heat transfer between TE legs and the ambient does lead to a decrease in both energy efficiency and exergy efficiency.And the rate and magnitude of the decrease in exergy efficiency are greater than the rate and magnitude of the decrease in energy efficiency, especially when convective heat transfer is in the natural convective heat transfer stage (h is within the range of 0-10 Wm −2 K −1 ).The energy efficiency can hardly be seen to be significantly decreased, which indicates that the share of energy loss due to irreversible convective heat transfer is small, but the exergy efficiency caused by the convective heat transfer is relatively significantly decreased.This suggests that the presence of convective heat transfer (even if its amount is small) has a nonnegligible effect on system performance, especially on exergy efficiency.It is obvious, illustrated in Figure 4, that in the scenario of low-temperature thermal energy utilization by TEGs, the T A B L E 1 Effect of h on TEG energy efficiency.exergy efficiency is relatively high despite the low energy efficiency, reflecting the important characteristics of TEGs in the utilization of low-grade energy.In addition, consistent with the usual characteristics of thermoelectric generators, both system energy efficiency and exergy efficiency are improved as the temperature of the high-temperature heat source increases (temperature difference increases).
As a special class of batteries, the TEG has the voltage-current characteristics of general-purpose batteries, as shown in Figure 5.And it is easy to find from Figure 6 that the TEG also has a feature of impedance matching: when the load resistance value is equal to the internal resistance value (about 2.7 Ω), the power output reaches the maximum.In addition, the effect of load characteristics on the efficiency of the TEG system can be illustrated in Figure 6.Obviously, characteristics of impedance matching also apply to efficiencies, with the slight difference that the maximum efficiencies occur when the load resistance value is slightly greater than the internal resistance value of 2.7 Ω.Moreover, it is particularly important to note that once the load resistance value deviates from the matched one, especially when the load resistance value is smaller, the rate and magnitude of the decrease in exergy efficiency is much greater than that of the decrease in energy efficiency.The above characteristics are essential for the application of TEGs.

| Entransy analysis
The analysis of efficiency is founded on energy conservation principles, whereas the examination of exergy relies on the differences in energy grades, that is, the convertibility of energy.Nevertheless, both approaches fall short in tackling the problem posed by energy transferability.Based on the analogy between electrical and heat conduction, a new physical quantity known as entransy is proposed. 18The entransy of an object reflects its ability to transfer thermal energy, similar to how the electrical energy in a capacitor relates to the charge transfer ability.An analysis of the heat transfer from an object indicates that the entransy, akin to the electric potential energy in a capacitor, exhibits the nature of "energy."Objects can be characterized as thermal capacitors, capable of storing both heat and "thermal potential energy."It is widely accepted that electric charge is non-dissipative and remains constant during conduction, yet electric energy is partially transformed into thermal energy due to electric resistance.The dissipation of electric energy is a measure of irreversibility for electric conduction that is unrelated to the conversion of electric energy and other types of energy, except for thermal energy.Heat transfer is always accompanied by entransy transfer.However, while thermal energy is conserved, entransy is not due to dissipation.The concept of entransy dissipation can define the efficiency for heat transfer processes and optimize these processes.The dissipation of entransy during the transfer of thermal energy is a quantification of the irreversibility of the heat transfer process.The concepts of entransy and entransy dissipation were used to develop the extremum principle of entransy dissipation for the optimization of heat transfer.When the boundary heat flux is fixed, conduction process optimization occurs when the entransy dissipation is minimized, while when the boundary temperature is fixed, optimization is achieved through maximization of the entransy dissipation. 26While the concept of entransy has been proposed for over a decade, and its analytical theory has been developed and applied to some extent, 19,27 no research on the application of entransy analysis to thermoelectric energy transfer has been published.Hence, this section of the work represents a preliminary attempt and exploration.
In this study, since the heat involved is in the form of heat flow (Q H and Q C ), it is more convenient to use entransy flow instead of entransy.The entransy flows at the hot and cold ends of the thermoelectric generator are defined, respectively, as follows: Thus, the entransy flow transfer efficiency η e of the TEG can be defined as the ratio of the cold-end output entransy flow to the hot-end input entransy flow, where E ˙ϕ is the entransy dissipation flow.
To provide a concise and intuitive understanding of the various concepts and their relationships related to entransy based on Equation (41), consider heat conduction in a large flat plate as an extreme example, that is, one-dimensional steady-state heat conduction.Note a large flat plate doesn't have to be very large, as long as it's thin enough, it can be considered as a large flat plate.Heat conduction in a large flat plate can be approximated as being one-dimensional since heat conduction through a large flat plate will be dominant in one direction (through the plate) and negligible in other directions (within the plate).Consequently, input heat flow at the hot side of the large flat plate is equal to output heat flow at the cold side, that is, Q H = Q C , and the physical meaning of Equation (41) becomes obvious: the greater the temperature drop, the greater the entransy dissipation flow, and the lower the entransy flow transfer efficiency.
Consider the following expression for energy efficiency η, Then, Equation (41) can be rewritten as It can be found from Equation ( 43): (a) when energy efficiency is constant, the lower the temperature drop, the lower the entransy dissipation flow, the greater the entransy flow transfer efficiency; (b) when temperature drop is constant, the lower the energy efficiency, the lower the entransy dissipation flow, the greater the entransy flow transfer efficiency.Point (a) is well understood, similar to the flat plate heat conduction problem; point (b) seems to make energy efficiency "contradict" entransy flow transfer efficiency, which is due to the fact that energy efficiency itself couples temperature drop, but this provides a pair of "contradictory" parameters that can be used for dealing with the optimization problem, which is one of the elements that the authors plan to further develop in their subsequent work.
As shown in Figure 4, as the irreversible loss increases, the energy efficiency gradually decreases slightly, while the entransy flow transfer efficiency gradually increases slightly.It should be noted that as the hot end temperature (or temperature difference between the two ends) increases, both energy efficiency and exergy efficiency increase nearly linearly, but the entransy flow transfer efficiency decreases, and the magnitude of the decrease is significantly larger than that of the increase in energy efficiency and exergy efficiency, which means the entransy flow transfer efficiency is more sensitive to temperature changes.For a power system, the characteristic of impedance matching is still valid for entransy flow transfer efficiency, which is illustrated in Figure 6.However, the entransy flow transfer efficiency reaches the minimum when the load resistance is matched to the internal resistance, rather than the maximum like power output, energy efficiency, and exergy efficiency.And as with energy efficiency, as long as the load resistance is large XIAO | 49 enough, the entransy flow transfer efficiency remains at an almost constant extreme value.
In addition, according to the defining equation of Carnot cycle efficiency, that is, Equation (36), entransy flow transfer efficiency, Equation (43), can then be rewritten as ).
e C a r n o t (44) It is no coincidence that exergy efficiency can also be expressed, according to Equation (35), by the same two parameters: energy efficiency η and Carnot cycle efficiency η Carnot , as follows η η η = .
x Carnot (45) Equations ( 44) and (45) are very concise, and importantly, reflect the intrinsic link between Carnot cycle efficiency, energy efficiency, exergy efficiency, and entransy flow transfer efficiency.The authors plan to dig further into the efficiency analysis (based on energy efficiency), the exergy analysis (based on exergy efficiency), and the entransy analysis (based on entransy flow transfer efficiency), explore the intrinsic mechanism and unique characteristics of thermoelectric conversion and establish a multidimensional evaluation system for thermoelectric conversion.

| CONCLUSION
Based on basic thermoelectric effects and fundamental theories of heat transfer, a detailed derivation of the revised generalized thermoelectric energy equations considering convective heat transfer between TE legs and the ambient has been carried out.Irreversible heat transfer processes have been analyzed by employing energy analysis based on the first law of thermodynamics and exergy analysis based on the second law of thermodynamics, and the results show that convective heat transfer between TE legs and the ambient leads to a decrease in both energy efficiency and exergy efficiency: the rate and magnitude of the decrease in exergy efficiency are greater than the rate and magnitude of the decrease in energy efficiency.
It is found that the exergy efficiency is relatively high despite the low energy efficiency in operation, revealing the features and advantages of TEGs in low-grade energy utilization.For efficient operation of the TEG system, the load resistance value should match the system's internal resistance, or at least be greater than that, to avoid a sharp drop in system power output and efficiencies.
In an attempt at theoretical analysis, the concept of entransy was first introduced into thermoelectric analysis, yielding two concise relational equations which reflect the intrinsic link between Carnot cycle efficiency, energy efficiency, exergy efficiency, and entransy flow transfer efficiency.The entransy analysis based on the index of entransy flow transfer efficiency, together with energy analysis and exergy analysis, may be a novel and valuable guideline for the operation and optimization of thermoelectric generators, which needs to be further investigated.

F I G U R E 4
Effect of h and ΔT on TEG efficiencies characteristics.F I G U R E 5 Effect of R L on TEG voltage and current characteristics.F I G U R E 6 Effect of R L on TEG power output and efficiencies characteristics.