## Abstract

Entropy generation is recognized as a common measurement of the irreversibility in diverse processes, and entropy generation minimization has thus been used as the criterion for optimizing various heat transfer cases. To examine the validity of such entropy-based irreversibility measurement and its use as the optimization criterion in heat transfer, both the conserved and non-conservative quantities during a heat transfer process are analysed. A couple of irreversibility measurements, including the newly defined concept *entransy*, in heat transfer process are discussed according to different objectives. It is demonstrated that although thermal energy is conserved, the accompanied system entransy and entropy in heat transfer process are non-conserved quantities. When the objective of a heat transfer is for heating or cooling, the irreversibility should be measured by the entransy dissipation, whereas for heat-work conversion, the irreversibility should be described by the entropy generation. Next, in Fourier’s Law derivation using the principle of minimum entropy production, the thermal conductivity turns out to be inversely proportional to the square of temperature. Whereas, by using the minimum entransy dissipation principle, Fourier’s Law with a constant thermal conductivity as expected is derived, suggesting that the entransy dissipation is a preferable irreversibility measurement for heat transfer.

## 1. Introduction

Heat transfer occurs in about 80 per cent of all the energy utilization systems, so improving the heat transfer performance significantly promotes the energy conservation in most thermal systems, through either increasing the heat flow rate for a given volume of facility, or reducing the cost of equipment with given heat load (Webb & Bergies 1983; Bergles 1997, 1988). Besides, enhancing the heat transfer effectively raises the operation reliability of electronic devices where electricity-generated heat has frequently posed a serious problem (Chen 1996). Thus, heat transfer improvement (or optimization) has become one of the critical issues for the efficiency of energy utilization, and heat transfer research has propelled rapid advancement in various heat transfer enhancement technologies, including using extended surfaces, stirrers and external electric or magnetic fields (Bergles 1988, 1997; Webb 1994; Karcz *et al.* 2005; Schäfer *et al.* 2005).

In the theoretical study of heat transfer, based on the concepts of entropy and entropy generation, Gyarmati (1970) derived Fourier’s Law with the criterion of minimum entropy generation and showed that entropy generation is the irreversibility measurement for any heat transfer process. Bejan (1979, 1996) and Charach & Rubinstein (1989) used this criterion for the optimization of convective heat transfer process and heat exchangers for different applications. On the other hand, there are some scholars who questioned if the entropy generation is *the universal* irreversibility measurement for heat transfer or if the minimum entropy generation is *the general* optimization criterion for all heat transfer processes, regardless of the nature of the applications. For instance, Bertola & Cafaro (2008) found that when satisfying the Onsager reciprocal relation, the principle of minimum entropy production (Prigogine 1967) could be tenable only if there is zero generalized flow under a non-zero generalized force, or the thermal conductivity should be inversely proportional to the square of the absolute temperature during steady-state heat conduction. By analysing the relationship between the efficiency and the entropy generation in 18 heat exchangers with different structures, Shah & Skiepko (2004) demonstrated that even when the system entropy generation reaches the extremum, the efficiency of the heat exchangers can be at either the maximum or the minimum, or anything in between. In addition, the so-called ‘entropy generation paradox’ (Bejan 1996; Hesselgreaves 2000) exists when the entropy generation minimization is used as the optimization criterion for counter-flow heat exchanger. That is, enlarging the heat exchange area from zero simultaneously increases the heat transfer rate and improves the heat exchanger efficiency, but does not reduce the entropy generation rate monotonously—the entropy generation rate increases at first and then decreases. Therefore, it was speculated that the optimization criterion of minimum entropy generation is not always consistent with the heat transfer improvement.

Recently, Guo *et al.* (2007) introduced the concepts of entransy and entransy dissipation to measure, respectively, the heat transfer capacity of a system and the loss of such capacity during the process. Moreover, Guo *et al.* (2007) proposed the entransy dissipation extremum as an alternative optimization criterion for a heat transfer process not involved in a thermodynamic cycle, and, consequently, developed the extremum principle of entransy dissipation to optimize the processes in heat conduction (Guo *et al.* 2007; Chen *et al.* 2009*a*), heat convection (Meng *et al.* 2005; Chen *et al.* 2007) and thermal radiation (Wu & Liang 2008).

The contribution of this present paper is to further examine the physical implications of both entropy generation and entransy dissipation, compare their differences and, more importantly, examine their applicability to heat transfer optimization in applications of different nature.

## 2. Irreversibility of heat transfer

### (a) Conserved and non-conserved quantities in a transport process

After intensive study, we found that all transport processes contain two different types of physical quantities owing to the existing irreversibility, i.e. the conserved ones and the non-conserved ones, and the loss or dissipation in the non-conserved quantities can then be used as the measurements of the irreversibility in the transport process. Taking an electric system as an example, although both the electric charge and the total energy are conserved during electric conduction, the electric energy, however, is not conserved and it is partly dissipated into the thermal energy owing to the existence of the electrical resistance. Consequently, the electrical energy dissipation rate is often regarded as the irreversibility measurement in the electric conduction process. Similarly, for a viscous fluid flow, both the mass and the momentum of the fluid, transported during the fluid flow, are conserved, whereas the mechanical energy, including both the potential and kinetic energies, of the fluid is turned into the thermal energy owing to the viscous dissipation. As a result, the mechanical energy dissipation is a common measure of irreversibility in a fluid flow process. The above two examples show that the mass, or the electric quantity, is conserved during the transport processes, while some form of the energy associated with them is not. This loss or dissipation of the energy can be used as the measurement of irreversibility in these transport processes. However, an irreversible *heat* transfer process seems to have its own particularity, for the *thermo-energy* always remains constant during transfer and it does not appear to be readily clear what the non-conserved quantity is in a heat transfer process. Non-equilibrium thermodynamics (Gyarmati 1970; Kreuzer 1981) seems to offer an answer by suggesting that the entropy or available energy (exergy) is the non-conserved quantity in a heat transfer process; that is, entropy can be generated or exergy can be dissipated during the process. Also it is interesting that, in general, all other physical energies (mechanical, electric, acoustic, etc.) can turn into thermal energy as the final form, except we rarely, if ever, use entropy or exergy to measure the irreversibility in other physical processes besides heat.

### (b) Irreversibility measurements in heat transfer

Before discussing the irreversibility in a heat transfer process, it is necessary to distinguish the objectives of heat transfer, for, as shown below, the irreversibility may have different implications for different objectives. The objectives in heat transfer can be classified into two categories: one is to use thermal energy as a form of energy to perform work, and the other is directly using thermal energy for warming up or cooling down the temperature. When performing work, i.e. in heat–work conversion, the heat transfer process is a link in a thermodynamic cycle. Whereas for heating or cooling, the heat transfer process is apparently much simpler. We will demonstrate that it is necessary to use different concepts and quantities to describe the irreversibilities in two such distinctive processes.

Based on the analogy between heat conduction and electric conduction, Guo *et al.* (2007) introduced a physical quantity, termed *entransy*, to study a heat transfer process not involved in heat–work conversion. The definition of entransy *G* is
2.1where *T* is the temperature and *U* the internal energy of the system. The expression of entransy is analogous to that of electric energy *E*_{e} in a capacitor,
2.2where *Q*_{e} and *V* are the electric quantity and the electric potential, respectively. As shown in equations (2.1) and (2.2), the entransy *G* in the heat transfer should be viewed as a special form of energy with the dimension (J⋅K). Further, accompanying the electric charge, the electric energy is transported during electric conduction. Similarly, along with the heat, the entransy is transported during heat transfer too. For example, in a heat conduction process, the thermal energy conservation equation can be expressed as
2.3where *ρ* is the density, *c*_{v} the constant-volume specific heat, *t* the time, the heat flow density and the internal heat source. Multiplying both sides of equation (2.3) by temperature *T* gives an equation that can be viewed as the conservation equation of the *entransy* in the heat conduction:
2.4that is
2.5where *g* = *G*/*V* = (1/2) *uT* is the specific entransy, *V* the volume, *u* the specific internal energy, the entransy flow density, the entransy change owing to heat source and *ϕ*_{h} can be taken as the entransy dissipation rate per unit volume, expressed as
2.6The left term in either equation (2.4) or (2.5) is the time variation of the entransy stored per unit volume, consisting of three items shown on the right: the first represents the entransy transferred from one (or part of the) system to another (part), the second term can be considered as *the local entransy dissipation* during the heat conduction and the third is the entransy input from the internal heat source. It is clear from equation (2.5) that the entransy is dissipated when heat is transferred from high temperature to low temperature. Thus, heat transfer is irreversible from the viewpoint of entransy, and the dissipation of entransy can hence be used as a measurement of the irreversibility in heat transfer.

Figure 1 is a schematic diagram of a case of one-dimension steady-state heat conduction. During the heat transfer, the entropy (or exergy) and entransy are transported with the heat. Among the system parameters, the thermal energy is conserved during the entire process, i.e. , but neither the entropy nor the entransy is conserved—the entropy is generated and the entransy is dissipated, i.e. and . Now we have two physical quantities for measuring the irreversibility in heat transfer, and we will demonstrate that, because of the intrinsic complex nature of heat transfer, we need both in dealing with the aforementioned different objectives in heat transfer. For heat–work conversion, the entropy generation or the exergy dissipation is a better irreversibility measurement, whereas for heating or cooling, the entransy dissipation is preferable. The physical distinctions between them will be further elucidated later in this article.

## 3. Irreversibility of heat transfer and Fourier’s Law

In non-equilibrium thermodynamics, the thermodynamic force and thermodynamic flow for heat transfer are chosen to ensure that the scalar product of them equals the entropy generation rate *S*_{gen} = *k*|∇*T*|^{2}/*T*^{2}. Thus, based on the Onsager theory, the linear phenomenological heat transfer law can be generally expressed as
3.1where *L* is the phenomenological coefficient for heat transfer process and a constant unrelated to temperature. The phenomenological law in equation (3.1) is also termed the entropy picture of heat transfer by Gyarmati (1970), which is different from the Fourier picture of heat transfer, i.e. Fourier’s Law:
3.2where *k* is the thermal conductivity. Comparison between equations (3.1) and (3.2) gives the relationship between the different phenomenological coefficients in the two representations
3.3

That is, if the phenomenological law of heat transfer in non-equilibrium thermodynamics is to be consistent with Fourier’s Law, the thermal conductivity *k* has to be proportional to 1/*T*^{2}.

According to the entropy picture of heat transfer, Prigogine (1967) proposed in 1947, the least energy dissipation principle, i.e. the minimum entropy generation principle, which is expressed for systems that satisfy the linear phenomenological law and the Onsager reciprocal relations as: ‘during evolution towards a stationary state the entropy production decreases and takes its lowest value compatible with external constraints when this stationary state is reached’ (p. 85). From this minimum entropy generation principle, Fourier’s Law was then derived. However, in the derivation, the phenomenological coefficient *L* in the entropy picture was treated as a constant, requiring the thermal conductivity be inversely proportional to the square of the absolute temperature during steady-state heat conduction. But in actuality, the thermal conductivity of most materials used in engineering is largely a constant, independent of temperature under normal conditions. This led to the conclusion by Prigogine (1967) himself that Fourier’s Law derived using the minimum entropy production principle is ‘not the most desirable’.

In contrast, according to the expression of entransy dissipation in equation (2.6), the temperature gradient is thought as the generalized force and the heat flux as the generalized flow for heat conduction, i.e.
3.4and
3.5Here, the scalar product of the ‘generalized force’ and ‘generalized flow’ equals to the entransy dissipation rate. Then based on the Onsager theory, the generalized force representation of the dissipation function *ψ* can be expressed as
3.6where the constant, *L** is the heat transfer phenomenological coefficient. Moreover, the linear phenomenological law of the generalized force and generalized flow is
3.7Substituting equation (3.7) into equation (3.6) yields
3.8Comparing equation (2.6) with equation (3.8) gives
3.9

In order to search for the extremum of the dissipation function in equation (3.6) with the constraint from equation (3.9), a Lagrange function, *F*, is constructed as
3.10where *A* is the Lagrange multiplier. The variation of *F* with respect to the general force, as shown in equation (3.4), is
3.11For *F* to be stationary, the term within the square bracket must vanish, that is,
3.12Meanwhile, the partial differential of the entransy dissipation and the dissipation function, as shown in equations (2.6) and (3.6), with respect to the temperature gradient ∇*T* are, respectively
3.13and
3.14

Substituting equations (3.13) and (3.14) into equation (3.12) yields 3.15

Equating equation (3.9) to equation (3.15), we obtain the Lagrange multiplier *A* = 1. Then equation (3.11) can be rewritten as
3.16

This is the requirement for the extreme value of the force representation in the dissipation function and satisfies both the linear phenomenological law and the Onsager reciprocal relation. Because *ψ* is a positive-definite function and *δ*^{2}*F* = *δ*^{2}(*ψ*−*ϕ*_{h})_{J} = *L** > 0, the extreme value determined by equation (3.16) is the minimum, i.e.
3.17

Since the temperature gradient ∇*T* is arbitrary, the term in the parentheses should be equal to 0.
3.18Now when the linear phenomenological coefficient *L** is chosen to equal the thermal conductivity *k*, equation (3.18) is exactly Fourier’s Law for heat transfer. In other words, for a given heat flux distribution, of all the possible temperature distributions, the actual solution is the one that satisfies the minimum entransy dissipation. Similarly, from the ‘general flux’ representation of the dissipation function, we can also prove that for a given temperature distribution, compared with all the other possible heat flux distributions, the actual solution satisfies the least action principle based on the entransy dissipation. Therefore, for an arbitrary boundary condition, heat will always flow along the path with the least action of the minimum entransy dissipation—a truly natural law as expected.

The aforementioned deduction process is similar to that based on non-equilibrium thermodynamics, and the differences between them lie in that the least action here refers to the entransy dissipation rate, while the least action used in non-equilibrium thermodynamics is in terms of the entropy generation rate. Furthermore, Fourier’s Law derived here requires a constant thermal conductivity, whereas in non-equilibrium thermodynamics, the thermal conductivity has to be inversely proportional to the squared temperature—something deviating grossly from existing facts.

## 4. Optimization of heat transfer with different objectives

### (a) Entransy dissipation extremum principle

Heat transfer optimization aims for minimizing the temperature difference at a given heat transfer rate, 4.1or maximizing the heat transfer rate at a given temperature difference 4.2In conventional heat transfer analysis, it is difficult to establish the relationship between the local temperature difference (or local heat transfer rate) and the other related physical variables over the entire heat transfer area, so the variational methods in equations (4.1) and (4.2) are not practically usable. However, in addition, to be the expression of least action in heat transfer, the entransy dissipation in equation (2.6) is also a function of the local heat flux and local temperature gradient in the heat transfer area, and thus the variational method will become usable if written in terms of the entransy dissipation (Cheng 2004).

Integrating the conservation equation of the entransy equation (2.4) over the entire heat transfer area gives 4.3For a steady-state heat conduction problem, the left term in equation (4.3) vanishes, i.e. 4.4If there is no internal heat source in the heat conduction domain, equation (4.4) is further reduced into 4.5

By transforming the volume integral to the surface integral on the domain boundary according to Gauss’s Law, the total entransy dissipation rate in the entire heat conduction domain is deduced as
4.6where *Γ*^{+} and *Γ*^{−} represent the boundaries of the heat flow input and output, respectively.

The continuity of the total heat flowing requires a constant total heat flow , 4.7

We further define the ratio of the total entransy dissipation and the total heat flow as the heat flux-weighted average temperature difference Δ*T*
4.8

For one-dimensional heat conduction, equation (4.8) is reduced into Δ*T* = (*T*_{in}−*T*_{out}), exactly the conventional temperature difference between the hot and cold ends. Using the heat flux-weighted average temperature difference defined in equation (4.8) and applying the divergence theorem, a new expression for optimization of a steady-state heat conduction at a given heat flow rate can be constructed as
4.9

It shows that when the boundary heat flow rate is given, minimizing the entransy dissipation leads to the minimum in temperature difference, that is, the optimized heat transfer. Conversely, to maximize the heat flow at a given temperature difference, equation (4.9) can be rewritten as 4.10showing that maximizing the entransy dissipation leads to the maximum in boundary heat flow rate.

Similarly, for a steady-state heat dissipating process with internal heat source in equation (4.4), the total entransy dissipation rate in the entire heat conduction domain is derived as 4.11The heat generated in the entire domain will be dissipated through the boundaries, i.e. 4.12

Again the heat flux-weighted average temperature is defined as the entransy dissipation over the heat flow rate 4.13And thus the optimization of the process is achieved when 4.14which means that in a heat dissipating process, minimizing the entransy dissipation leads to the minimum-averaged temperature over the entire domain.

Based on the results from equations (4.9), (4.10) and (4.14), it can be concluded that the extremes in entransy dissipation lead to the optimized heat transfer performance at different boundary conditions.

### (b) Application to a two-dimensional volume-point heat conduction

We will apply our proposed approach to practical cases where heat transfer is used for heating or cooling such as in the so-called volume-point problems (Bejan 1997) of heat dissipating for electronic devices as shown in figure 2. A uniform internal heat source distributes in a two-dimensional device with length and width of *L* and *H*, respectively. Owing to the tiny scale of the electronic device, the joule heat can only be dissipated through the surroundings from the ‘point’ boundary area such as the cooling surface in figure 2, with the opening *W* and the temperature *T*_{0} on one boundary. In order to lower the unit temperature, a certain amount of new material with high thermal conductivity is introduced inside the device. As the amount of the high thermal conductivity material (HTCM) is given, we need to find an optimal arrangement so as to minimize the average temperature in the device.

According to the new extremum principle based on entransy dissipation, for this volume-point heat conduction problem, the optimization objective is to minimize the volume-average temperature, the optimization criterion is the minimum entransy dissipation, the optimization variable is the distribution of the HTCM and the constraint is the fixed amount of the HTCM, i.e. 4.15

By the variational method, a Lagrange function, *Π*, is constructed
4.16where the Lagrange multiplier *B* remains constant because of a constant thermal conductivity. The variation of *Π* with respect to temperature *T* gives
4.17

Because the boundaries are either adiabatic or isothermal, the surface integral on the left side of equation (4.17) vanishes, that is, 4.18

Moreover, owing to a constant entransy output and a minimum entransy dissipation rate, the entransy input reaches the minimum when 4.19

Substituting equations (4.18) and (4.19) into equation (4.17) in fact gives the thermal energy conservation equation based on Fourier’s Law
4.20This result shows again that the irreversibility of heat transfer can be measured by the entransy dissipation rate *Φ*_{h}. The variation of *Π* with respect to thermal conductivity *k* gives
4.21

This means that in order to optimize the heat dissipating process, i.e. to minimize the volume-average temperature, the temperature gradient should be uniform. This in turn requires that the thermal conductivity be proportional to the heat flow in the entire heat conduction domain, i.e. the HTCM be placed at the area with the largest heat flux.

As an example, the cooling process in a low-temperature environment is analysed here. For the unit shown in figure 2, *L* = *H* = 5 cm, *Q* = 100 W cm^{−2}, *W* = 0.5 cm and *T*_{0} = 10 K. The thermal conductivity of the unit is 3 W (mK)^{−1}, and that for the HTCM is 300 W (mK)^{−1} occupying 10 per cent of the whole heat transfer area. The implementary steps are as follows:

(A) Local optimization

(1) Divide the entire heat transfer region into several parts; (2) set all the parts composed by the original substrate material with low thermal conductivity; (3) numerically simulate the temperature field and the entransy dissipation rate in the entire heat transfer region; (4) find the part with the highest temperature gradient, and fill it with the HTCM; (5) numerically simulate again the temperature field and the total entransy dissipation rate in the heat transfer region; (6) compare the total entransy dissipation rates with and without the HTCM element just filled in. If the total entransy dissipation rate is reduced, go to step 8. Otherwise go to step 7; (7) remove the HTCM element from the part that is just filled in and fill it in the part where the temperature gradient is the next largest. Then go to step 5; (8) judge whether all the HTCM has been used up. If so, go to ‘(B) global optimization’. Otherwise, go to step 4.

(B) Global optimization

In the above steps, the HTCM is distributed by iteration. Since local optimum may not assure a global optimum, the results obtained above need further adjustment. The detailed steps are:

(9) Choose one of the parts filled with the HTCM one after another; (10) remove the HTCM element from the chosen part and fill it in with the original substrate material; (11) numerically simulate the temperature field and the entransy dissipation rate in the entire heat transfer region; (12) find the part with the highest temperature gradient, and fill it with the HTCM; (13) numerically simulate again the temperature field and the total entransy dissipation rate in the heat transfer region; (14) compare the total entransy dissipation rates with and without using the HTCM. If the total entransy dissipation rate is reduced, go to step 16. Otherwise go to step 15; (15) remove the HTCM element from the part that is just filled in and fill it in the part where the temperature gradient is the next largest. Then go to step 13; (16) judge if all the HTCM filled parts in series have been checked and the arrangement of them has not been changed: if not, go to step 9 and continue the optimization steps, or else end the optimization.

For example, after dividing the whole heat transfer area into 40 × 40 parts, figure 3*b* shows the distribution of the HTCM according to the extremum principle of entransy dissipation, where the black area represents the HTCM—the same hereinafter. The HTCM with a tree structure absorbs the heat generated by the internal source and transports it to the isothermal outlet boundary—similar in both the shape and function of actual tree roots.

For a fixed amount of HTCM, figure 4*a*,*b* compares the temperature distributions between a uniform distribution of HTCM shown in figure 3*a*, and the optimized distribution in figure 3*b* based on the extremum principle of entransy dissipation. The average temperature in the first case is 544.7 K while the temperature in the second-optimized case is 51.6 K, a 90.5 per cent reduction! It clearly demonstrates that the optimization criterion of entransy dissipation extremum is highly effective for such applications. Furthermore, as shown in figure 4*b*, the temperature gradient field is also less fluctuating in the optimized case.

### (c) The same case optimized using the minimum entropy approach

For comparison, we also treated the same problem using the minimum entropy generation principle. Again, the constraint is , and what we seek is also the optimal HTCM arrangement, except in this case that: (i) the optimization criterion is the minimum entropy generation and (ii) the corresponding energy conservation equation should be added as a constraint, because it is not implied in the principle of minimum entropy generation when the thermal conductivity is constant.

Introducing the corresponding Lagrange function
4.22where *B*′ and *C*′ are also the Lagrange multipliers. The constraint of thermal conductivity is the isoperimetric condition, and consequently *B*′ is a constant. *C*′ is a variable related to space coordinates. The variation of *Π*′ with respect to temperature *T* gives
4.23While the variation of *Π*′ with respect to thermal conductivity *k* yields
4.24

Likewise, equation (4.24) gives the guideline for optimization based on the criterion of minimum entropy generation. Meanwhile, the most optimization steps used are the same as in the first case, except that: (i) change ‘finding the part with extreme temperature gradient’ to that with the extreme in absolute value of ∇*A*_{3}⋅∇*T*−(∇*T*^{2}/*T*^{2}); (ii) replace the criterion of the total entransy dissipation rate by that of the total entropy generation rate.

Figure 3*c* shows the distribution of HTCM based on the principle of minimum entropy generation. Comparison of figure 3*b*,*c* shows that although the distributions of HTCM are similar between the two results in most areas, the root-shape structure from the extremum principle of entransy dissipation is not directly connected to the heat flow outlet, leaving some parts with the original material in between them so that the heat cannot be transported smoothly to the isothermal outlet boundary. Figure 4*c* gives the optimized temperature distribution obtained by the minimum entropy generation. Because the low-thermal conductivity material is adjacent to the heat outlet, the temperature gradient grows larger and thus lowers the entire heat transfer performance. The averaged temperature of the entire area is 150.8–99.2 K higher than that obtained by the extremum principle of entransy dissipation. From the definition, it is easy to find that in order to decrease the entropy generation, we have to both reduce the temperature gradient and raise the temperature, thus leading to the arrangement of HTCM shown in figure 3*c*.

In addition, according to the principle of minimum entropy generation, the optimization objective of a steady-state heat dissipating process can be expressed as:
4.25where (Δ(1/*T*))_{m} = ((1/*T*)−(1/*T*_{0}))_{m} is the equivalent thermodynamics potential difference that represents the generalized force in the entropy picture for heat transfer. Thus, minimizing the entropy generation equals to minimizing the equivalent thermodynamics potential difference (Δ(1/*T*))_{m}, leading to the highest exergy transfer efficiency. That is, the minimum entropy generation principle is equivalent to the minimum exergy dissipation during a heat transfer process.

To facilitate the comparison between the two results from figure 4*b*,*c*, table 1 lists the key findings side by side, obtained, respectively, by the optimization criteria of the minimum entropy generation and the entransy dissipation extremum. It indubitably shows in the table that the proposed entransy-based approach is more effective than the entropy-based one in heat transfer optimization, for the former leads to a result with significantly reduced mean temperature than that by the latter (51.6 versus 150.8 K), and much lower maximum temperature (83.0 versus 194.9 K). Whereas the entropy-based approach is preferred in exergy transfer optimization, as it results in a significantly lower equivalent thermodynamic potential (7.1 × 10^{−3} K^{−1} versus 2.2 × 10^{−2} K^{−1}).

Besides heat conduction, we (Chen *et al*. 2009*b*) also compared the two criteria in heat convection optimization. Our results indicate that both principles are applicable to convective heat transfer optimization, subject, however, to different objectives. The minimum entropy generation principle works better in searching for the minimum exergy dissipation during a heat–work conversion, whereas the entransy dissipation extremum principle is more effective for processes not involving heat–work conversion, in minimizing the heat transfer ability dissipation.

In addition, based on the concept of the entransy dissipation rate, we (Chen *et al*. 2009*a*) introduced the non-dimensional entransy dissipation rate and employed it as an objective function to analyse the thermal transfer process in a porous material. Moreover, some of us (Guo *et al.* 2010) defined the equivalent thermal resistance of a heat exchanger to measure the irreversibility of heat transfer in the processes of heating or cooling. After establishing the relationship between the heat exchanger effectiveness and the thermal resistance, Guo *et al.* found that reducing the thermal resistance leads to a monotonic increase in the heat exchange effectiveness. Guo *et al.* also demonstrated that the irreversibility in a heat exchanger is more effectively represented by its thermal resistance, while the so-called entropy generation paradox occurs if using the entropy generation criterion.

## 5. Conclusions

— In an electric conduction process or in a fluid flow process, although the electron or the fluid mass is conserved, the electric energy or the mechanical energy is dissipated. These two non-conserved quantities are used as the irreversibility measurements for their respective processes. Similarly, in heat transfer processes, the thermal energy is conserved. When the heat transfer involves heat–work conversion, exergy is the dissipated quantity, whereas in the heating or cooling process, we have demonstrated that the newly defined entransy is the dissipated quantity.

— Correspondingly, the objectives of heat transfer are classified into two different categories: one is direct heating or cooling and the other is for heat–work conversion. The irreversibility of the former case should be measured by the entransy dissipation rate while the latter should be measured by the rate of entropy generation or exergy dissipation.

— More generally, Fourier’s Law derived by the minimum entransy dissipation principle satisfies a constant thermal conductivity, whereas by the principle of minimum entropy production, the thermal conductivity has to be inversely proportional to the squared temperature—something that contradicts the existing facts. Thus, the entransy dissipation is a better alternative for the measurement of irreversibility in heat transfer.

— For the testing case of a volume-point heat conduction problem with uniform internal heat source, if a fixed amount of HTCM is used to minimize the volume-average temperature, the entransy dissipation extremum has shown to be a more effective optimization criterion than the minimum entropy generation.

## Acknowledgements

The present work is supported by the National Natural Science Foundation of China (grant no. 51006060) and the Postdoctoral Scientific Fund of China (grant no. 200902080).

- Received June 8, 2010.
- Accepted September 14, 2010.

- © 2010 The Royal Society