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Entransy theory for the optimization of heat transfer – A review and updateReview


International Journal of Heat and Mass Transfer 63 (2013) 65–81

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International Journal of Heat and Mass Transfer
j

ournal homepage: www.elsevier.com/locate/ijhmt

Entransy theory for the optimization of heat transfer – A review and update
Qun Chen, Xin-Gang Liang, Zeng-Yuan Guo ?
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

a b s t r a c t
Heat transfer optimization methods to effectively improve heat transfer performance is of great importance for energy conservation and pollution reduction. A recently developed heat transfer optimization method based on entransy theory and related peer-reviewed papers published between 2003 and 2010 are reviewed and updated in this paper to describe entransy, entransy dissipation, optimization criteria and optimization principles and their applications to different heat transfer modes (thermal conduction, convection and radiation) and to different levels (heat transfer element, heat exchanger, and heat exchanger network). Entransy theory is then compared with entropy theory in several aspects, including the heat transfer purpose, irreversibility and optimization principle for energy savings or weight reductions of thermal facilities. Finally, entransy theory is also compared with constructal theory in terms of optimization objective, optimization method and optimized results. ? 2013 Elsevier Ltd. All rights reserved.

Article history: Received 4 December 2012 Received in revised form 25 February 2013 Accepted 2 March 2013 Available online 22 April 2013 Keywords: Heat transfer Optimization Entransy Entropy Constructal law

Contents 1. 2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Origin of entransy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Deduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entransy dissipation and entransy balance equation for heat transfer processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Heat convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Heat transfer processes in two-fluid heat exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization of heat transfer processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Optimization objectives of heat transfer processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Optimization principles for heat transfer processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Heat convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Optimization criteria for heat transfer processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Heat convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Heat transfer processes in two-fluid heat exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of the entransy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Volume-point heat conduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Laminar forced convection in a round tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Forced turbulent convection between two parallel plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Thermal radiation between parallel plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differences between the entransy theory and the entropy theory for optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 67 67 68 68 68 69 69 69 70 70 70 70 71 71 71 71 72 73 73 73 73 73 74 75 75

3.

4.

5.

6.

? Corresponding author. Tel.: +86 10 62782660.
E-mail address: demgzy@tsinghua.edu.cn (Z.-Y. Guo). 0017-9310/$ - see front matter ? 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.03.019

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Nomenclature A; B ; C ; c cp cv Ee Eg Eh eh _h e F Ft f G g _ g _r g H K k ke M _ M Mh Nu ~ n P Pr Qve Qvh _ Q _t Q _ q C 0 ; A0 ; B0 ; C 0 ; C 00 Lagrange multipliers speed of light in vacuum, m s?2 constant pressure speci?c heat capacity, J kg?1 K?1 constant volume speci?c heat capacity, J kg?1 K?1 electrical potential energy in a capacitor, J gravitational potential energy of ?uid in a vessel, J potential energy of a phonon gas, J density of potential energy of a phonon gas, J m?3 pontential energy ?ux of a phonon gas, W m?2 additional volume force for laminar heat transfer, N m?3 additional volume force for turbulent heat transfer, N m?3 friction factor entransy, J K speci?c entransy, J K m?3; gravitational acceleration, m s?2 entransy ?ux, W K m?2 radiative entransy ?ux, W2 m?4 height, m heat exchanger heat transfer coef?cient, W m?2 K?1 thermal conductivity, W m?1 K?1 electrical conductivity, S m?1 mass, kg mass ?ow rate, kg s?1 thermomass, kg Nusselt number unit vector pressure, Pa Prandtl number electrical charge stored in a capacitor, C thermal energy stored in an incompressible body, J heat transfer rate, W total heat exchange rate in a heat exchanger, W heat ?ux, W m?2 _e q Re Rh _q S _ S hg _g S _g s T U Ue Uh Uhr ! U ?! Uh V u, v, w x, y, z electrical current density, A m?2 Reynolds number thermal resistance, K/W internal heat source, W m?3 internal entransy source, W K m?3 entropy generation rate, W K?1 entropy generation rate per unit area during thermal radiation, W K?1 m?2 temperature, K internal energy, J electrical potential, V thermal potential, K radiative heat ?ux potential, W m?2 velocity vector, m s?1 velocity vector of a phonon gas, m s?1 volume, m3 velocity components in x, y and z directions, m s?1 Cartesian coordinates, m

Greek symbols e emissivity g uniformity factor for a temperature difference ?eld k Lagrange multiplier l dynamic viscosity, kg m?1 s?1 q density, kg m?3 qh thermomass density, kg m?3 r Stefan–Boltzmann constant, W m?2 K?4 s time, s; viscous force, N m?2 _ Ug entransy dissipation rate, W K _e / dissipation rate per unit volume of the potential energy of a phonon gas, W m?3 _g / entransy dissipation rate per unit volume, W K m?3 _ /m viscous dissipation rate per unit volume, W m?3

7. 8.

6.1. Volume-point heat conduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Convective heat transfer in a square cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Thermal radiation between parallel plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differences between the entransy theory and the constructal law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 76 77 78 78 79 79

1. Introduction Heat transfer is thermal energy in transit due to a temperature difference, one of the most common physical phenomena in the world, especially in energy systems. Estimates of all the worldwide energy utilization suggest more than 80% involve heat transfer processes. Thus, improved heat transfer performance offers a huge potential for conserving energy and reducing CO2 emission so as to reduce global warming [1,2]. During the past several decades, heat transfer science, dealing with analyses of heat transfer rates taking place in systems, has been well developed with a large number of heat transfer enhancement techniques developed to improve the performance of energy generation, conversion, consumption and conservation [1–13]. However, unlike thermodynamics, another thermal science subject, there is no concept of ef?ciency but only the concept of the heat transfer rate, so scientists and engineers often focus on heat transfer enhancement which is usually associ-

ated with increased pumping power and in turn usually reduces the energy utilization ef?ciency. Heat transfer is an irreversible, non-equilibrium process from the thermodynamic viewpoint. In a key part of non-equilibrium thermodynamics [14], Onsager [15,16] set up the famous reciprocal relationships for non-equilibrium processes including heat transfer and derived the principle of the least dissipation of energy in terms of the entropy balance equation, where entropy generation is considered to be a measure of irreversibility. Prigogine [17] developed the minimum entropy production principle based on the idea that the entropy production in a thermal system at steady state had to be at the minimum. Both of these principles, however, did not deal with how to improve the heat transfer performance until Bejan [18–23] deduced an expression for the entropy generation to measure the irreversibility of convective heat transfer in viscous ?uid ?ows. Thereafter, many scholars [24–46] have employed the minimum entropy generation as an objective function, and some of them also combined it with optimization

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methods, including simulated annealing and genetic algorithms, to optimize heat transfer processes. At the same time, Andresen and his colleagues [47–50] developed the theory of ?nite-time thermodynamics and introduced such concepts as the maximum power, generalized potential, and thermodynamic length to optimize practical thermodynamic cycles involving heat transfer processes with ?nite temperature differences. Both the minimum entropy generation principle and the theory of ?nite-time thermodynamics have been applied to optimize heat transfer processes with the objective of minimizing the entropy generation (or the exergy destruction) so as to maximize the power output of a thermodynamic cycle. However, Bertola and Cafaro [51] found that when satisfying the Onsager reciprocal relations, the minimum entropy production principle could only be tenable if there was zero generalized ?ow for a non-zero generalized force. To obtain Fourier’s Law from the minimum entropy production principle, the thermal conductivity must be inversely proportional to the square of the absolute temperature [52–54]. However, in reality, the thermal conductivity of most materials used in engineering is essentially constant, independent of temperature for normal conditions. This led to the conclusion by Prigogine himself that Fourier’s Law derived using the minimum entropy production principle is ‘‘not the most desirable’’ [17]. In addition, the so-called ‘‘entropy generation paradox’’ [21,55–57] exists when the entropy generation minimization is used as the optimization objective for counter-?ow heat exchangers. That is, enlarging the heat exchange area from zero continuously improves the heat exchanger ef?ciency, but does not uniformly reduce the entropy generation rate. The entropy generation rate increases at ?rst to a maximum, and then begins to decrease [57]. Furthermore, Shah and Skiepko [58] analyzed the relationship between the effectiveness and the entropy generation in 18 different types of heat exchangers to demonstrate that the heat exchanger effectiveness can be at the maximum, the minimum, or any value in between when the system entropy generation is minimized. Therefore, the minimum entropy generation does not always correspond to the best heat transfer performance. Bejan [59,60] inspired by the tree-like structure in both nature and engineering [61–66], then developed the constructal law to minimize the maximum temperature difference to guide the construction of tree-like structures for solving volume-point heat conduction problems. The constructal law provides some insights for geometric optimization of volume-point ?ow problems encountered not only in engineering and physics [59,67–87] but also in biology [88–93] and other branches of science [94–97]. Nevertheless, Kuddusi and Egrican [98] reviewed fourteen different applications involving tree shaped ?ow networks to show that the constructal theory result does not necessarily improve the ?ow performance if the internal branching of the ?ow ?eld is increased. In reality, the performance will often be lowered if the internal ?ow ?eld branching is further increased. Therefore, increasing the internal ?ow ?eld branching does not always improve the ?ow performance, hence, the generality of constructal theory is in question [98–102]. In view of the fact that for the existing techniques of heat transfer enhancement it lacks a way to understand the universal mechanism of various techniques of heat transfer enhancement, and a large additional pumping power is usually associated with the heat transfer enhancement techniques, the ?eld synergy principle proposed by Guo et al. [103–106] and Tao et al. [107] indicates that the strength of convective heat transfer depends not only on the velocity and the temperature gradient ?elds, but also on their synergy, that is, the better the synergy of velocity and temperature gradient ?elds, the higher the convective heat transfer rate under the same other conditions. Consequently, we can have not only a understanding of the universal mechanism of various heat transfer

enhancement techniques, but also can develop novel heat transfer enhancement structures with a smaller increase of pumping power. Recently, aware of the ineffectiveness of entropy analyses for pure heat transfer problems, Guo et al. [108] introduced the physical quantity of entransy, originally termed heat transfer potential capacity [109,110], which represents the heat transfer capacity of an object during a time period. Entransy dissipation can be used to measure the irreversibility of pure heat transfer processes. Furthermore, they [108] proposed the entransy dissipation extremum principle, i.e., the minimum entransy dissipation-based thermal resistance principle, for optimization of heat transfer processes not involved in a thermodynamic cycle. Methods were then developed to optimize the three basic heat transfer modes of thermal conduction [54,111–116], convection [117–121] and radiation [122,123]. This principle has also been successfully used in practical engineering applications for the optimal design of heat transfer elements [110,115], heat exchangers [124–128], and heat exchanger networks [129–133]. Moreover, Chen et al. used the analogy between heat and mass transfer to extend entransy theory to optimize mass transfer [134–136] and coupled heat and mass transfer processes [137–141]. Li and Guo [142] conducted a review on the ?eld synergy principle for the optimization of convective heat transfer and applied the entransy concept to reveal its physical essence. This paper comprehensively reviews the existing literatures related to entransy to further illustrate the origin in terms of both induction and deduction, the physical nature of entransy including its relation to the thermomass potential energy, the minimum entransy dissipation-based thermal resistance principle and its practical applications with the optimization objectives, the optimization principles and the optimization criteria of heat transfer. In addition, entransy theory is also systemically compared with the minimum entropy generation principle and the constructal law to clarify their differences from the aspects of heat transfer purpose, optimization objectives, criteria, methods and results. 2. Origin of entransy 2.1. Induction Since entropy is not suitable for irreversibility analyses of some types of heat transfer problems [17,51,55–58], a new physical quantity is needed to measure the irreversibility of heat transfer problems not related with heat-work conversion. The analogies among electrical conduction, ?uid ?ow, and heat conduction listed in Table 1 were used to introduce that the temperature, as the thermal potential, corresponds to the electrical potential and the gravitation potential, and Fourier’s law corresponds to Ohm’s law and Newton’s law. Therefore, the thermal energy stored in an incompressible object should correspond to the electrical charge stored in a capacitor and the mass of a ?uid in a vessel. However, there is no quantity in heat transfer theory corresponding to the electrical potential energy of a capacitor or the gravitation potential energy of a ?uid in a vessel. Hence, Guo et al. [108] de?ned the quantity, G, with the differential form

dG ? Mcv TdT
and the integral form

?1?

G?

1 1 UT ? Mcv T 2 ; 2 2

?2?

where cv is the constant volume speci?c heat capacity, T is the temperature, M is the mass, U is the internal energy of an object and G is a state quantity called entransy, which was referred to as the heat

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Q. Chen et al. / International Journal of Heat and Mass Transfer 63 (2013) 65–81

Table 1 Analogies among electrical conduction, ?uid ?ow and heat conduction theories. Electrical conduction Fluid ?ow Heat conduction
a b

Electrical charge stored in a capacitor Qve (C) Fluid mass in a vessel M (kg) Thermal energy in an incompressible object Qvh = U = McvT (J)

Electrical potential Ue (V) Gravitation potential gH (m2/s2) Thermal potential Uh = T (K)

e _ e ? ?ke dU Ohm’s law q d~ n

Electrical potential energy in a capacitor Ee = QveUe/2 (J) Gravitation potential energy of ?uid in a vesselb Eg = MgH/2 (J) ?

Newton’s lawa

s ? ?l du d~ n

_ ? ?k dT Fourier’s law q d~ n

s ? ?l du is the constitutional relation of ?uid for the simple shear ?uid ?ow, which is de?ned as Newtonian ?uid. d~ n
H is the height of the ?uid level in a vessel. Since the ?uid mass is the function of its height, the overall gravitation potential energy of ?uid in a vessel is Eg = MgH/2.

transport potential capacity in an earlier paper [109]. Dividing the entransy by the volume gives the entransy density,

dEh ?

U d?cv T ?: c2

?6?

g?

G 1 ? qcv T 2 ; V 2

? 3?

The integral form of Eq. (6) is

where q is the density and V is the object volume. 2.2. Deduction Besides induction based on the analogy among heat conduction, ?uid ?ow and electric conduction, the de?nition of entransy can also be set up within the frame of continuum mechanics. The thermal energy induced by lattice vibrations in a dielectric is quantized with the energy quantum called a phonon. The state of the lattice vibrations may be characterized by a phonon gas consisting of large numbers of phonons moving randomly. According to the Einstein’s mass-energy equivalence [143,144], the equivalent mass of a phonon gas, Mh, called the thermomass stored in an object, is

Eh ?

Ucv T qc2 T2V ? v 2 ; 2 2c 2c

?7?

if cv = const. Eqs. (2) and (7) lead to the relation between the entransy and the potential energy of the thermomass,

G ? Eh

c2 : cv

?8?

Mh ?

U qcv TV ? c2 c2 Mh qcv T ? 2 ; c V

? 4?

and its density is

qh ?

? 5?

Both cv and c are normally assumed to be constant, so the entransy is in fact a simpli?ed expression for the potential energy of the phonon gas (thermomass). Hence, the entransy analysis in fact corresponds to a thermomass energy analyses without the factor cv/c2 for convenience. Biot [145] suggested a similar concept in his derivation of the differential conduction equation using the variational method. Eckert and Drake [146] pointed out that ‘‘Biot de?nes a thermal potenRRR 2 tial E ? 1 X qcT dV . . .. The thermal potential E plays a role 2 analogous to a potential energy . . ..’’ However, Biot did not further explore the physical meaning of the thermal potential and it was not used later except in approximate solutions of anisotropic conduction problems.

where c is the speed of light in vacuum. Noted that unlike in caloric theory, the thermal energy in a dielectric is regarded as a weighable phonon. Hence, the heat transfer process in dielectrics is in fact a ?ow of a weighable phonon gas in the medium. As shown in Fig. 1, since the thermomass (the equivalent of a phonon gas) in a thermal ?eld has a speci?c thermal potential, cvT, its potential energy change dEh required to transfer an additional element of thermomass dM is

3. Entransy dissipation and entransy balance equation for heat transfer processes As is well-known, electric charge is non-dissipative and is conserved during electric conduction, while the electric energy is partially dissipated into thermal energy due to the electric resistance. Hence, the dissipation of electric energy is a measure of the irreversibility for electric conduction that is not related to the conversion of electric energy and other types of energy except for thermal energy. Likewise, the ?uid mass is non-dissipative and conserved during ?uid ?ow, while the mechanical energy is dissipated due to ?uid friction. The dissipation of mechanical energy is a measure of the ?uid friction induced irreversibility of the ?uid ?ow that is not involved in a thermodynamic cycle. 3.1. Heat conduction For heat transfer processes in system without volume variation, energy being transferred or the equivalent mass of the phonon gas is conserved. There must be some other quantity that is dissipated during irreversible heat transfer processes. Using the concept that a heat transfer process is equivalent to the ?ow of a phonon gas, Cao and Guo [144] developed the continuity equation for a phonon gas

Fig. 1. Sketch of thermomass potential energy variation by an additional element of thermomass.

?! @ qh ? r ? ?qh U h ? ? 0; @s

?9?

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?! _ =qcv T is the phonon gas velocity or the mean drift where U h ? q _ is the heat ?ux, and s is the time. velocity of phonons, q Multiplying both sides of Eq. (9) by the thermal potential cvT leads to the balance equation for the potential energy of a phonon gas:

! where cp is the speci?c heat at constant pressure, U is the velocity _ q is the internal heat source generated by viscous fricvector, and S tion, chemical reaction or Joule heating. Multiplying Eq. (18) by the temperature T leads to [119]

@ eh _ e ? 0; _h ? / ?r?e @s

?10?

@g ! _ hg ? / _ g: _ ?S ? U ? rg ? ?r ? g @s

?19?

where eh = qhcvT/2 is the potential energy density of the phonon gas, ?! ! _e ? q ? _ h ? qh U h cv T is the potential energy ?ux, and / e h U h ? r?cv T ? is the dissipation rate of potential energy in the phonon gas. Combining Eq. (10) with Fourier’s heat conduction law and dropping the factor, cv/c2 in all terms gives the entransy balance equation [54,108]:

@g _ g; _ ?/ ? ?r ? g @s

The terms on the left side of Eq. (19) describe the convective transport of entransy with the ?uid ?ow. On the right side, the ?rst term is the entransy ?ux into the differential element from adjacent parts of the ?uid due to diffusion, the second term is the internal entransy source due to internal heat sources, and the third term is the entransy dissipation rate per unit volume. Therefore, entransy dissipation is also a measure of the irreversibility of convection processes. 3.3. Thermal radiation Cheng and Liang [122] and Cheng et al. [123] used the product of the Stefan–Boltzmann constant and the fourth power of temperature as the radiative heat ?ux potential, Uhr,

?11?

_ g is the entransy dissipation _ is the entransy ?ux and / _ ? qT where g rate per unit volume caused by heat conduction, expressed as

_ g ? ?q _ ? rT : /

?12?

Eqs. (10) and (11) once again show that entransy posses the nature of the potential energy of the phonon gas. Heat, like mass or charge, is not dissipated during transport process, whereas entransy, like the gravitational potential energy or the electric potential energy, is dissipated. The entransy dissipation rate, consequently, is a measure of the irreversibility of a pure heat transfer process. As a simple example, when two bodies, A and B, with initial temperatures TA and TB, are in contact with each other, thermal energy will ?ow from the higher temperature body to the lower temperature one, and the two objects will come to an equilibrium temperature, TAB, after a suf?ciently long time. The initial entransies in the two objects are [108]:

U hr ? rT 4
_ r , is and the radiative entransy ?ux, g

?20?

_ hr ; _ r ? qU g

?21?

where r is the Stefan–Boltzmann constant. For an enclosure involving n grey surfaces, the emitted heat transfer rate for surface i is

_ i ? ei Ai U hri ; Q

?22?

GA ? GB ?

1 M A cv A T 2 A; 2 1 M B cv B T 2 B: 2 1 ?MA cv A ? M B cv B ?T 2 AB ; 2 M A cv A T A ? M B c v B T B : MA cv A ? M B cv B

?13? ?14?

where ei is the emissivity of surface i whose area is Ai with the ther_ i emitted from Ai will be mal radiation potential Uhri. The energy Q absorbed by all the surfaces in the enclosure. Absorption factors can be de?ned for each surface as

_ ij =Q _ i; Bij ? Q

?23?

_ ij is the energy absorbed by Aj from all the energy emitted where Q by Ai. With consideration of the energy balance
n X Bij ? 1 and j?1

The ?nal total entransy of the two objects after equilibrium is

ei Bij Ai ? ej Bji Aj

?24?

GAB ?

?15?

where

and the energy balance equation of surface j is

T AB ?

?16?

_ j;net ? Q _j? Q

n X j? 1

_ i Bij ; Q

?25?

Hence, during this heat transfer process, the entransy difference before and after equilibrium is

DG ?

1 M A M B cv A cv B ?T A ? T B ?2 > 0: 2 M A cv A ? MB cv B

?17?

_ j;net is the net heat ?ow leaving Aj. where Q Multiplying Eq. (25) by Uhrj and summing over all the surfaces yields the entransy balance equation for radiative heat transfer in enclosures,
n n X n X X _ j;net U hrj ? _ i Bij ?U hri ? U hrj ?; Q Q j?1 j?1 i?1

Eq. (17) shows that the total entransy is reduced after the two bodies reaching equilibrium due to the entransy dissipation associated with the thermal transport. Cheng et al. [147] proved that the total entransy of an isolated system always decreases for a heat transfer process without heat generation in the domain due to electrical, nuclear or chemical heating. 3.2. Heat convection For an incompressible convection process, conservation of thermal energy is expressed as

?26?

where the left term is the total net radiative entransy ?ow out of all the surfaces, which is not conserved and equals the total radiative entransy dissipation due to the radiative heat transfer between surfaces in the enclosure shown by the right term. Therefore, radiative entransy dissipation can then be used to measure the irreversibility of thermal radiation processes. 3.4. Heat transfer processes in two-?uid heat exchangers For heat transfer in a two-?uid heat exchanger, the heat transfer rate between the hot and cold ?uids over a differential element is

qcp

! @T _ q; _ ?S ? qcp U ? rT ? ?r ? q @s

?18?

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Q. Chen et al. / International Journal of Heat and Mass Transfer 63 (2013) 65–81

optimization involves such four fundamental elements as specifying the purpose of the heat transfer process, choosing the objective function, applying the optimization method, and ?nally deducing the optimization principle and criterion. If the purpose of the heat transfer is to heat or cool an object, the optimization objective is to maximize the heat transfer coef?cient, which is equivalent to minimizing the average temperature difference at a given heat transfer rate or to maximizing the heat transfer rate at a given temperature difference. The variation of this objective is

_ ; q; cv ; . . .? ? 0 d?DT ? ? df ?x; y; z; s; k; q
or

?32?

_ ? ? dg ?x; y; z; s; T ; k; q; cv ; . . .? ? 0: d?Q

?33?

_ diagram of heat transfer processes in two-?uid heat exchangers. Fig. 2. T-Q

_ ? ?M _ h dhh ? M _ c dhc ; dQ

?27?

_ is the mass ?ow rate, h is the speci?c enthalpy, and the where M subscripts h and c represent the hot and cold ?uids. If the speci?c heats of the ?uids are constant, Eq. (27) can be rewritten as

_ ? ?M _ h cp;h dT h ? M _ c cp;c dT c : dQ

?28?

Fig. 2 illustrates the temperature variations of hot and cold ?uids versus the heat transfer rate given by Eq. (28), which is actually a heat transfer property diagram [148]. All points on lines ab and cd represent states of motion for the heat, while the slopes of lines ab and cd are the reciprocals of the heat capacity rates of the hot and cold ?uids. The shaded area in Fig. 2 is then

Conventional heat transfer analyses do not have quantitative relationships between the boundary temperature difference or the boundary heat transfer rate and other physical parameters over the entire heat transfer area, so the variational methods in Eqs. (32) and (33) are not practically usable. Thus, engineers can only apply the ‘‘trial-and-error’’ method or combine it with other optimization methods, such as the gradient free algorithm, the gradient-based algorithm, the genetic algorithm and the neural network algorithm, to improve the heat transfer performance. The entransy dissipation given in Eq. (12), which is a function of the local heat ?ux and local temperature gradient, can be used with the variational method to introduce a new optimization principle for heat transfer processes if the entransy dissipation can be used as the optimization objective. 4.2. Optimization principles for heat transfer processes 4.2.1. Heat conduction For a steady state heat conduction process without internal heat sources, integration of the entransy balance equation, Eq. (11), over the entire heat conduction domain gives [54,108]:

_ ? T c dQ _: dS ? T h dQ

?29?

The ?rst term on the right side of Eq. (29) represents the entransy _ leaving from the ?ow rate accompanying the heat transfer rate dQ hot ?uid, while the second term is the entransy ?ow rate accompa_ into the cold ?uid. Therefore, the nying the heat transfer rate dQ shaded area represents the entransy dissipation rate per unit heat ?ux due to heat transfer from the hot ?uid to the cold ?uid:

ZZZ
X

_ ?dV ? ?r ? ?qT

ZZZ
X

_ ? rTdV : ?q

?34?

_: _ h ? ?T h ? T c ?dQ /

?30?

Transforming the volume integral to a surface integral on the domain boundary by Gauss’s Law gives the total entransy dissipation rate in the entire heat conduction domain as

Integrating Eq. (30) gives the total entransy dissipation in the heat exchanger

Uh ?

ZZZ
X

_ ? rTdV ? ?q

ZZ
C
?

_ in T in dS ? q

ZZ
C?

_ out T out dS; q

?35?

_g ? U

Z
0

_t Q

_ ? ?T h;a ? T c;c ? ? ?T h;b ? T c;d ? Q _ t ? DT AM Q _ t; _ g dQ / 2

?31?

where DTAM is the average arithmetic temperature difference between the hot and cold ?uids in the heat exchanger. From Eq. (31), the area between the temperature variation curves for the hot and cold ?uids, i.e., the trapezoid abdc, represents the total entransy dissipation rate in the heat exchanger. Therefore, the heat transfer property diagram can be used to conveniently analyze the irreversibility for heat transfer in a heat exchanger and even complex heat exchanger networks to optimize their heat transfer performance [129,130,133]. 4. Optimization of heat transfer processes 4.1. Optimization objectives of heat transfer processes Optimization is the procedure of obtaining the best performance, i.e., maximizing or minimizing a desired objective function, while satisfying the prevailing constraints [149]. Heat transfer

where C+ and C- represent the in?ow and out?ow boundaries for the thermal transport. Multi-dimensional heat conduction processes need a heat ?uxweighted average temperature difference, DT , de?ned as the ratio of the total entransy dissipation rate and total heat transfer rate,

DT ?

_g U _t Q

?

ZZ
C
?

_ in q T dS ? _ t in Q

ZZ
C
?

_ out q T dS: _ t out Q

?36?

The divergence theorem is then used to derive the minimum entransy dissipation principle for steady-state heat conduction optimization at a given heat transfer rate, which can be expressed as:

_ t d?DT ? ? d Q

ZZZ
X

_ g ?: kjrT j2 dV ? d?U

?37?

When the boundary heat transfer rate is given, minimization of the entransy dissipation leads to minimization of the temperature difference. Conversely, for heat conduction at a given temperature difference, the maximum entransy dissipation principle can be written as:

Q. Chen et al. / International Journal of Heat and Mass Transfer 63 (2013) 65–81

71

_ t? ? d DT d?Q

ZZZ
X

1 2 _ g ?: _ j dV ? d?U jq k

?38?

For a steady state heat removal process with a prescribed internal heat source [54], the process optimization is achieved when

_ in and Q _ out are the total net amounts of radiant energy rewhere Q leased and absorbed, i represents the surfaces with a negative net radiant heat transfer rate and j represents the surfaces with a positive net radiant heat transfer rate. Then, an average radiative thermal potential difference can be de?ned as

_ q d ? DT ? ? d S

ZZZ
X

_ g ?; _ ? rTdV ? d?U ?q

?39?

DU hr ?

_ ini XQ
i

_t Q

U hri ?

_ outj XQ
j

_t Q

U hrj :

?44?

which means that minimizing the entransy dissipation in a heat removal process leads to the minimum average temperature over the entire domain. Eqs. (37)–(39) show that the entransy dissipation extremum leads to the best heat conduction performance with the maximum effective thermal conductance, i.e. the maximum heat ?ux for proscribed boundary temperature difference, or the minimum temperature difference for a given boundary heat ?ux or a prescribed internal heat source. 4.2.2. Heat convection As with the derivation of the entransy dissipation extremum principle for heat conduction, for steady-state convective heat transfer in a tube with a constant boundary heat ?ux, integrating the entransy balance equation, Eq. (19), transforming the volume integral to a surface integral on the domain boundary and ignoring the heat generated by viscous dissipation yields [120]:

For a prescribed total heat ?ux, substituting Eq. (44) into the radiative entransy balance equation, Eq. (26), yields

" # n X n X _ t d?DU hr ? ? d 1 _ g ?; ei Ai Bij ?U hri ? U hrj ?2 ? d?U Q 4 j ? 1 i? 1

?45?

which shows that minimizing the radiative entransy dissipation leads to the minimum average radiative thermal potential difference. Similarly, if the average radiative thermal potential difference is given,

" # n X n 1X 1 _2 _ _ g ?: DU hr d?Q t ? ? d Q ? d?U 4 j?1 i?1 ei Ai Bij ij

?46?

  _ t d T w ? T in ? T out ? d?U _ g ?: Q 2

?40?

Since the inlet and outlet ?uid temperatures, Tin and Tout, are ?xed for a given boundary heat ?ux, Eq. (40) indicates that the minimum entransy dissipation in the domain corresponds to the minimum wall temperature. In the same way, for a prescribed boundary temperature Tw (Tw > Tin),

Eq. (46) demonstrates that the maximum radiative entransy dissipation corresponds to the maximum heat ?ow for a prescribed average radiative thermal potential difference. Thus, the radiative entransy dissipation extremum also corresponds to the best thermal radiation performance with the maximum heat transfer coef?cient for various boundary conditions. In summary, for all three heat transfer modes including heat conduction, heat convection, and thermal radiation, an entransy dissipation-based thermal resistance can be de?ned as the total entransy dissipation rate over the square of the heat transfer rate,

Rh ?

_g U _2 Q t

:

?47?

_ _ t ? Q t dQ _ ? d?U _ g ?: ?T w ? T in ?dQ qcp t

?41?

Eq. (41) illustrates that for boundary heat ?uxes from 0 to the maximum, qVcp(Tw ? Tin), the entransy dissipation increases linearly with increasing boundary heat ?ux, which means that the maximum entransy dissipation results in the maximum boundary heat ?ux. In addition, for convective heat transfer processes in rectangular closed systems with an internal heat source, the entransy balance equation, Eq. (19), gives

The entransy dissipation extremum principle for all three heat transfer modes can then be written as the minimum entransy dissipation-based thermal resistance principle, which is also valid for heat exchanger optimization [124,125]. 4.3. Optimization criteria for heat transfer processes Chen and Tso [150] used entransy theory to classify the heat transfer regimes for forced convection in a porous medium and identi?ed the region where the thermal resistance to the wall heat ?ux changes drastically due to the effect of viscous dissipation. Many others [124–128,151–154] have optimized the structural and operating parameters of heat exchangers based on the entransy dissipation extremum principle or the minimum entransy dissipation-based thermal resistance. However, they have not described the optimization criterion. Since the entransy dissipation is a function of the local heat ?ux and the temperature gradient over the entire heat transfer area, it is possible and necessary to deduce the corresponding optimization criteria by using the variational method as shown in Eqs. (32) and (33) not only for the trial-and-error method, but also for comparing the performance characteristics of various thermal devices. 4.3.1. Heat conduction A volume-point heat conduction problem [59] such as shown in _ q , distributed uniformly in a Fig. 3, has an internal heat source, S two-dimensional region with length L and width H. Assuming a tiny device, the Joule heat can only be transferred to the surroundings from a ‘‘point’’ boundary such as the cooling surface in Fig. 3

_ q d?DT f ? ? d?U _ g ?; S

?42?

where DT f is the average temperature difference between the ?uid and the boundary. Eq. (42) show that the minimum entransy dissipation for this condition leads to the minimum average temperature difference in the heat transfer domain. Eqs. (40)–(42) are called the optimization principle of entransy dissipation extremum for convective heat transfer. These show that the entransy dissipation extremum results in the best convective heat transfer performance with the maximum convective heat transfer coef?cient for different boundary conditions. 4.2.3. Thermal radiation In all enclosures [122,123], some surfaces receive a positive net amount of radiant energy, while others releasing a net amount. The total heat exchange for the system can be de?ned as

_t ?Q _ in ? Q

X X _ ini ? ?Q _ out ? ? Q _ outj ; Q
i j

?43?

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Q. Chen et al. / International Journal of Heat and Mass Transfer 63 (2013) 65–81

The variational method gives a Lagrange function, P, for heat transfer with laminar ?ow [110,119,120,142]

Y

?

ZZZ j
X

k _ q ? qcp U ~ ? rT ? ? Br ? qU ~ dV ; _ m ? A?kr2 T ? S kjrT j2 ? C 0 / ?53?

_ m represents the where A, B and C0 are the Lagrange multipliers. / viscous dissipation rate in the ?uid ?ow. The ?rst term in the bracket of Eq. (53) is the entransy dissipation rate, the second is the constraint on the prescribed viscous dissipation rate, the third and fourth terms are the constraints on the energy and continuum equations. The variational of Eq. (53) yields

! ?qcp U ? rA ? r ? krA ? 2r ? krT
and
Fig. 3. Two-dimensional heat conduction with a uniformly distributed internal heat source [54].

?54?

q U ? r U ? ?rP ? lr2 U ? F ;
where F is an additional volume force given by

!

!

!

?55?

with opening W and temperature T0 on one boundary. In order to lower the device temperature, a certain amount of high thermal conductivity material (HTCM) will be inserted in the device. As the amount of the HTCM is given, we need to ?nd an optimal distribution of thermal conductivity so as to minimize the average temperature in the device. Assuming that the thermal conductivity may vary continuously in the domain, the constrain for heat conduction optimization is

! ! F ? C U ArT ? q U ? r U
and CU is a parameter related to the viscous dissipation as

?56?

CU ?

qc p : 2C 0

?57?

ZZZ
X

k?x; y?dV ? const :

?48?

The entransy dissipation extremum principle, i.e., the minimum entransy dissipation-based thermal resistance principle, and the variational method can be used to construct a Lagrange function, P, to minimize the average temperature with the constraint of a ?xed amount of the HTCM [54]

P?

ZZZ j k _ q ? dV ; kjrT j2 ? Bk ? C ?r ? krT ? S
X

?49?

where B and C are the Lagrange multipliers. B remains constant because of the constant overall thermal conductivity, while C varies with the spatial coordinates. The variation of P with respect to the thermal conductivity, k, gives

Eq. (55) is the momentum equation with an additional volume force. For a laminar forced convection process with prescribed inlet velocity or Re, a given CU determines a speci?c additional volume force in Eq. (55), which leads to the optimal ?uid velocity ?eld with the largest heat transfer coef?cient for a speci?c viscous dissipation rate. For given different values of CU, that is, for different viscous dissipation rates, solving Eqs. (54) and (55) together with the continuity and energy conservation equations can obtain different optimal ?uid velocity ?elds. Meanwhile, Eq. (55) is also called the synergy equation for the ?uid velocity and the temperature gradient ?elds [110] because the entransy dissipation extremum corresponds to the best ?eld synergy [103,120], where the ?eld synergy degree is quantitatively described by the ?eld synergy number. For a two-dimensional boundary heat convection process, the integral of the energy conservation equation over the boundary layer thickness, dt, is

Z
0

dt

@T _ w: qcp ?U ? rT ?dy ? ?k ?q @y
w



?58?

With the following dimensionless variables,

jrT j ? ?B ? const;

?50?

which means that to minimize the volume-averaged temperature, the temperature gradient in the domain should be uniform, which implies that the thermal conductivity should be proportional to the local heat ?ux over the entire conduction domain. Therefore, the uniformity of the temperature gradient is the optimization criterion for heat conduction problems. 4.3.2. Heat convection For convective heat transfer, pumping power must be consumed to maintain the ?uid ?ow, which should be used as an additional constraint with the continuity equation

! ! U U ? ; U1

rT ?

rT ; ?T 1 ? T w ?=dt

y?

y ; dt

?59?

Eq. (58) can be rewritten in dimensionless form as:

Nux ? Rex Pr

Z
0

1

  ! U ? rT dy;

?60?

~?0 r ? qU
and the energy conservation equation

?51?

! where U is the dimensionless velocity, rT is the dimensionless temperature gradient and y is the dimensionless length. Thus, the performance of convective heat transfer depends not only on the velocity and the temperature gradient, but also on their included angle, i.e., their synergy. The integral of the dimensionless convection term over the entire thermal boundary layer in Eq. (60) has been named the ?eld synergy number, Fc, [103]

Fc ? ?52?

_ q; ~ ? rT ? kr2 T ? S qcp U
to maximize the convective heat transfer coef?cient.

Nux Nux ? ? Rex Pr Pex

Z
0

1

  ! U ? rT dy;

?61?

which indicates the degree of synergy between the velocity and the temperature gradient ?elds. The ideal case with the best synergy is

Q. Chen et al. / International Journal of Heat and Mass Transfer 63 (2013) 65–81

73

! that U and rT are uniform and their included angle is zero everywhere in the heat transfer domain, so that Fc = Nu/Pe = 1. Therefore, the largest ?eld synergy number is the optimization criterion for convection heat transfer performance. Similarly, Chen et al. [117] derived the ?eld synergy equation for turbulent heat transfer:

Moreover, Cheng et al. [154] derived the relation between the uniformity factor and the entransy dissipation-based thermal resistance as:

g ? p??????????? ;
KARh

1

?69?

q U ? r U ? ?rP ? leff r2 U ? F t ;

!

!

!

?62?

where leff is the effective viscosity and Ft is the additional volume force for turbulent heat transfer. Solving Eq. (62) together with the continuity and energy conservation equations will also lead to the optimal ?uid ?ow ?eld with a larger ?eld synergy number than any other ?eld for a given set of constraints. Hence, the largest ?eld synergy degree is the optimization criterion for turbulent heat transfer as well. 4.3.3. Thermal radiation Consider two in?nite plates in parallel with emissivities of e1 and e2 [122]. The radiative thermal potential of Plate 2, Uhr2, is gi_ , is ?xed. Then, the disven. The total net heat transfer on Plate 1, Q tribution of the heat ?ux on Plate 1 can be optimized to reduce the average radiative thermal potential. Consider a small element, dA, on Plate 1, there is

where K is the overall heat exchanger heat transfer coef?cient. As shown in Eq. (69), when the thermal conductance of a two-?uid heat exchanger is given, a smaller entransy dissipation-based thermal resistance will lead to a larger temperature difference ?eld uniformity factor. The minimum entransy-dissipation-based thermal resistance then corresponds to a uniform temperature difference ?eld, that is, the maximum temperature difference ?eld uniformity factor. Therefore, the uniformity factor for the temperature difference ?eld is the optimization criterion for two-?uid heat exchangers. 5. Applications of the entransy theory 5.1. Volume-point heat conduction The volume-point heat conduction process shown in Fig. 3 [54] _ q ? 100 W=cm2 , is analyzed as an example, where L = H = 5 cm, S W = 0.5 cm and T0 = 10 K. The thermal conductivity of the base material is 3 W/(m K), while that for the HTCM is 300 W/(m K), which occupies 10% of the entire heat transfer area. The heat conduction area is divided into 40 ? 40 elements to use the uniform temperature gradient optimization criterion with the implementation as shown in Ref. [54]. Compared with the uniform HTCM distribution shown in Fig. 4(a), Fig. 4(b) shows the optimized distribution of the HTCM for a ?xed amount of HTCM, where the black area represents the HTCM. The HTCM arrangement with a tree structure absorbs the heat generated by the internal source and transports it to the isothermal outlet boundary with a shape similar to that of actual tree roots. Figs. 5(a) and (b) compare the temperature distributions between the uniform distribution of HTCM shown in Fig. 4(a) and the optimized distribution in Fig. 4(b). The average temperature with the uniform case is 544.7 K while the optimized result is 51.6 K, a 90.5% reduction. This shows that entransy theory is very effective for such applications. 5.2. Laminar forced convection in a round tube The solution of Eqs. (51), (52), (54), and (55) for a given CU will give the optimal ?uid ?ow and temperature ?elds for laminar heat transfer in a circular tube, which have the highest heat transfer coef?cient for a given viscous dissipation and boundary conditions. For instance, forced laminar convection in a straight circular tube, 20 mm in diameter, is optimized here for a wall temperature of 310 K, an average inlet temperature of 300 K, and an inlet Reynolds number of 400 [110]. Fig. 6 shows a typical numerical result for the cross-sectional ?ow ?eld for CU = ?0.01 with eight longitudinal vortices. The Nusselt number is increased by 313% for the case of Fig. 6, compared with fully-developed laminar ?ow without vortices in a circular tube ((fRe)s = 64, Nus = 3.66), while the pumping power is only increased by 17%. This result indicates the optimal ?ow pattern for laminar heat transfer in circular tube has multiple longitudinal vortices. This optimal ?ow ?eld led to the discrete double-inclined ribs tube (DDIR tube) design to generate such longitudinal vortices in practice, as shown in Figs. 7(a) and (b) [110]. The outer diameter of the DDIR tube is 20 mm. There are three pairs of double-inclined ribs in the cross-section of the DDIR tube. The heights of the ribs are 0.85 mm, the lengths in axial direction are 6 mm, the inclined an-

ZZ

U hr1 ? U hr2 _ ? const ; dA ? Q RdA?2

?63?

where RdA?2 is the radiative thermal resistance between the small element and Plate 2,

RdA?2 ?

1

e1 dA

?

1 ? e2 ; e2 A2

?64?

where A2 is the area of Plate 2. The radiative entransy dissipation of the system is

_g ? U

ZZ

_ dA DU hr dA ? Q

ZZ

?DU hr ?2 dA; RdA?2

?65?

_ dA is the net heat transfer rate for the where DUhr = Uhr1 ? Uhr2 and Q element. Based on Eqs. (63) and (65), a function can then be written for radiative heat transfer by using the Lagrange multiplier method as,

P?

# ZZ " ?DU hr ?2 DU hr dA: ?k RdA?2 RdA?2

?66?

Then, the variational method gives

_ k RdA?2 Q DU hr ? ? ? ? const : 2 A

?67?

This equation shows that the optimum radiative thermal potential distribution on Plate 1 is uniform, which means that the net heat ?uxes and the temperatures of each area unit on Plate 1 are the same. This is the optimization criterion for this type of thermal radiation processes. 4.3.4. Heat transfer processes in two-?uid heat exchangers For two-?uid heat exchangers, the optimization criterion of uniform temperature difference ?eld has been employed and proved to be applicable by optimizing several kinds of heat exchangers [155,156], where the uniformity factor for the temperature difference ?eld, g, is de?ned as
c C h ?: g ? q????????????????????????????????????? RR

RR

jT ? T jdA

A

C ?T h

? T c ?2 dA

?68?

Song et al. [157] theoretically validated that this optimization criterion is consistent with the entransy dissipation extremum principle.

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Q. Chen et al. / International Journal of Heat and Mass Transfer 63 (2013) 65–81

Fig. 4. Different HTCM arrangements [54]. (a) Uniform HTCM arrangement. (b) HTCM arrangement optimized by entransy theory.

Fig. 5. Temperature ?elds obtained from different HTCM arrangements [54]. (a) From the uniform HTCM arrangement in Fig. 4(a). (b) From the optimized HTCM arrangement of Fig. 4(b).

gels are 45°, and the pitch between every two ribs is 12 mm. Experiments have demonstrated that the DDIR tube has better heat transfer performance and lower ?ow resistance increase than other kinds of enhanced tubes. 5.3. Forced turbulent convection between two parallel plates For turbulent ?ows in a parallel plate channel with a height of 20 mm. Water ?owing between the parallel plates is assumed to be fully developed with a Reynolds number of 20,000. The inlet water temperatures is 300 K and the wall temperature is 350 K [117]. In this case, the heat ?ux is 712.8 kW/m2 and the mechanical energy dissipation rate is 92.2 W/m3. For heat transfer with fully-developed turbulent ?ow between two parallel plates, the velocity vectors and the temperature gradients are nearly perpendicular to each other, leading to a relatively small heat transfer coef?cient. Solving Eq. (62) together with the continuity equation and the energy equation gives the optimized velocity ?eld (C0 = ?1 ? 107), with the ?ow near the upper plate shown in Fig. 8. Unlike the multiple longitudinal vortices in lami-

Fig. 6. Optimum ?ow ?eld for laminar heat transfer in a circular tube (Re = 400, CU = ?0.01) [110].

Q. Chen et al. / International Journal of Heat and Mass Transfer 63 (2013) 65–81

75

(a)
5%U m

0
Fig. 9. A radiative heat transfer system of four in?nite plates in parallel [122].

o

(b)
Fig. 7. Structure and ?ow pattern of the DDIR-tube [110]. (a) Photo of the DDIR tube. (b) Cross-sectional ?ow ?eld in the DDIR-tube.

m2. In this case, the thermal energy absorbed by the right plate with a lower temperature is 157.23 W/m2, while that absorbed by the left plate with a higher temperature is only 22.77 W/m2, which demonstrates that this emissivity distribution lets more thermal energy ?ow into the lower potential plate. 6. Differences between the entransy theory and the entropy theory for optimization Chen et al. [54,119] indicated that the various heat transfer processes have two different purposes as for heat-work conversion and object heating or cooling only, and consequently it is need to have different irreversibility measures for process optimization. For the ?rst purpose, the entropy generation rate is the best measure of the irreversibility and its optimization should apply the minimum entropy generation principle, while for the second purpose the entransy dissipation rate is the best measure of the irreversibility and the entransy dissipation extremum principle, i.e., the minimum entransy dissipation-based thermal resistance, should by applied for optimization. Use of these different principles to optimize the same heat transfer process will lead to different optimization results as the following examples illustrate. 6.1. Volume-point heat conduction Reconsider the volume-point problem shown in Fig. 3. The optimal HTCM arrangement will be found using the minimum entropy _ g ? ? min?RRR kjrT j2 =T 2 dV ? [54]. The generation principle, min?S X optimization uses the Lagrange function

Fig. 8. Optimized velocity ?eld near the wall for turbulent heat transfer [117].

nar tube ?ow, turbulent ?ow has several small counter-clockwise eddies near the plate. In this case, the heat ?ux is 978.1 kW/m2 and the mechanical energy dissipation rate is 158 W/m3. Thus, the heat ?ux is increased by 37% compared with the results before optimization, while the mechanical energy dissipation rate is increased by 72%, so the performance evaluation criteria (PEC), de?ned as ?Nu=Nu0 ?=?f =f0 ?1=3 , is 1.15. Calculations of the optimal ?ow and temperature ?elds for various C0 show that the optimal ?n heights for different Reynolds numbers are equal to half of the turbulent ?ow transition layer thickness [117], which has also been validated experimentally [158]. 5.4. Thermal radiation between parallel plates For radiative heat transfer between four large plates in parallel, as shown in Fig. 9, where the temperature of the left and right plates, T01 and T02, are 300 K and 200 K, the net heat ?uxes released _ 1 and Q _ 2 , are 80 W/m2 and 100 W/m2. When from Plates 1 and 2, Q the total emissivity of Plates 1 and 2, e1 ? e2 , is equal to 1, Cheng and Liang [122] found that the radiative entransy dissipation per unit area is minimized when e1 ? 0:24, with an average radiative thermal potential difference reaching the minimum of 274.43 W/

# ZZZ " jrT j2 0 0 _ P? k 2 ? B k ? C ?r ? krT ? Sq ? dV ; T
X

?70?

where B0 and C0 are the Lagrange multipliers. The variational method then gives

?r ? ?krC 0 ? ?
and

2kjrT j2 T3

?

_q 2S T2

?71?

rC 0 ? rT ?

rT 2
T2

? B0 ? const:

?72?

Eq. (72) is then the optimization criterion for heat transfer corresponding to the minimum entropy generation principle: the HTCM should be inserted into the domain to make rC 0 ? rT ? rT 2 =T 2 constant. Fig. 10(a) shows the optimized HTCM distribution based on the minimum entropy generation principle. Comparison of Figs. 4(b) and 10(a) shows that the root-shaped

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1 where D 1 ?T ? T10 . Thus, the minimum entropy generation leads to T the minimum average difference in the reciprocal of the temperature. Eqs. (39) and (73) indicate that the minimum average temperature in the domain corresponds to the best heat transfer performance while the minimum difference in the reciprocal of the temperature corresponds to the maximum heat to work conversion ef?ciency. Hence, the entransy dissipation extremum principle should be used rather than the minimum entropy generation principle to optimize heat conduction processes not involved in a thermodynamic cycle.

6.2. Convective heat transfer in a square cavity Stirring is a typical method to produce convection so as to boost heat transfer by constantly changing the temperature pro?le and ?uid velocity ?eld. Chen et al. [119] studied a two-dimensional stirring process in a square cavity with side length L = 100 mm as shown in Fig. 11. There is a uniform heat source in the cavity and the overall heat generation rate is 70 W. The lateral boundary temperatures are T1 = 300 K and T2 = 550 K, while the top and bottom are adiabatic. The ?uid inside the cavity is air with the physical properties, q = 1.225 kg/m3, l = 1.7894 ? 10?5 kg/(m s), k = 0.0242 W/(m K), and cp = 1006.43 J/(kg K), which are all constant during the process. Gravity is not considered in this case. The problem is to ?nd the optimal air velocity distribution to minimize the average temperature in the cavity for a given pumping power. Eqs. (51), (52), (54), and (55) (CU = 9.26 ? 10?9) can be solved simultaneously to yield the optimal velocity ?elds and temperature contours in the cavity shown in Fig. 12(a) and (b). The ?ow ?led has four vortexes in the cavity as shown in Fig. 12(a) to enhance the convection. As shown in Fig. 12(b), the heat transfer rates on the left and right boundaries are 44.0 W and 26.0 W. The average temperature in the domain is 580 K. The minimum entropy generation principle can also be used to obtain the optimization equation by the variational method as [119]:

q U ? r U ? ?rP ? lr2 U ? ?C 0U A0 rT ? q U ? r U ?
and

!

!

!

!

!

?74?

  ! k rT : ?qcp U ? rA0 ? kr ? rA0 ? 2 r ? T T

?75?

Simultaneously solving these two equations with the continuity and energy conservation equations (C 0U ? 3:19 ? 10?3 ) give the optimized velocity and temperature ?elds shown in Fig. 13(a)

Fig. 10. Optimized results using the minimum entropy generation principle [54]. (a) HTCM arrangement. (b) Temperature ?eld.

structure is quite different though the HTCM distributions in some areas seem to be similar for the two results. The average temperature of the entire area shown in Fig. 10(b) is 150.8 K, 99.2 K higher than that obtained using the entransy dissipation extremum principle. As shown by Eq. (39), the entransy dissipation extremum principle gives the optimized result with the minimum average temperature and a uniform temperature gradient in the domain, while the minimum entropy generation principle uses the optimization objective for a steady-state heat removal process expressed as:

_ td D 1 Q T

! _ g ?; ? d?S ?73?
Fig. 11. Sketch of the square cavity geometry and boundary conditions [119].

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77

Fig. 12. Optimized velocity and temperature ?elds based on the entransy dissipation extremum principle. (a) Velocity ?eld. (b) Temperature ?eld.

and (b). There are only two vortexes in the cavity, and their positions are close to the right higher temperature boundary, meaning more heat is moved to the right boundary. As shown in Fig. 13(b), the heat transfer rate at the right boundary is 33.6 W, 7.6 W higher than that optimized by the entransy dissipation extremum principle, and the average temperature in the cavity is 614 K, 34 K higher than that given by the entransy dissipation extremum principle. For a given heat transfer rate and temperature gradient, the entropy generation principle suggests that a higher absolute temperature will lead to a smaller entropy generation rate for a given heat ?ux, so the entropy generation induced by the heat transfer at the right boundary with the higher temperature will be smaller. Therefore, the vortexes are required near the right boundary to transfer more heat to the right boundary so as to decrease the total entropy generation in the entire cavity. The average temperature in the cavity is the not minimized by the minimum entropy generation principle. 6.3. Thermal radiation between parallel plates For the radiative heat transfer system shown in Fig. 9, Cheng and Liang [122] further derived the expression for entropy generation per unit area for the system as

Fig. 13. Optimized velocity and temperature ?elds based on the minimum entropy generation principle. (a) Velocity ?eld. (b) Temperature ?eld.

_ g ? U hr1 ? U hr01 S 1

   1 1 U hr1 ? U hr2 1 1 ? ? ? 1 1 T 01 T 1 T2 e1 e1 ? e2 ? 1 T 1   U hr2 ? U hr02 1 1 ? ? 1 T 02 T 2 e
2



?76?

and studied the variations of the entropy generation per unit area with variation of the emissivity of Plate 1. They found that the entropy generation per unit area is minimized when e1 ? 0:89. In this case, the thermal energy absorbed by the right plate is only 76.81 W/m2, while that absorbed by the left plate is 103.19 W/m2. Most thermal energy ?ows into the higher radiative thermal potential environment, so the average thermal potential difference is 391.80 W/m2, 117.37 W/m2 higher than that obtained by the entransy dissipation extremum principle. Thus, the differences between entransy theory and entropy theory for heat transfer optimization are related to the heat transfer purpose, the optimization objective, the optimization principle and the optimization criterion, as listed in Table 2. The purposes of heat transfer can be classi?ed into two different categories: one is the object heating or cooling only, and the other is for heat-to-work conversion, that is, heat transfer process is involved in a thermodynamic cycle. The optimization objective of the former case is the maximum heat transfer coef?cient and the corresponding optimization principle is the principle of minimum entransy dissipation-based thermal resistance, while the latter is the maximum heat-work conversion ef?ciency and the corresponding optimization principle is the principle of minimum entropy generation or exergy destruction. Moreover, based on the principle of minimum entransy dissipation-based thermal resistance together with the variational method, three optimization cri-

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Table 2 Major differences between the entransy and entropy theories for heat transfer optimization. Entransy theory Heat transfer Optimization Optimization Optimization purpose objective principle criterion Object heating or cooling Maximum heat transfer coef?cient Minimum entransy dissipation-based thermal resistance Uniformity of temperature gradient for heat conduction; ?eld synergy degree for heat convection; uniform thermal potential for thermal radiation. Entropy theory Heat-work conversion Maximum heat-work conversion ef?ciency Minimum entropy generation None

teria have been deduced for heat conduction, heat convection and thermal radiation, respectively, which are convenient for the implementation of heat transfer optimization. In contrast, there is no optimization criterion corresponding to the minimum entropy generation principle in the existing literature. 7. Differences between the entransy theory and the constructal law The constructal law was stated by Bejan in 1996 [60] as: ‘‘For a ?nite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that ?ow through it.’’, which is considered, in some way, as a summary of all design generation and evolution phenomena in nature. This indicates that shape and structure arise to facilitate ?ow. The designs that arise spontaneously in nature re?ect this tendency, that is, they allow entities to ?ow more easily - to measurably move more current farther and faster for less unit of useful energy consumed [89]. This principle can be applied for optimizing engineering systems to ?nd the optimal shape and structure of ?ow channels, leading to not only tree-shape ?ows but also other geometric forms encountered in engineering [159]. However, the construtal law, as Bejan stated, is not about ‘‘dissipation’’ [160], and thus it is not concerned with the irreversibility during spontaneous processes. Unlike the constructal law, the entransy theory starts from an analyses of the heat transfer irreversibility and introduces a new physical quantity, entransy. Since the entransy is not conserved during heat transfer and its dissipation indicates the irreversibility in a heat transfer process, the minimum entransy dissipationbased thermal resistance principle can be established using the variational method for the optimization of heat transfer processes not involved in thermodynamic cycles. Hence, this principle can be applied not only to optimizing the shape and structure of ?ow systems but also to optimizing ?ow ?elds inside tubes and the distribution of heat transfer area or ?ow rates inside a heat exchanger network [129,130,133]. Even for the shape and structure optimization of high conductivity material in the volume-point heat conduction process shown in Fig. 3, the entransy theory is quite different from the constructal law in many aspects including the optimization objective, characteristic temperature difference, optimization principle and criterion, and optimal results, as listed in Table 3. Furthermore, in the
Table 3 Optimization of volume-point heat conduction problems. Entransy theory Optimization objective Characteristic temperature difference Thermal resistance de?nition Optimization principle Optimization criterion Optimal construct Reduction of the mean temperature in the domain Average temperature difference

constructal law, the structures are always self-similar, the conductivities of the structures are assumed to be much larger than those of the substrate, and the aspect ratio and angle of the structures are the only adjustable variables to reduce the peak temperature in the domain. Thus, increasing the internal branching in the ?ow ?eld will not always improve the heat transfer performance [98–102]. There is no such assumption in entransy theory. Chen and his co-workers [111,112,161–173] combed the minimum entransy-based thermal resistance principle with the constructal method to optimize several heat conduction problems, especially some volume-point problems to show that the constructs based on the minimum entransy dissipation-based thermal resistance reduce the mean temperature difference more than those based on the minimization of the maximum temperature difference, so they further improve the heat conduction performance greatly [111,162,164,167,168,171].

8. Conclusions The physical quantity, entransy, has been developed by both inductive and deductive reasoning. Entransy represents the heat transfer ability of an object or a system during a time period because the entransy is a simpli?ed expression of the thermomass potential energy, which is different from thermal energy. During a heat transfer process, thermal energy, which is conserved, is a non-dissipative quantity, while entransy, which is partly dissipated, is a dissipative quantity. Therefore, the entransy dissipation rate, like the entropy generation rate, is able to represent the irreversibility of heat transfer processes. Heat transfer processes can be classi?ed into two categories according to their purposes: for heat-work conversion in thermodynamic cycles and for pure object heating or cooling. For the ?rst category, the entropy generation rate is the measure of the heat transfer irreversibility, so thermodynamic optimization using entropy applies the minimum entropy generation principle. For the second category, the entransy dissipation rate measures the heat transfer irreversibility so heat transfer optimization uses the minimum entransy dissipation-based thermal resistance principle. Entransy theory also differs from the constructal theory for heat conduction optimization in their optimization objective, optimization method, optimization principle and, consequently, their optimal results.

Constructal law Reduction of the peak temperature in the domain Largest temperature difference (DTmax = Tmax ? Tmin) Largest temperature difference over the heat transfer _ t) rate (DT max =Q Largest temperature difference based thermal resistance minimization None Self-similar structure [59]

_ 2) _ g =Q Entransy dissipation rate over the square of the heat transfer rate (U t Entransy dissipation-based thermal resistance minimization or entransy dissipation extremum Uniform temperature gradient in the domain Without self-similar assumption (Fig. 4(b))

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The minimum entransy dissipation-based thermal resistance principle has been used to optimize different heat transfer modes, including heat conduction, heat convection, and thermal radiation problems, and heat transfer facilities at different levels including heat transfer elements, heat exchangers, and heat exchanger networks for energy saving or weight reduction of thermal facilities. Acknowledgment The present work was supported by the National Natural Science Foundation of China (Grant Nos. 51006060 and 51136001). References
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