Constructal Law and Its Corollaries of Extremum Principles for Energy Expenditure and Entropy Production | A. Heitor Reis

Table of Contents

Use and validity of principles of extremum of entropy production in the study of complex systems

A. Heitor Reis^∗

University of Évora, Department of Physics and Geophysics Center of Évora, Colégio Luis Verney, R. Romão Ramalho, 59, 7002-554, Évora, Portugal

^∗Tel.: +351 967324948; fax: +351 266 745394.

E-mail addresses: ahr@uevora.pt, ahrgoo@gmail.com.
http://dx.doi.org/10.1016/j.aop.2014.03.013
0003-4916/© 2014 Elsevier Inc. All rights reserved.

Annals of Physics 346 (2014) 22–27

Highlights

The principles of extremum of entropy production are not first principles.
They result from the maximization of conductivities under appropriate constraints.
The conditions of their validity are set explicitly.
Some long-standing controversies are discussed and clarified.

Article Info

Article history:
Received 6 February 2014
Accepted 29 March 2014
Available online 4 April 2014

Keywords:
Entropy production
Extremum principles
Non-equilibrium

Abstract

It is shown how both the principles of extremum of entropy production, which are often used in the study of complex systems, follow from the maximization of overall system conductivities, under appropriate constraints. In this way, the maximum rate of entropy production (MEP) occurs when all the forces in the system are kept constant. On the other hand, the minimum rate of entropy production (mEP) occurs when all the currents that cross the system are kept constant. A brief discussion on the validity of the application of the mEP and MEP principles in several cases, and in particular to the Earth’s climate is also presented.

1. Introduction

In the celebrated textbook ‘‘The Feynman Lectures on Physics’’ [1] the authors stated that: ‘‘minimum principles sprang in one way or another from the least action principle of mechanics and electrodynamics. But there is also a class that does not. As an example, if currents are made to go through a piece of material obeying Ohm’s Law, the currents distribute themselves inside the piece so that the rate at which heat is generated is as little as possible. Also, we can say – if things are kept isothermal – that the rate at which heat is generated is as little as possible’’.

The above statement marks just one moment in the long-standing debate about the legitimacy of using principles of extremum of entropy production rate as a basis to explain the behavior of certain systems out of equilibrium.

The principle of Minimum Entropy Production rate (mEP) was first proposed by Prigogine [2,3] as a rule governing open systems at nonequilibrium stationary states. The justification of mEP presented by Prigogine still continues to be the subject of heated controversy (see Refs. [4,5]).

Maximum Entropy Production rate principle (MEP) was proposed in 1956 by Ziman [6] in the form: ‘‘Consider all distributions of currents such that the intrinsic entropy production equals the extrinsic entropy production for the given set of forces. Then, of all current distributions satisfying this condition, the steady state distribution makes the entropy production a maximum’’. A practical application was put forward by Paltridge [7–9] who proposed that the Earth’s climate structure could be explained through the MEP principle. As in the case of the mEP principle, comments have appeared in the literature suggesting that the MEP principle as a basis for understanding the Earth-Atmosphere system is far from simple and universal (see Refs. [10,11]).

Recent work on mEP and MEP principles has focused either on the conceptual development and foundations of both principles [12–16], or on their applications to various systems, and namely the Earth’s climate [17–19].

2. Extrema of entropy production

In this paper we show that the so-called mEP and MEP principles may be derived (each one under two different sets of constraints) from the maximization of the conductivities that couple flows with the forces that drive such flows. The maximization of the conductivities follows from the general principle of maximization of ‘‘global flow access’’, known as the Constructal Law, which was first put forward in 1997 by Bejan [20] in the form: ‘‘For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed (global) currents that flow through it’’. The Constructal Law entails generation of flow configuration in such a way that it provides the highest global conductivity compatible with the existing constraints, and has successfully explained shapes and patterns of the animate [21,22] and inanimate [23–25] systems (see also some Constructal Law reviews [26–28]). In the general case of N forces F_i and N flows J_i (forces and flows are represented here as vectors), the entropy production rate σ is given by [3]:

where the symbol ◦ means dot product, i, j = 1, . . . , N, and F_k = ∇Φ_k, i.e. the gradients of the potentials Φ = {T , P, µ, . . . , φ}, are the forces that drive the flows J_i, which in the linear regime is read [3]:

where L_ik is the phenomenological coefficient that couples force F_k with flow J_i. In Eqs. (1) and (2) as well in the following we consider that the indices i and j run from 1 to N. With the help of Eq. (2), the entropy production rate – Eq. (1) – may be rewritten as:

which, according to the Second Law is always positive or null. Here, because we are dealing with non-equilibrium systems we consider σ > 0. Moreover, from Eq. (3) one concludes that the matrix [L] is symmetric, i.e., L_ik= L_k_i, and positive definite [29].

2.1. Fixed forces

First, let us consider the case when all the forces F_k are kept constant, and then take the differential of both members of Eq. (3):

which, according to the Constructal Law must be null. In fact, ‘‘easiest flow access’’ occurs when all L_ik conductivities are at their maxima, i.e. when

Hence, for the second order differential of σ one must have:

Therefore, from Eqs. (4) and (6) one can draw the following conclusion: ‘‘In the case when all the forces driving the flows are kept constant the entropy production rate reaches the maximum value that is compatible with the existing constraints’’.

Moreover, from Eq. (6) one also concludes that the matrix [−d²L] is positive definite.

2.2. Fixed flows

Now, let us consider the case when all the flows J_i are kept constant, together with Eq. (3). Because the matrix [L] is symmetric, i.e., L_ik = L_ki, and positive definite, has an inverse matrix [Λ] = [L]⁻¹ that is also positive definite [29]. Therefore, from Eq. (2) we can write:

Hence, in this case, Eq. (1) reads:

By taking the differential of both sides of Eq. (8), and noting that:

and invoking the Constructal Law (Eqs. (5)) again, one obtains:

Now, by taking the second order differential of both sides of Eq. (10), and using Eq. (9) to notice that dL_ik = 0 ⇒ dΛ_ik = 0, and considering Eqs. (5) one has:

In Eq. (11) we have used the property that the product of two positive definite matrices, [Λ] and [−d²L] is also positive definite [30].

Therefore, Eqs. (10) and (11) prove that ‘‘In the case when all the flows are kept constant, the entropy production rate reaches the minimum value that is compatible with the existing constraints’’.

In the more general case when F_k, k = l, . . . , m, and J_i, i = n, . . . , r are kept fixed, and taking into account Eqs. (1)–(5) and (7)–(10), the entropy production rate reaches an extremum if any of the following conditions is met:

Hence, the more the first members of Eqs. (12) deviate from zero, the less the system shall be in line with any of the principles of extremum of the entropy production rate. This could help explain the problems with the use of both the mEP and MEP principles in some complex cases (see Refs. [4,5,10,11]). However, in the cases when Eqs. (5) are met, then by using equations analogous to Eqs. (6) and (11) one can verify if systems are governed either by mEP or MEP principles.

3. MEP AND mEP principles as corollaries of the Constructal Law

The extrema of entropy production rate, which are often presented as principles governing the behavior of certain systems, are in fact corollaries of the Constructal Law (Eq. (5)) in certain specific situations.

Therefore, if all the forces are kept fixed, the Constructal Law (maximal flow access, or equivalently, maximal global conductivity arrangement) entails maximal rate of entropy production (MEP). This is for instance the case of the Sun–Earth system. The blackbody temperature at which the Sun emits the energy that reaches the Earth is constant (T_S = 5778 K) while the blackbody temperature of the Earth, T_E is about 254.4 K and constant at the climatic scale [31]. Hence, according to the Constructal Law, because the Earth system ‘‘processes’’ the energy coming from the Sun at a fixed ‘‘force’’, which is proportional to (T_S − T_E )(T_S T_E )⁻¹, the global Earth system must operate under the rule of maximum entropy production rate (MEP). This is a general rule for the global system, and namely shall hold at least approximately in the Climate system, therefore clearing up the relative success of Paltridge’s explanation [7–9] of the global state of the Earth’s climate by MEP principle. As regards subsystems of the climatic system only approximately we can consider that some of the mesoscale forces are kept constant, instead multiple forces and flows are not constant. This poses the problem of verifying the conditions of applicability of MEP principle at the pertinent scale in terms of Eqs. (12), and might help explain why MEP appears not to hold for certain subsystems [10].

Despite Constructal Law requires maximum rate of global entropy production in the Earth, it does not forbid that this objective is reached through the global arrangement of entropy production that includes processes in which entropy production rate is at its minimum, together with other processes in which it is at its maximum. For example, in the cases in which the flow rate is fixed, as it happens when water inflow through precipitation over a territory balances water outflow carried by the streams, at least approximately, it is expected that the flow organizes itself in such a way that the overall conductivity arrangement leads to a minimum of the entropy production rate [24]. In this case, Bejan [26] showed that the flow organization leads to a minimum of the driving force (water pressure head over the territory). Because the flow rate is fixed, minimal pressure head entails minimal entropy production rate, therefore confirming the general rule for this case.

At a larger scale one finds another process in which entropy generation rate is at its maximum: that of transport of the excess heat from the Earth’s equator (at temperature T_H ) to the poles which are at a lower temperature T_L. In this case, the global flow organizes itself in patterns that enable maximal ‘‘global heat conductivity’’ [23], and because heat flows under a fixed force proportional to (T_H − T_L)(T_H T_L)⁻¹, maximal entropy production rate is expected to occur.

It is also interesting to analyze here the ‘‘Theorem of Minimum Entropy Production’’ in Prigogine’s formulation [3]: ‘‘In the linear regime, the total entropy production in a system subject to flow of energy and matter, reaches a minimum value at the nonequilibrium stationary state’’. In the proof offered for the case of two forces and two flows, though Prigogine fixes one of the forces, he really fixes both flows thus reaching the conclusion that entropy production rate is at its minimum. As we have shown before, for the case in which all flows are fixed, the same result of minimum entropy production rate is anticipated for the much general case of any number of forces and flows.

The problem that puzzled Feynman [1]: ‘‘. . . if currents are made to go through a piece of material obeying Ohm’s Law, the currents distribute themselves inside the piece so that the rate at which heat is generated is as little as possible. Also, we can say – if things are kept isothermal – that the rate at which heat is generated is as little as possible’’, finds a straightforward explanation in the context developed here. In fact, Feynman fixes the ‘‘currents (that) are made to go through a piece of material’’ therefore it follows that in such conditions, because the entropy production rate is at its minimum, hence as a result ‘‘the rate at which heat is generated is as little as possible’’.

Surely it is not easy to find ‘‘pure’’ cases in which all forces, or all flows are fixed, instead mixed conditions are more likely to occur. Mixed combinations in which only a part of the forces and flows are fixed are also possible to be considered in terms of Eqs. (12). However, these cases require a more complex treatment, for the reason that the respective entropy production regimes do not appear so clearly as in the extreme cases in which either all forces are fixed, or that all flows are fixed.

4. Conclusions

We have shown that either the principle of minimum entropy production rate (mEP), or that of maximum entropy production rate (MEP), can be derived by maximizing the overall conductivity associated with each flow that crosses the system, under specific constraints. Thus, the entropy production rate is minimal when all the flows are kept constant. Likewise, the entropy production rate is maximal when all the forces that generate the flows are kept constant.

The principle of ‘‘maximum flow access’’, known as the Constructal Law appears as the theoretical foundation behind the mEP and MEP principles. It becomes clear that the Constructal Law is distinct from the Second Law, and is a fundamental law that rules the generation of configuration and structure in systems out of equilibrium.

Acknowledgments

The author acknowledges the funding provided by the Évora Geophysics Centre, Portugal, under the contract with Fundação para a Ciência e a Tecnologia (the Portuguese Science and Technology Foundation), Pest/OE/CTE/UI0078/2014.

References

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INTERNATIONAL JOURNAL OF HEAT AND TECHNOLOGY
Volume 34 (2016), Special Issue 1, pp.S147-S150
http://dx.doi.org/10.18280/ijht.34S119

AD-HOC PRINCIPLES OF “MINIMUM ENERGY EXPENDITURE” AS COROLLARIES OF THE CONSTRUCTAL LAW
THE CASES OF RIVER BASINS AND HUMAN VASCULAR SYSTEMS

A. Heitor Reis

Department of Physics and Institute of Earth Sciences (ICT), University of Évora, R. Romao Ramalho, 59, 7002-554 Evora, Portugal

Email: ahr@uevora.pt

ABSTRACT

In a recent paper [1] Reis showed that both the principles of extremum of entropy production rate, which are often used in the study of complex systems, are corollaries of the Constructal Law. In fact, both follow from the maximization of overall system conductivities, under appropriate constraints. In this way, the maximum rate of entropy production (MEP) occurs when all the forces in the system are kept constant. On the other hand, the minimum rate of entropy production (mEP) occurs when all the currents that cross the system are kept constant.

In this paper it is shown how the so-called principle of “minimum energy expenditure” which is often used as the basis for explaining many morphologic features in biologic systems, and also in inanimate systems, is also a corollary of Bejan’s Constructal Law [2].

Following the general proof some cases namely, the scaling laws of human vascular systems and river basins are discussed as illustrations from the side of life, and inanimate systems, respectively.

Keywords: Flow systems, Ad-hoc principles; Entropy production rate, Energy expenditure, Constructal Law.

1. SYSTEM DYNAMICS, ENERGY EXPENDITURE, ENTROPY PRODUCTION, AND THE CONSTRUCTAL LAW

Dynamics and evolution of complex systems are not easy to predict. This is why many studies of such systems generally invoke either principles of extremum of entropy production rate or extremum of “energy (power) expenditure” as the theoretical ground for explaining their dynamics.

The principle of Minimum Entropy Production rate (mEP) was first proposed by Prigogine [3,4] as a rule governing open systems at nonequilibrium stationary states: “In the linear regime, the total entropy production in a system subject to flow of energy and matter, reaches a minimum value at the nonequilibrium stationary state” [4]. The justification of mEP presented by Prigogine still continues to be the subject of heated controversy.

The principle of Maximum Entropy Production rate (MEP) was proposed in 1956 by Ziman [5] in the form: “Consider all distributions of currents such that the intrinsic entropy production equals the extrinsic entropy production for the given set of forces. Then, of all current distributions satisfying this condition, the steady state distribution makes the entropy production a maximum”.

The principle of “Minimum Energy Expenditure” has also been widely used. Classical examples from the side of living systems are the derivation of the scaling laws of branching in vascular systems by C. Murray [6-8], and from the inanimate side the studies on the tri-dimensional structure of river basins by Rodríguez-Iturbe et al. [9]. Interestingly, in the later work the authors also used the additional principle of “Equal Energy Expenditure”.

Being taken as “principles” none of the above statements has been demonstrated, and therefore should be considered as “rational beliefs” whose validity and utility have to be inferred from the adequacy of their predictions to observational data. However, it is beyond doubt that so many “ad-hoc principles” generate some intellectual discomfort, and so it becomes clear that an effort is needed to unify the underlying theoretical framework for the study of complex systems.

In a recent paper [1] Reis showed how both mEP and MEP principles stem from Constructal Law as corollaries. The Constructal Law, which states: “For a ﬁnite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed (global) currents that ﬂow through it” [2], was translated into the mathematical form:

where L_ik is conductivity that couples force F_k with flow J_i. The forces F_k = ∇Φ_kare the gradients of the potentials Φ = {T , P, µ, . . . , φ}. In fact, “easiest flow access” occurs when all conductivities L_ikare at their maxima. The results may be summarized as follows [1]:

“In the case when all the forces driving the flows are kept constant the entropy production rate reaches the maximum value that is compatible with the existing constraints”.
“In the case when all the flows are kept constant, the entropy production rate reaches the minimum value that is compatible with the existing constraints”.

Therefore, the extrema of entropy production rate, which are often presented as principles governing the dynamics of certain systems, are in fact corollaries of the Constructal Law in certain specific situations. Moreover, in each situation the Constructal Law requires a specific flow organization as a fundamental condition for achieving either maximum, or minimum entropy production rate.

In what follows we show that the so-called “principles of extremum of energy expenditure” also follow as corollaries of the Constructal Law.

2. THE “PRINCIPLES OF EXTREMUM OF ENERGY EXPENDITURE” AS COROLLARIES OF THE CONSTRUCTAL LAW

Though no unified formulation is found in the works that use these principles, the general idea is that many biologic and inanimate systems behave as if the whole “energy expenditure” required to sustain the internal processes was minimum. The derivation of the known scaling laws of dichotomous branching flow systems is such an example. For instance, Murray’s law for diameter scaling reads:

where the subscript 0 stands for parent vessel and subscripts 1 and 2 for daughter vessels. Murray [6, 7] derived his law using biological considerations, but Sherman [8] showed that it could be derived by considering minimum total power expenditure. Total power was considered to be the power needed to drive the flow in the Poiseuille regime that scales with D^-4L – where L is vessel length – plus the “metabolic power” required to maintain the volume of blood and vessel tissue involved in the flow, which in a cylindrical vessel scales with D²L. In fact, in Sherman’s minimization, the consideration of a “metabolic power” in addition to power for driving the flow is mathematically equivalent to minimizing the power needed to drive the flow under total constant vessel volume (vessel volume =πD²L/4), as Bejan et al. [10] did by using the Constructal Law.

The scaling laws of branching pulsatile flows have been studied by Silva and Reis [11], who showed that they reduce to Murray’s law at zero pulse frequency. Moreover, by using Constructal Law these authors were able to explain some physiological features such as the elongation of the ascending aorta with age [12], and also the different behaviour of the radial and carotid arteries with pulse frequency [13].

In their studies on the tri-dimensional structure of river basins Rodríguez-Iturbe et al. [9] found that known empirical scaling laws could be explained if the following theoretical framework was assumed: “(1) the principle of minimum energy expenditure in any link of the network, (2) the principle of equal energy expenditure per unit area of channel anywhere in the network, and (3) the principle of minimum total energy expenditure in the network as a whole”. With respect to the scaling laws of river basins, studies published after the paper by Rodríguez-Iturbe et al. may be found in the literature that show that the Constructal Law alone was able to anticipate the empirical scaling laws [14, 15].

To understand the connection between the Constructal Law and ad-hoc “minimum energy expenditure” principles first let us note that “minimum energy expenditure” actually means “minimum flow exergy destruction”. In fact the streams carry a flow exergy potential (per unit mass) that reads

where h stands for specific entalphy (Jkg^-1), T₀ for ambient temperature (K), s for specific entropy (JK^-1kg^-1), v for velocity (ms^-1), g for acceleration due to gravity (m²s^-1), and z for height above a reference level (m).

According to the Gouy-Stodola theorem: “In any open system, the rate of flow exergy lost for irreversibility $\dot{\Psi}=\dot{m}\psi$ , (which is negative, and where $\dot{m}$ is mass flow rate) and the entropy generation rate $\dot{S}_{gen}$ are related each another as $\dot{\Psi} = -T_0 \dot{S}_{gen}$ , where T₀ is the ambient temperature” (see for instance [16]).

Therefore by applying the Gouy-Stodola theorem to a flow tree with N channels, one has

In view of eq. (4), and the statements A and B, it follows:

A1 – When the entropy production rate reaches the minimum value, the rate of “energy expenditure” is also at the minimum;

B1 – When the entropy production rate reaches the maximum value, the rate of “energy expenditure” is also at the maximum.

Therefore it is clear that to each extremum of rate of “energy expenditure” corresponds the respective extremum of entropy generation rate. In this way, minimum “energy expenditure” is equivalent to minimum entropy production rate. As referred above, the so-called mEP principle is a corollary of the Constructal Law that is applicable when all currents are fixed, therefore the so-called principle of minimum “energy expenditure” is also a corollary of the Constructal Law.

In this context, the successful application of the principle of minimum “energy expenditure” to scaling of branching vessels (e.g. Murray’s Law) in the case of biologic systems finds its justification in the fact that blood flow rate is set by the fixed needs of the organs and tissues supplied by the vascular tree. Hence, because the flow is fixed, the entropy generation rate is at its minimum, and same occurs with the rate of “energy expenditure”.

Analogously, in the case of river basins, the global flow is fixed by the precipitation regime, hence the streams distribute in the basin in such a way that the entropy generation rate, and also the rate of “energy expenditure” are at their minima. This explains the successful application of the principle of minimum “energy expenditure” by Rodríguez-Iturbe et al. It is worth to say that Reis [14] and Bejan et al., [15] have anticipated the empirical scaling laws of river basins by using the Constructal Law alone, without needing to invoke any of the above mentioned ad-hoc principles.

On the other side, some authors found that another ad-hoc principle, the “principle of maximum energy dissipation” was useful in describing patterns of water infiltration in cohesive soil with different populations of worm burrows for a range of rainfall scenarios [17]. They found that “flow in connected worm burrows allows a more efficient redistribution of water within the soil, which implies a more efficient dissipation of free energy/higher production of entropy”. They explained that “this is because upslope run-off accumulates and infiltrates via the worm burrows into the dry soil in the lower part of the hillslope, which results in an overall more efficient dissipation of free energy”. In fact, it is not surprising that the pattern corresponds to the maximum entropy production rate, given that water infiltrates into the soil under a fixed force (i.e. the gradient of water potential).

A classical application of the “maximum entropy production rate” was carried out by Paltridge [17-20] who proposed that the Earth’s climate structure could be explained through the MEP principle. Also in this case, we can show that all the processes occurring in Earth that are powered by solar radiation occur in such a way that the net (whole) result is maximal entropy production rate.

For example, the processes involved in the transport of the excess heat absorbed at the equator (at temperature) T_H to the poles, which are at a lower temperature T_L, develop to jointly produce maximum entropy per unit time. In fact, in this case, the global flow organizes itself in patterns that enable maximal “global heat conductivity” [1, 2], and because heat flows under a fixed force proportional to (T_H − T_L)(T_H T_L)⁻¹, maximal entropy production rate is expected to occur.

Many other cases in which the ad-hoc “principles of extremum of energy expenditure” are invoked as the theoretical basis for understanding system dynamics could be presented here. In all such cases it would be possible to show that they stem from a unique principle: the Constructal Law.

3. CONCLUSIONS

“Energy expenditure” in the sense of exergy destruction is related to entropy generation. Therefore, ad-hoc “principles of extremum of energy expenditure” are equivalent either to the “principle of minimum entropy production rate” – mEP, or to the “principle of maximum entropy production rate” – MEP.

Both the mEP, and MEP principles are shown to be corollaries of the Constructal Law. Therefore, both the “principle of minimum energy expenditure”, and the “principle of maximum energy expenditure”, also are corollaries of the Constructal Law. In fact, taking together the statements A, A1, B, B1, from the Constructal Law it follows:

A2 – “In the case when all the forces driving the flows are kept constant the “energy expenditure” reaches the maximum value that is compatible with the existing constraints”.

B2 – “In the case when all the flows are kept constant, the “energy expenditure” reaches the minimum value that is compatible with the existing constraints”.

As the main conclusion: there is no need for using ad-hoc “principles of extremum of energy expenditure”, because a unique principle – the Constructal Law – provides the theoretical basis for describing the dynamics of flow systems.

ACKNOWLEDGMENT

The author acknowledges the support provided by ICT under contract with FCT (the Portuguese National Science Foundation).

REFERENCES

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