Guest Post by Willis Eschenbach
One of the most fundamental and far-reaching discoveries in modern thermodynamics is the Constructal Law (see the wiki entry as well). It was first formulated by Adrian Bejan in 1996. In one of his descriptions, the Constructal Law is:
For a finite-size (flow) system to persist in time (to live), its configuration must evolve such that it provides easier access to the imposed currents that flow through it.
Figure 1. An example of the Constructal Law processes at work in a river system. Formation of meanders, followed by cutting through a meander to form an oxbow lake. Image Source.
The Constructal Law has been described as being as important as the first two Laws of Thermodynamics, but most people have never heard of it. What does the Constructal Law mean in plain English, and what does it have to do with the climate?
Here is a different statement (pdf) of the Constructal Law, again from Bejan:
In 1996, the constructal law was formulated and proposed to expand thermodynamics in a fundamental way.
First was the proposal to recognize that there is a universal phenomenon not covered by the first law and the second law. That phenomenon is the generation of configuration, or the generation of ‘design’ in nature.
All thermodynamic systems in nature are flow systems (i.e. live, non-equilibrium systems), and they all have configuration. If they do not have it, then they acquire it, in time. The generation of configuration is ubiquitous, like other phenomena covered by other ‘laws’ in physics. Biological systems are configured. Geophysical systems are configured. Engineering and societal systems are configured. The configuration phenomenon unites the animate with the inanimate. All the other phenomena of physics (i.e. of ‘everything’) have this unifying power. Falling rocks, like falling animals, have weight, conserve energy, generate entropy, etc.
Second was the statement that this universal phenomenon should be covered by the constructal law. This law accounts for a natural tendency in time (from existing flow configurations, to easier flowing configurations). This tendency is distinct from the natural tendency summarized as the second law.
Again not necessarily the clearest statement, but the general idea of the Constructal Law is that flow systems continually evolve, within the physical constraints of the particular system, in order to maximize some variable(s).
A meandering river in bottomland is a good physical example to understand what this means. In the case of a river, what is being maximized by the flow system is the length of the river. However, this ideal condition is never achieved. Instead, the river length oscillates above and below a certain value.
As shown in Fig. 1, in an “S” shaped river, the moving water erodes the outside of the bends and deposits silt on the inside of the bends. Of course, this inevitably makes the river longer and longer. But when the river does this for a while, it gets too stretched out for the land to bear. At some point, the river cuts through and leaves an island and what will become an oxbow lake.
That leaves the river shorter. Again the lengthening process continues, until the river cuts through some other bend and shortens again. And as a result, the length of the river oscillates around some fixed value. It is constantly evolving to maximize the length, an ideal which it never attains.
Now, here’s the point of this whole example. Suppose I didn’t know about this active, evolutionary, homeostatic characteristic of rivers. If someone asked me if a river could be shortened, I’d say “Sure. Just cut through a meander.”. And if I cut through the bend I could physically measure the river length and prove that indeed, the river was shorter.
But would that really make the river shorter?
Of course not. Soon the relentless forces of flow would once again increase the length of the river until the next cutoff forms another oxbow lake, and the cycle repeats.
Net effect of my cut on the length of the river? None. The length of the river continues to oscillate around the same fixed value.
The key to understanding flow systems is that they are always “running as fast as they can”. They are not just idling along. They are not at some random speed. They are constantly evolving to maximize something. The Constructal Law ensures that they are up against the stops, so to speak, always going flat out.
What does all of this have to do with climate? The Earth’s climate is a huge flow system. It circulates air and water from the tropics to the poles and back. As a result the climate, like the river, is subject to the Constructal Law. This means that climate is constantly evolving to maximize something. Climate, like the river, is also “running as fast as it can”.
What does the climate flow system maximize? Because it is a heat engine (converting sunlight into the physical work of the planetary circulation), Bejan says (pdf) that it is doing a dual maximization. It maximizes the sum of the work done driving the planetary circulation, and the heat rejected back to space at the cold end of the heat engine. Again in Bejan’s words:
The earth surface model with natural convection loops allows us to estimate several quantities that characterize the global performance of atmospheric and oceanic circulation. We pursue this from the constructal point of view, which is that the circulation itself represents a flow geometry that is the result of the maximization of global performance subject to global constraints.
The first quantity is the mechanical power that could be generated by a power plant operating between Th and Tl, and driven by the heat input q. The power output (w) is dissipated by friction in fluid flow (a fluid brake system), and added fully to the heat current (qL) that the power plant rejects to Tl.
where Th and Tl are the temperatures of the hot and cold ends of the system. The system is maximizing the sum of work done and heat rejected.
There is a most fascinating interplay between those two. When the speed of the planetary circulation is low, so are the turbulent losses. So as speed increases, up to a certain point the sum of work done (circulation speed) and heat rejected is also increasing.
But as the speed increases further, the turbulence rapidly starts to interfere with the circulation. Soon, a condition exists where further speed increases actually decrease the total of work done and heat rejected. That is the point at which the system will naturally run. This is why nature has been described in the past as running at “the edge of turbulence”.
What does that mean for understanding the climate? This is a new area of scientific investigation. So I don’t know what all of that means, there’s lots of ramifications, some of which I may discuss in a future post. However, one thing I am sure of.
If we want to understand the climate, or to model the climate, we have to explicitly take the Constructal Law into account.
We are not modeling a simple system with some linear function relating forcing and response. That kind of simplistic understanding and modeling is not valid in the type of system where, for example, cutting a river shorter doesn’t make it any shorter. We are modeling a dynamic, evolving system which may not be affected by a given forcing. The modelers claim (falsely, but we’ll let that be) that their models are based on “physical principles”.
However, they have left one central, vital, physical principle out of the mix, the Construcal Law. And at the end of the day that means that all of their modelling is for naught. Sure, they can tweak the model so that the output resembles the actual climate. But the actual system does not change over time in a random way. It is not driven here and there by forcing fluctuations. It changes in accordance with the Constructal Law. The future evolution of the climate, what Bejan calls the “generation of configuration”, is ruled by the Constructal Law. It cannot be understood without it.
PS – For those that think that the Constructal Law is some crackpot theory, it is not. Bejan is one of the 100 most cited engineering authors of our time, and the results of the Constructal Law have been verified in a host of disciplines. It is indeed a new fundamental law of thermodynamics, one which we cannot ignore.
Joel, it isn’t a double standard. It’s human nature to instinctively listen to and follow a person that strikes a chord in your own brain. The trick is to know whether or not you are being led around by a ring in your nose. That includes questioning the nose ring someone else may have put there, as well as your own brain you put there.
Joel Shore says:
November 16, 2010 at 4:18 pm
And you are impressed by awards from someone’s employer and AAAS and APS? I guess we live in parallel scientific universes … Hansens’s science is good. But it is not stellar. It is not extraordinary. He is not even listed on the ISI HighlyCited site, and Bejan is in the top 100.
If you can’t see the difference … well, perhaps that’s why you believe so strongly in the AGW hypothesis.
Feet2theFire says:
November 16, 2010 at 6:10 pm
There are plenty of people with papers and awards. What Bejan has is achievements that have led to breakthroughs. I respect that more than Hansen’s Heinz Ketchup Award, but you are free to differ …
Feet2theFire says:
November 16, 2010 at 6:10 pm
I know people don’t like it.
But I don’t like it when people ask questions without first investigating what they are talking about. If you want to be spoon-fed, you’ve come to the wrong place. Either do your homework or move on. I’m not here to force anyone to learn something. If you are interested in e.g. the math of the Constructal Law, as some people have asked about, then all I can tell you is to read the relevant documents so we can have an interesting discussion. This is not a course in the mathematical or other details of the Constructal Law. It is intended as a discussion of the implications of the Constructal Law for climate science.
RockyRoad says:
November 16, 2010 at 4:21 pm
Phil. says:
November 16, 2010 at 3:57 pm
“Are you sure, isn’t the river attempting to achieve a brachistocrone profile and thereby maximize flowrate? If at a suboptimal profile it might achieve that by lengthening. If the meandering river is already at the optimum flowrate then shortening it or lengthening it will slow down the flow.”
We assume the elevation difference between two points along a river is fixed, whether the stream is meandering or not–hence the river would have a steeper gradient the shorter it is, reaching maximum gradient in a straight course.
Water flow would be fastest for the steepest gradient; it would be slower for anything less than the steepest gradient.
Not true, the fastest descent would be down the brachistocrone, not the steepest gradient, it will be slower for any curve not conforming to the brachistocrone.
Hence, decreasing the length between two points by straightening the river will speed the river up (it corresponds to the steepest gradient); increasing the length will slow it down (it corresponds to less than the steepest gradient), provided the same volume of water is passing through.
Since the foregoing isn’t true this doesn’t hold either.
Phil. says:
I love your weightless, massless, frictionless explanation regarding a brachistocrone, Phil.
I find the definition of a brachistocrone, or “a curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and is constrained to move along the curve to the second point, under the action of constant gravity and assuming no friction.”
http://en.wikipedia.org/wiki/Brachistochrone_curve
However, you conveniently forget that streams are real world environments with intial velocity, have significant mass, and also considerable friction. Indeed, that’s where it all starts to fall apart, since the definition of a brachistocrone also includes the disclaimer that if the body we’re considering (the stream) is given an initial velocity or if friction is taken into account, the curve that minimizes time will differ from a brachistocrone.
So you can stay in your hypothetical world and believe in brachistocrones all you want. Not me. I’ll take a straight river course with steepest gradient any day–but then, that’s just the engineer speaking.
@tallbloke
…if these things bore you, or you feel the odds are too low to bother, feel free to go and get cynical about the next thread along the line. 😉
witty but unfair. I did consider it & I’m not a cynic. I don’t want to be a me-too poster on topics I and everyone else likes. I just read it and move on when I like something.
This brachistochrone would appear as a curve to the east or west of a river flowing due south???
Or, is it a curve in the vertical profile of the river? Id est carving a deeper bed in the upper stretch and depositing sediment for a shallower course lower down.
If all that’s being claimed is the latter, then its course would still be a straight line from north to south.
I would love to hear of an actual river that exhibits anything like this characteristic. Even a diagram with dimensions would be impressive.
@Steve Garcia –
I forgot to mention that the Constructal law talks about maximising its processes, but what I’ve observed in nature is that it attempts to minimise losses. The two are not the same even though they may have similar outcomes.
So I guess GDP is like a hot air balloon… you have to keep burning fuel to stay aloft.
This is where the Constructal Law becomes interesting because it facilitates change… usually small variations around the optimum as the flow peaks and trough…. like the bends in the Mississippi river… but sometimes the changes may be more catastrophic – like Hurricane Katrina breeching the levees in New Orleans… so it is important to remember that flow systems are subject to Destructive Events… especially manmade flow systems.
If you take inland transportation, as an example, then the increasing utilisation of fuels has enabled societies to evolve from using horse and carts… first they built canals… then railways… then tarmac roads… and finally airports… this is the evolutionary history of the industrial revolution which has steadily increased the flow of people and goods to every corner of the land.
The point being that it has taken generations for this system to evolve and that, by definition, there is no fallback plan… we drive cars to the supermarkets – not horse & carts… the supermarkets are restocked by road – not railways. The system is only as resilient as its weakest link… this resilience can be counted in days on one hand… so it is always good to remember that our energy dependent society is always only five days away from collapse…. Man Proposes But Nature Disposes.
Willis Eschenbach says:
November 16, 2010 at 1:02 pm
Paul Birch says: November 16, 2010 at 4:30 am …
5) I’m afraid that Bejan, in falsely believing he has found a great new organising principle, is indeed promoting something of a “crackpot theory”, and Willis, in attempting to apply that notion to the climate, is falling into the same error.
“Sometimes you guys crack me up. First, Bejan applied the idea of the Constructal Law to the climate, not me, ”
You are quite obviously attempting to apply his ideas to the climate through your own pet theories. I never said Bejan didn’t do much the same.
“More to the point, Bejan has a PhD from MIT and is a full professor at Duke University … … … … [ and more and more ad nauseam)].”
I’m astonished you have the gall to perpetrate so egregious an example of the fallacy of “argument by authority”. The number of PhDs and respected scientists who have promoted “crackpot theories” (your words) is legion. Even the great Newton, Einstein and Galileo were not immune.
As stated, the so-called Constructal Law is not a scientific law. It is not falsifiable, because one can pick and choose what flow variable is the one it maximises. The specific example you gave (that of a meandering river) was false. Rivers do not do what you claim, as many commenters have pointed out to you. Nor does the climate. To treat this vague “organising principle” as a newly established fundamental law of thermodynamics is “crackpot”. At most, it is a potentially enlightening way of looking at old problems.
@Willis
“What Bejan has is achievements that have led to breakthroughs.”
I can’t find any specifics on those “breakthroughs”. I even did a patent search which turned up empty. That’s highly unusual in engineering.
Surely you must know some specific practical things that have come about due to these breakthroughs. Please share some examples with us.
I believe there may indeed a Constructal thermodynamic principle in nature that perhaps explains why radiation from the sun and the stars eventually caused the construction of all the intricate complexities on our planet, but I am highly skeptical that Adrian Bejan’s wording is the proper expression of the principle of that process.
“Willis Eschenbach says:
November 16, 2010 at 12:37 pm
OK, Jose, try this link …
files.me.com/williseschenbach/y1gqp0
It seems they have decided to password protect all my public files … I am very unhappy about that.
w.”
That worked, thanks for your time.
Jose
“In an open system, unrestrained by ACOE dikes, erosion from the land surrounding the defined bed and bank river path will creat gulleys and perennial or intermittent streams.
In a straight system, erosion along all sides of the stream should happen equally, maintaining straightness. This is the principle. If areas of erosion differ (hard rock in one section as opposed to soil) then that’s the river reacting to the environment and attempting to find a new optimum.
As described by willis the river WANTS to be as long as possible. BS. The river doesn’t desire to meander. It MAY meander due to the nature of the physical systems and what it runs into, but ultimately the river follows topography. The river WANTS as easy a system as possible to get from a high energy level to a low energy level. The environment change change, causing the optimum path to vary over time, but that’s not because the river desires to be longer and take a circuitous siteseeing path.
It does not WANT to maximize length. If that were true then instead of a single river you’d have an entire landscape covered with streams spreading out in every direction. The construction law is basically a BS rule-of-thumb, that merely observes that order can arise out of even chaotic systems. No great observation there.
Could Bejan’s Constructal law just be describing the Lyapunov stability of attractors in the phase space of a non-equilibrium non-linear pattern system? Phase space can be almost anything, thus the universality of the statement that “certain parameters are maximised”.
On the other hand, what if attractors are themselves an attractor? One of the outcomes of Lyapunov stability is the unexpected pervasiveness and persistence of emergent pattern at the boundary of linearity and chaos, where Willis states that Constructal law tends to confine a dynamic flow system. For instance why are all coastlines fractal, rather than just a subset of them? Why also do most rather than just some biological structures show fractal and emergent nonlinear pattern? Why do nonlinear interactions between clouds make them more persistent than they “should” be?
There is something there, we are smelling the smoke but maybe haven’t quite found the fire.
RockyRoad says:
November 16, 2010 at 9:00 pm
Phil. says:
Not true, the fastest descent would be down the brachistocrone, not the steepest gradient, it will be slower for any curve not conforming to the brachistocrone.
I love your weightless, massless, frictionless explanation regarding a brachistocrone, Phil.
I find the definition of a brachistocrone, or “a curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and is constrained to move along the curve to the second point, under the action of constant gravity and assuming no friction.”
http://en.wikipedia.org/wiki/Brachistochrone_curve
However, you conveniently forget that streams are real world environments with intial velocity, have significant mass, and also considerable friction. Indeed, that’s where it all starts to fall apart, since the definition of a brachistocrone also includes the disclaimer that if the body we’re considering (the stream) is given an initial velocity or if friction is taken into account, the curve that minimizes time will differ from a brachistocrone.
It will still be a brachistocrone, just a different one (an additional friction term), it will not be a straight line!
So you can stay in your hypothetical world and believe in brachistocrones all you want. Not me. I’ll take a straight river course with steepest gradient any day–but then, that’s just the engineer speaking.
One who ignores the discoveries of Newton, Bernoulli et al., I would hope that most engineers know that a straight channel is not the fastest way to conduct water from A to B. Here’s an illustration for you: http://curvebank.calstatela.edu/brach/Straight2NickForever.gif
“Paul Birch says:
November 16, 2010 at 4:30 am
1) The so-called “Constructal Law” is not a scientific law, because it does not state what specific physical property of a system will be maximised or minimised. In general terms, it is a behaviour that can be seen in every conceivable “flow system”, because that is the mathematical nature of variables; in any mathematically definable differentiable process or procedure something (some variable, or some combination of variables) must be maximised (or its reciprocal minimised). In the expression a+b=c, what is being minimised is the difference c-(a+b). This is not some deep quiddity; it’s a tautology.”
files.me.com/williseschenbach/y1gqp0
Paul, read the paper – the link is now open (above). Yes it does, the “specific physical property” can be called Optimal Flow. I think the confusion is coming from the word “maximized” which after reading the paper I believe it to mean “optimized”, and as such it makes complete sense. What I get now is that flow systems, as in heat engines, are constantly “hunting around” for this optimal flow. As such then “optimal flow” is not rhetorical. The earth atmosphere is a perfect analog for testing this because it has all the necessary components, with ample descriptors supplied by other previous studies that can all be expressed mathematically.
From the horse’s mouth (the last two paragraphs – but read the whole paper):
“To see why the constructal law of optimization of flow architecture is distinct from the second law, let us review the essence of thermodynamics. Thermodynamic theory was developed in order to account for the functioning and improvement of heat engines. The first law accounts for the conservation of energy, and serves as definition of energy as a concept. The second law accounts for the generation of entropy (irreversibility), on the one-way nature of currents that overcome resistances, and serves as definition of entropy as a concept.
The first law and the second law account for the functioning of a given (observed, assumed) heat engine configuration. The history of flow systems (e.g. heat engines) shows what the first law and the second law are not covering: the case-by-case increase in performance, in time. Every class of flow systems exhibits this behaviour, from river basins to animals and heat engines. Each system in its class represents a flow architecture. New flow systems coexist with old systems, but persist in time if they are better, while older systems gradually disappear. This never-ending parade of flow systems represents the generation of flow architecture-the generation of geometric form as the clash between objective and constraints in flow systems. This is the phenomenon summarized in the constructal law.”
Paul Birch
November 16, 2010 at 4:30 am
Good Post.
This so called “Constructal Law of Flow Systems” is not a real physical law derived from accepted principles of physics and mathematics, as far as I can tell. There is no mathematical proof of anything.
In fact there are examples where the flow of a river has been altered in a significant way. The construction of a dam can do this. The flow rate of the river will not be the same. A dam can create a situation where evaporation reduces the flow rate of the river downstream.
In fact, the dam analogy is appropriate here because John Tyndall, who discovered the role of greenhouse gases 150 years ago, by measuring the IR absorption of gases, made the analogy that the earth’s atmosphere was a dam which causes the level of water behind the dam to rise. The water level is analogous to the temperature. Arrhenius pointed out in 1896, that the burning of coal would make the dam higher and increase the water level behind the dam.
If we are going to use a river as an analogy this would seem to be the one to use.
Every scientist knows that an analogy is a means of explaining the results of a physical and mathematical theory to laymen, and is not the basis of a scientific theory.
“”””” Phil. says:
November 17, 2010 at 6:07 am
RockyRoad says:
November 16, 2010 at 9:00 pm
Phil. says:
Not true, the fastest descent would be down the brachistocrone, not the steepest gradient, it will be slower for any curve not conforming to the brachistocrone.
One who ignores the discoveries of Newton, Bernoulli et al., I would hope that most engineers know that a straight channel is not the fastest way to conduct water from A to B. Here’s an illustration for you: http://curvebank.calstatela.edu/brach/Straight2NickForever.gif
Well why use a two syllable word; when a four syllable one will do just as well .
I’ll stay with Cycloid.
If memory serves me (you guys keep on dredging up ancient memories); it doesn’t matter where you start on the cycloid; or even on the brachistochrone; the time it takes to reach the bottom is exactly the same.
So if you did start with the straight line steepest slope, and the water was able to erode the bottom; we might as well make the material homogeneous, in line with all the other ideals: What shape would the water end up carving out for itself; assuming (another ideal) that it is constrained in the vertical plane ?
“”””” RockyRoad says:
November 16, 2010 at 9:00 pm
Phil. says:
Not true, the fastest descent would be down the brachistocrone, not the steepest gradient, it will be slower for any curve not conforming to the brachistocrone.
I love your weightless, massless, frictionless explanation regarding a brachistocrone, Phil.
I find the definition of a brachistocrone, or “a curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and is constrained to move along the curve to the second point, under the action of constant gravity and assuming no friction.”
http://en.wikipedia.org/wiki/Brachistochrone_curve
However, you conveniently forget that streams are real world environments with intial velocity, have significant mass, and also considerable friction. Indeed, that’s where it all starts to fall apart, since the definition of a brachistocrone also includes the disclaimer that if the body we’re considering (the stream) is given an initial velocity or if friction is taken into account, the curve that minimizes time will differ from a brachistocrone. “””””
Well I presume that the reason the word “brachistochrone” even exists; lies in that “curve of fastest descent”. (izzat Greek ?)
So the brachistochrone for the zero velocity start, and the frictionless fall happens to be the cycloid. Presumably under other non -ideal conditions the brachistochrone is still a brachistochrone ! Isn’t it ??
Interesting dialogs (polylogs?)
I think that we are not dealing with a law, but an example. or corollary to existing laws.
Fick’s First and Second Laws of Diffusion are overriding my thoughts about the Constructal Law. Fick says that diffusion, or trafficking of molecules from one place to another depends on the initial concentration of the particles. The initial concentration is a special example of average chemical potential, or enthalpy of the individual particles, driving things from high to low energy. According to thermodynamics, things move naturally from areas of high potential to low potential. For Fick, that is high concentration to low concentration, and for rivers, areas of high gravity potential to low. On the molecular scale, each particle moves in a random walk, or drunken sailor walk, as has been mentioned above. Since we have trillions of particles, or molecules, these statistically average to marching in a straight line until the chemical potential reaches equilibrium.
What is different here, is that we do not have an almost infinite number of rivers, although they are surely composed of particles, so we do not have a good statistical number to estimate the population (p) of all rivers possible. We only have (n) a singular case. We do not have (p-n) even, but we can approximate what (p) looks like if we imagine a hundred or a thousand rivers starting out spaced a few hundred yards apart. Ignoring coalescence for a moment, assuming each river stays separate, they proceed to the low potential river delta, each doing the random walk. On average, though, they move in a straight line. A Fickian result.
The random walks of droplets on a window pane have also been statistically modeled, [I think in Nature or somewhere recently]. Each droplet meanders like a river down the glass, but eventually each loses potential energy when it gets to the bottom. On average, with a million droplets, these average a sheet of water moving in a straight line from top to bottom.
So why do we call the Constructal Case a Law?
George E. Smith says:
November 17, 2010 at 9:01 am
Well I presume that the reason the word “brachistochrone” even exists; lies in that “curve of fastest descent”. (izzat Greek ?)
Indeed it is, ‘shortest time’.
So the brachistochrone for the zero velocity start, and the frictionless fall happens to be the cycloid. Presumably under other non -ideal conditions the brachistochrone is still a brachistochrone ! Isn’t it ??
Quite so, it’s more difficult to solve but can be done analytically for friction, you end up with the cycloid plus an additional friction term:
x=0.5k^2((Θ-sin(Θ) + mu(1-cos(Θ))
y=0.5k^2((Θ-cos(Θ) + mu(1+sin(Θ))
In the problem of the eroding river bed you referred to I’d expect optimization al la Bejan to tend towards the brachistocrone.
Jose Suro says:
November 17, 2010 at 6:25 am
Paul Birch says: November 16, 2010 at 4:30 am
1) The so-called “Constructal Law” is not a scientific law, because it does not state what specific physical property of a system will be maximised or minimised.
“Yes it does, the “specific physical property” can be called Optimal Flow.”
This is not a physical property. It is a subjective quality. “Optimal” means a state that a person prefers. Optimisation is economics, not physics. Rivers aren’t “trying” to flow fast or slow down, or to remove material from the river bed, or even to reach the sea. They just follow the mechanical laws of physics and do whatever they happen to do; even when this process maximises some interesting parameter (such as river length), it is not a universal law; other rivers will do something else, or maximise some other parameter. There is no such thing as an “optimal flow” for rivers – it is a meaningless descriptor, a particularly pathetic case of the pathetic fallacy!
Here is one counter-example which (perhaps) shows that the “Constructal Law” is not generally applicable to all ongoing, finite processes: growth of a protective oxide film on the surface of freshly-cut metal.
The driving force for this process is the free energy of oxidation of the metal. The “current that flows through the it” must be the rate of combination of oxygen with the metal. This rate declines over time (because oxygen and metal are separated by an ever-thickening layer of protective oxide), but does not necessarily tend to zero.
I daresay there is some explanation as to why such counter-examples don’t apply to the “Constructal Law”; and I expect that Warmists would be keen to get there hands on such a piece of reasoning.
By the way, there seem to be quite a few self-citations in Bejan’s paper – does the “Constructal Law” have anything to say about that? Does “flow” (rate of publication?) lead to “form” (having a lot of citations?), because this increases access to the flow (i.e. more self-citations become possible?). I think we should be told.