I propose a challenge to Maximal Flow theories by a new theory:Indifferent Time

I'm giving people an early peek at a new theory that I have developed in response to essentially another kind of theory, (or a system of theories) which involves optimal energy dissipation and or system flow optimization. So called "Flow optimization" as well as optimal heat dissipation are related in principal, a principal that I specifically am addressing with my own work. I hope this derivation will serve as a chapter in a larger work.
The theory that I am specifically disproving here is a flow optimization theory, called "constructal theory." But this formal disproof would also apply to other maximization theories, such as maximal entropy production.


 [This discussion is relating to a very large kind of theory, a very broad theory about systems. It is not of course specific in any way to specific experiments or individuals or in any way related to specific engineering feats, but to a theory relating to how systems work which encompass behaviors/phenomenon at a very high level, say the "40,000 foot view" but possibly higher. It would not have anything to do with the quality of someones experiment they did on a Tuesday, this discussion relates to what that work itself might mean in terms of thermodynamics, the heat dissipation of the building etc., to use perhaps an awkward analogy which is partly true, as dissipation should be independent of "how" people are working as should the "optimized" work flow of their activities. After all, constructal theory HAS been applied itself to such wide ranging examples as to why a World Cup Football team wore white instead of a darker color..(*constructal.org)]


If my derivation is correct, then this notion of constructal law" and OTHER dissipative or optimizing theories, would not be relevant at this level, which is again very theoretical and broadly encompassing. The "constructal law" has been widely promoted by Dr Bejan as a thermodynamic law, with some 10,000 scientific references, and is accepted as a text book on thermodynamics (see link to Constructal.org). So my derivation challenging the constructal law, would certainly be significant.



One of the very difficult things about constructal theory is to understand what it is, in fact claiming. That is to say, to find a definitive scientific argument. This is unfortunate. In general, most scientific papers should not be so difficult to read, and it is not simply because it is technical. I would reference (by way of general comparison), Prigogine’s Nobel lecture, England’s paper (which I blogged on earlier), or any number of chemical papers that I read routinely where one can list the main arguments, the relevant data and the clear discussion about how the hypothesis might account for them. The thesis is easily understood, as are the claims. Sadly, this is not the case for constructal theory. It seems that some other standard of science applies to these papers, but what is striking are the few references to other obvious subjects, like biology and chemistry, and possibly if there were more of these, Dr. Bejan would not make claims like "Science generally believes that evolution cannot be observed in our lifetime..." a claim that seems truly uninformed about papers in biology that study mutation in bacteria. But there is also the general claim of a theory linking the inanimate with the animate that never mentions genetics or biochemistry? One of the terms I have attempted to understand in constructal theory is Dr Bejan’s use of “evolution”. It is not clear at all what this term means physically in the larger “constructal theory”.  I have at least three text books of advanced chemistry on my desk, and none of these ever talks about “evolution” as a chemical descriptor of chemical behavior. So what exactly is this meaning? I highly doubt you will find, this theory in any physics text book. And for good reason. The particle theories have been formulated to descrirbe these very behaviors discussed in "constructal law", in chemical and physical terms. I hope that my disproof of theories like Constructal law will help rather than hurt progress forward. I cannot imagine that a misinformed theory at best, taught in schools, should in any way help progress forward in science. It appears that its proponents are not too keen to share their knowledge or answer questions. What nonsense is this? And what does it say about the status of theoretical science?

At the Constructal.org Blog itself, I posted [1][2] what I thought was a fair question on this point, attempting to get more description or clarification about constructal paper. Unfortunately, the Constructal.org blog editors it seems, are not really interested in replying to scientific inquiries or even clarifying what they mean. You can read the exact post I made here, which is not on the Constructal site because it is still “awaiting moderation”.  The Constuctal.org blog is said to be “an intimate” blog of Dr Adrien Bejan, who postulated constructal theory in 1996.

Let me state very generally, that I simply don’t agree with constructal theory’s misconstruction or reconstruction of physics, whatever the case may be. It is a reversal in my opinion, of progress in science.This kind of physics, in which inanimate nature is deliberately imbued with function and purpose in order to operate, was moved away from thousands of years ago. The paper itself is laden with anthropomorphic presumptions. There is a very big difference between using expressions or metaphors, and invoking these as direct mechanisms ,or even "laws". I note also that Dr Bejan claims in some recent papers that the Constructal Law is (comparable) to the Second Law of Thermodynamics. [I have pointed this out specifically in a previous reply (see this one) as I believe many of these arguments essentially “beg the question,” that is, they must be first accepted in order to make the premise itself, tenable. That is another point I explore previously, and if that is really the case, then that is truly unfortunate. What are the objectives of this essay? They are very clear: to explore the SCIENCE of these notions expressed by constructal theory, they are obviously defining in their numerous scientific papers and teaching books, a "territory" of science. So the exploration of a scientific hypothesis, is in a sense, an exploration of science itself, that is the objective on equal grounds with virtually any other scientific pursuit. The other objective is to delineate and differentiate my own theories from what is extant.]

 I have, in reply to this paper specifically, a hypothesis which is a stand alone theory, but coincidentally, disproves the constructal theory premise, which I call an “indifferent time” theory. "Indifferent time" states that at any given time point the flow through the system, is indifferent, it is neither optimized nor unoptimized in terms of maximal flow or minimal flow. (1)
I have derived this from Newton’s laws of motion, and it can be shown at microstates or macrostates. The burden as I see it, is on the constructal theory to provide evidence that there is some physical data, or data suggesting that there is an imbalance of forces at some time point. If not, then we can only assume that the normal forces (due to flow) are in perfect equilibrium with the system, be it a river delta or any other inanimate physical system at any time t. The basic premise of a system changing to “optimize” flow would essentially be false as it is always in perfect equilibrium with its surrounding forces. Since the presumption of change (evolved or optimized) can’t be substantiated, the data itself would be challengeable. I object to the use of partial differential equations in this paper (and as I've indicated elsewhere, without physically defined units, it's not at all clear what they might actually reference). In my opinion, one would not provide a formal equation or analysis, until the problem area is clearly established.  What you will find however, in this review, is that I am not actually showing the conflict here between constructal theory itself, but this is more broad in scope and likely encompasses other optimization and/or dissipation theories, really more broadly, to the concept of ones physical state “leading” towards another because it is more preferred. In that sense there are many theories which propose optimization in different forms either via entropy gain or heat dissipation.

According to constructal theory, the earth is an “engine”.

As Bejan states : “Because the flowing Earth is a constructal heat engine, its flow configuration has evolved towards paths that generate less irreversibility.”

The problem with this claim-i.e. the assumption that a physical system will “evolve” is that to evolve, it is implied that such a system would need to be in a less preferred state (at some initial time) and then alter itself (at a later time) to become a more preferred state. That is the understanding that the word “evolve” conveys. The statement..”evolved towards” some path” implies that it is heading in a direction and is obviously, not yet there. And actually, Bejan does indicate that this is what these systems do, and in fact some are not “alive” but will be once they take on the property of maximal flow.

As he indicates in Fig. 4, “(Bejan & Paynter 1976; Bejan 1982, 2006). The constructal law governs how the system emerges and persists: by generating a flow architecture that distributes imperfections through the flow space and endows it with configuration. The ‘engine’ part evolves in time towards generating more power (or less dissipation), and as a consequence, the ‘brake’ part evolves towards more dissipation.” There is an implication in that statement, that there is a physical difference that can be made, tested or proven to exist, between for example, a system which dissipates “less”, and a system that dissipates more. The further implication is that at some point in time t, the physical system is not optimized, or not configured” to dissipate. In reality, there is no physical distinction here, which is what we would expect of a natural system. (2)
As I have just stated in Indiffernt Time: any system, is already dissipating its "ideal" amount of energy in accordance with the localized physical laws that apply. The physical laws are not "behaving" differently in one region or another, thus we have no reason to believe that these laws would "care" to increase or decrease optimization. Nor would we be able to state that at one time point the system was NOT optimized and at another point it WAS optimized. (3)
It is not clear how one would go about showing what this difference is. (4)
What is clear, and what I believe constructal theory is not realizing, is that even when a system doesn't APPEAR to be optimized for heat dissipation, it actually is. Again, it is indifferent, and Nature cares not if you view a system perfectly "functioning" hurricane, or a light trade wind, each of these systems is in perfect harmonic balance with the localized forces that apply. And we further can see the universality of this statement and the implications of my hypothesis, Indifferent Time, to other theories which portend to operate upon the assumption of "preferred" or optimized dissipation of energy. (5)
The surprising result is that there is no optimal dissipation of energy in a given system it is always optimized because the forces are in balance at any time t. Further, the notion that a system is not optimally dissipative at one time point, and more optimally dissipative or flowing at another time point is anthropomorphically applying rules that aren't there. You can do that with equations, you can make various relationships and possibly make the claim, but what I'm saying is that you won't be able to prove this with the physical model. (6)

Constructal theory apparently claims that there is some distinction (be it flow optimization or heat dissipation) that can be made between a system at different times, which we might call configuration I, II, III etc. This is the problem specifically with the implication, the strong implication of animation and of anthropomorphing something like a river delta or erosion patterns on a mountain. Unless some argument is provided to the contrary, or some evidence is provided, we must assume the system is the same, (in terms of "optimal" heat or mass flow) before there are any river patterns, in other words it is already “optimized” at every time point. System state “I” is identical to II and identical to III and so on. That is a fact about these physical systems that is incorrectly asserted in this paper. It is purely anthropomorphic to observe that the flow of the river is much higher and thus it is more efficient. Nature makes no such assessment of efficiency. (7)

TEST: One can also easily show that constructal theory cannot be a "physical law" that such a river pattern is an optimized physical dictate, like gravity or the second law of thermodynamics, both laws which are inviolable under ANY condition. We NOTE that the river actually can be made more efficient? How? By the use of machines, ships or other devices that pump water down stream, are far more efficient. Yes they take energy, but that is beside the point. We have deomonstrated that we can violate constructal law. We cannot do such an experiment with the 2nd law of thermodynamics, or with gravity, etc. [3]

Fig. 1 Showing a microstate of the river bank and relevant forces.

 

To evaluate the causality of a simple flowing system we can perform an operation in which we look at the various forces involved and by a simple process of addition, determine if anything is out of balance. The force of the water against the bank (black) is depicted by the vector in blue (proportional to its velocity), the force acting equally and opposite the force of water, i.e. the river bank, is in black. We also have a downward force of the water, its weight which is the pressure of a column of water.  Note that we are not adding up forces to create a vector, which would be a convention. We are actually doing something very different, this is adding up “causality”, again to simply see if there is some inbalance of cause or forces present in the system. (8)

A stagnant body of water has tremendous weight, if there was no normal force opposing this pressure, the water would break out.

(Force of water against bank, F_bank) + (forces of water weight, F_weight) =(Normal force of river bank opposing water N_bank)+ (normal force of river bottom against river weight N_bottom). Rearranging we have (Force of moving water) + Force of water weight+ Normal force of bank opposing moving water force + Normal force of river bottom opposing weight) OR, more simply…. F_bank  + F_weight  + -N_bank + -N_bottom= 0.  (9)
When F_bank =0, in the case of a non-moving body of water, then we see an identical expression where the normal forces of the river bottom are in balance with the weight of the body of water. (10) (*see note 21)It should be noted that we are neglecting the pressure against the bank, for simplicity, but could express this as F_{bank= (Sum) F pressure+ F bank} )
Now we can also examine this system in accordance to time or a possible change over time. According to Indifferent Time, meaning no preference, we can again look at the same equation of these forces we just did, but at different times: call these arbitrarily, time initial, I,… and II (a time later). What Indifferent Time means is that we do not see a difference between these states I and II in terms of their equilibrium as we just calculated for a micro condition somewhere on the river bank. Since there is no difference in force balance over time, meaning no preference, we state that the system is “indifferent” essentially to changing.  (11) How would this be disproven? In other words how do we test this condition? If we can show that in the equation above, there is in fact some slight change in a force, either the river force or the river weight, that is changing with respect to some other impetus, call it a “wild card” force, then this would in fact disprove my condition of indifferent time. (12)
The more simplified concept is to imagine that we see no direction here in how the equilibrium is fluctuating. In other words, since the forces are always in balance, according to Newton’s law of motion, “any force will see an opposing force directly against it” (i.e. with gravity or collisions of balls), and the additional stipulation I have introduced, “at any time” we can see that we could plot the relationship above and show that it has no trend. This has an implication to dissipative preferential equations and so-called selectively enhanced dissipative equations .(13)

 F_bank + F_weight  + -N_bank+ -N_bottom= Diff at I, and F_bank + F_weight + -N_bank + -N_bottom= Diff at II. Diff I- Diff II/ Diff (time I-time II) would be a formula for showing a rate of the change in the system (moving away from equilibrium, or towards it, greater or lesser). (14)
 Of course the “slight difference” we are defining here would be in theory, a difference between the F_weight and N_bottom, in other words one is out of harmony with the other. When would such a condition exist? It cannot. (And this is actually Newton’s third law of action-reaction F_w=-F_n ) (15)
(See note 28. We are showing with the difference formula (14) that the normalizing force, is the "optimization" intrinsically done by the microstate, or microstates and thus we cannot see optimization between arbitrary states I, II, ...III etc. Any "wild card" vector would have associated with it, a normalizing force. This rule applies as the river undergoes any slight deformation, as delta F= delta N] 

So we have also just proven that there is no causality to be found in a strict formal sense in a natural flowing stream or other natural system. Causality is purely artificial, and can only be demonstrated by artificially assuming that a force is entering the microstate (such as wind, water or other forces), but in reality this only appears to be causal, as these forces are not correctly accounted for and find imbalance only because the model of accounting forces them to be imbalanced. (16)

It is the convention in physics texts to assume that a free floating force will be examined and often times, its source is not relevant to the problem. But that assumption is not valid in these types of problems involving causality.

 

*in reality, these forces are not “entering” the system but are in fact, a part of the system. It is convention, to assume that “a wind blew over the tree”. A true statement in the proper context if one is simply force diagraming the problem. However in the causality problem here, and by the new causal analysis I’m proposing, such convention is misleading. We know that the wind is more accurately understood as a vortex that was resultant from a system of nano-states, molecules absorbing heat, pushed to disequilibrium. Thus their macroscopic vibrations, led to higher order vortices, eventually building to form the wind that blew down the tree.  (17)

**It is true that the ordered motion of a vortex or wave, is a response of the system to dissipate energy, like diffusion spreads out a cluster of molecules. (that is far-from equilibrium dissipation [Prigogine, 1977]). What is not correct is to assume that this is more efficient” or “less efficient” as there is no “less efficient” possibility in the physics. We would not say that diffusion occurs because molecules “need” to spread out their energies uniformly. (18)

[(29) I updated 5.5.15 the expression above to be inclusive of the non-equilibrium state and show its broader and surprising implications to the sun's system (see comment 29)] 

 

On the other hand, evolution at least as a theory, is a term that is clearly conveyed and defined in biology . And there is little point in defining it here, except to say that evolution implies that an organism is changing in some way that is a response to its environment or to its competitors. And so although Dr. Bejan clearly is making the argument that a river delta “ages” over time and might even be “born”, this statement lacks physical evidence to suggest that the rocks and dirt and the stream itself, are anything physically different than they were at any other time point (with respect to their capacity to carry flow, which [we recall] is a self defined instrinsic property of the system in question. (19)
The "flow capacity" of a river is obtained by formula, but this capacity alters proportionately to other parameters that define the physics. We can see with the aid of microstate analysis I perform here, that the river only increases flow  capacity as the volume of water increases, not before and not after, its physical shape is dictated by fundamental laws). (15)
 Are we to believe as constructal theory claims, that a river delta “ages” and if so, what physically differentiates it at a “young” age vs a much older age in terms of flow? Of course constructal theory suggests that rivers increase in flow over time, they get larger, so size would be a factor, but the inherent problem with this suggestion is that we know when a river was “born” and even where it was born… And what a river is, meaning what its boundaries are. Unlike particles with finite mass, and radius, rivers are highly ambiguous. Does the mississipi river delta really only encompass certain states or does it include every tributary, minor tributary, and even the tiny etchings on a mountain sides? What is a rivers’ boundaries? Chaos theory comes to mind. Neither of these physical boundaries issues are even discussed in the paper. But that is the trap that anthropomorphic theories fall into. (20)

 

 [note](There are basic problems with so called "optimization theories" which relate to flow. Of course, it many not even be true that a given river we are observing is at optimized flow, if we are to accept that the context is a valid physical model. What if the river that particular week or year, sub-optimal in terms of run-off? There would be no guarantee that the volume of water is moving at its theoretical capacity, whatever that capacity might be. In reality, the same problems can be raised with other optimization theories, such as Dewar's Entropy Maximization Theory (Dewar, 2006) which proposes to measure the optimal heat dissipation of an ATP processing enzyme. The paper speculates on how the function of the ATP enzyme is specifically designed for maximum entropy generation. Maximized" carries with it the implication that there are less than "optimal" entropy productions of the ATPase, and it is not clear how one would make this assertion, except to assume that the maximal level is somehow already known or pre-dictated. (21)

That carries a strong anthropomorphic implication to a natural system, not to mention that it fails to explain why the entropy production is not indifferent to maxima or minima, to a theoretical state that the enzyme is not "aware" of. It is not clear why the values given for the entropy generation are not precisely the values that are generated by the system at any time t, neither optimal nor sub-optimal. Furthermore, I can provide several tests, as entropy is a universal property of all molecular systems, not just ATPases (but also not just molecular systems). (22)

I do not run a cell toxicity experiment and consider if a drug species is "optimized" for cell toxicity or not, such labels would have no physical basis in chemistry. Nor would I expect that the combination of a certain inorganic salt with water, a process that creates entropy in a fixed molar quantity, is at an optimized level. (23)

Is there a sub-optimal generation of entropy in the dissolution of calcium carbonate in water? Again, for the given experimental conditions, it is exactly what the system generates. If we consider the experimental tables of Dewar in his paper, are we to assume that some of the values calculated for replicate entropy generation of ATPase were "less optimized" whereas other values of the entropy were "more optimized"? By the physical analysis I employ with "Indifferent Time" a working hypothetical test, each of these data points would be a precisely what was generated, not a maximum or minimum, which implicitly assumes that they should in fact be some other value than they are. ) [4]

Based on my theory of “indifferent time” , I believe there is no physical difference between a river system a thousand years ago, and today, in terms of its expected flow or any other normal forces it encounters. And this is completely independent of what age one arbitrarily assigns to the river. (24)

Again, I have searched this paper (and others) for evidence to the contrary, but according to my time indifference argument,(time is “indifferent” as are the normal forces, to what state a physical system is in). A natural system at time t is “optimized” for maximum flow of all its components flowing through it, and is not more optimized at a later time t=t2, in other words it is optimized” at any time t for any flow material passing through it. (11)
It appears as though the trenches are deeper, but we cannot claim that this system is more “optimized” than a prior system. Why? Because of the rather obvious fact that the prior system was already optimized for the given flow requirements that it encountered. This derivation I provide, stems directly from the basic laws of motion. (15)
But mathematically, it is a sound argument since the net flow capacity would always be normalized to what the capacity is of the system at any time point. We note that the capacity of a river might increase, but mathematically, so does the potential of the system, in exact proportion. Unexpectedly, we might see that this is a product of the system, it is the system generating the flow potential.

When Bejan claims that a system has in fact reached an “optimized configuration”, as he states here…“Because the engines of engineering and biology are constructal, they morph in time towards easier flowing configurations. They evolve towards producing more mechanical power (under finiteness constraints), which, for them, means a time evolution towards less dissipation, or greater efficiency.”) does he believe that this “snapshot” or drawing of the system, is really somehow different than its state at some other time point? Is it really optimized” now, when in fact it wasn’t optimized before? More importantly, how do we know this? What physical measurements indicate that the system is in fact “better at flowing” because implicit in this assumption is that it needs to be better than it is.

When I hike a dry gulch in Southern Utah and gaze up at trees and rocks that have been deposited high above, in the now dry banks, I consider F=ma, not constructual theory. I do not imagine for a moment that the river is animate and this is somehow designing itself to carry more jetsome or flotsome. Nor can we imagine that it is “designing” itself to carry more water. According to my theory of indifferent time, it carries precisely the amount of water it carries. If the flood waters are diverted then this canyon will erode (the dry gulch will fill with sand, and it would appear that it is no longer "optimized" for carrying water, though in this case we have now mentioned a wholly different force  that is at work opposing so-called "flow optimization".) (25)
 If constructal theory were to actually hold, I believe that Newton’s F=ma would need to be restated, such that masses would “want” to collide at maximum efficiency. And that really does pose mathematical problems. (26)
We do not justify such a statement because it is clear that the collision is not at disequilibrium. We should not be surprised that the flow in a natural river is optimized for the water that is coursing down it. Any more than we should be not surprised that an incline is not “optimized” for rolling rocks downslope. None of these “optimizations” are really true, and "optimization" is a term that is being introduced artificially, into the equation.

There is another issue too, in the constructal thesis that our engines and other designs are “evolving” towards “less dissipation.” In reality this runs contrary to the accepted notions of thermodynamics, since for engines to exist, greater dissipation of entropy must occur outside these systems to compensate for the increased order. This would occur in the burning of fuels and disorder of other ordered systems to make (highly ordered) engines. That is, the engine is at such a high state of order, that the cost of this lowered entropy is great on the system around it. Dissipation of NET energy must be greater for the system in this case, not less. This is the principal already espoused by Schrodinger, and many others. For example, cells must pump entropy out of their systems in accordance with this theory, so they in fact are dissipating more heat.

So I return to an argument I made earlier (here) that constructal theory is making numerous factual arguments, about how a system at some time point, i.e., is not yet optimized, but perhaps at a later time point is optimized. And yet, if this was the case, we would see disequilibrium. Looking at the system in terms of microstates, we realize from Newton’s laws, that the water against the bank applies a force, but the bank in turn applies an equal and opposite force against the water, thus the “optimization” of forces is in balance at all times. (14,15)
At any time point, the system is already optimized in an absolute sense, to the flowing material passing through it. It will not change or alter itself until it is met by a force which causes a change, at which point it will be instantly “optimized” again. To imply or infer that a river delta system is somehow “doing” anything, other than being molded by forces that are acting in its immediate time frame, strains incredulity.

The constructal theory, (like other optimization theories) has obviously concluded that if a system is evaluated at a later point in time, it will be apparent that it has achieved an “optimized” level of flow that it might not have had at an earlier point in time. Yet there is no physical reason for believing that such evidence does exist to support this alternate, physical view of a system. Moreover, it would not seem possible to envision what kind of data this would be? As I’ve argued previously, there is a distinct lack of any units of measure in which to qualitatively or quantitatively evaluate how a system might be more optimized at one time point and less optimaized at another for the given flow that passes through it. When this theory is evaluated causally, in terms of vectoral diagrams these issues I believe become more obvious.
My theory: "Indifferent time" states that at any given time point the flow through the system, is indifferent (to the normal forces opposing it), it is neither optimized nor unoptimized in terms of maximal flow or minimal flow. (1) Indifferent time would hold that the premise of a system “optimizing itself” or evolving to a more perfected form, is not evidenced and mathematically is not possible. Indifferent time posits that at any time t, the normalized forces are in perfect equilibrium and thus the system is at perfect optimization. Any flow equation that we might establish, just like any other equation resembling F=ma, would be de novo generated from the system itself, and therefore normalized against it. It is neither too optimized nor too underoptimized. This would disprove the contention that a change in optimization is occuring or an evolution, since if both states (I) and (II) for example are already 100% optimized, how can one observe a change towards greater optimization?