The Constructal Law of Flow Systems

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.