# An interesting problem from the classical mechanics of vibrations

Update on 18 June 2017:
Added three diagrams depicting the mathematical abstraction of the problem; see near the end of the post. Also added one more consideration by way of an additional question.

TL;DR: A very brief version of this post is now posted at iMechanica; see here [^].

How I happened to come to formulate this problem:

As mentioned in my last post, I had started writing down my answers to the conceptual questions from Eisberg and Resnick’s QM text. However, as soon as I began doing that (typing out my answer to the first question from the first chapter), almost predictably, something else happened.

Since it anyway was QM that I was engaged with, somehow, another issue from QM—one which I had thought about a bit some time ago—happened to now just surface up in my mind. And it was an interesting issue. Back then, I had not thought of reaching an answer, and even now, I realized, I had not very satisfactory answer to it, not even in just conceptual terms. Naturally, my mind remained engaged in thinking about this second QM problem for a while.

In trying to come to terms with this QM problem (of my own making, not E&R’s), I now tried to think of some simple model problem from classical mechanics that might capture at least some aspects of this QM issue. Thinking a bit about it, I realized that I had not read anything about this classical mechanics problem during my [very] limited studies of the classical mechanics.

But since it appeared simple enough—heck, it was just classical mechanics—I now tried to reason through it. I thought I “got” it. But then, right the next day, I began doubting my own answer—with very good reasons.

… By now, I had no option but to keep aside the more scholarly task of writing down answers to the E&R questions. The classical problem of my own making had begun becoming all interesting by itself. Naturally, even though I was not procrastinating, I still got away from E&R—I got diverted.

I made some false starts even in the classical version of the problem, but finally, today, I could find some way through it—one which I think is satisfactory. In this post, I am going to share this classical problem. See if it interests you.

Background:

Consider an idealized string tautly held between two fixed end supports that are a distance $L$ apart; see the figure below. The string can be put into a state of vibrations by plucking it. There is a third support exactly at the middle; it can be removed at will.

Assume all the ideal conditions. For instance, assume perfectly rigid and unyielding supports, and a string that is massive (i.e., one which has a lineal mass density; for simplicity, assume this density to be constant over the entire string length) but having zero thickness. The string also is perfectly elastic and having zero internal friction of any sort. Assume that the string is surrounded by the vacuum (so that the vibrational energy of the string does not leak outside the system). Assume the absence of any other forces such as gravitational, electrical, etc. Also assume that the middle support, when it remains touching the string, does not allow any leakage of the vibrational energy from one part of the string to the other. Feel free to make further suitable assumptions as necessary.

The overall system here consists of the string (sans the supports, whose only role is to provide the necessary boundary conditions).

Initially, the string is stationary. Then, with the middle support touching the string, the left-half of the string is made to undergo oscillations by plucking it somewhere in the left-half only, and immediately releasing it. Denote the instant of the release as, say $t_R$. After the lapse of a sufficiently long time period, assume that the left-half of the system settles down into a steady-state standing wave pattern. Given our assumptions, the right-half of the system continues to remain perfectly stationary.

The internal energy of the system at $t_0$ is $0$. Energy is put into the system only once, at $t_R$, and never again. Thus, for all times $t > t_R$, the system behaves as a thermodynamically isolated system.

For simplicity, assume that the standing waves in the left-half form the fundamental mode for that portion (i.e. for the length $L/2$). Denote the frequency of this fundamental mode as $\nu_H$, and its max. amplitude (measured from the central line) as $A_H$.

Next, at some instant of time $t = t_1$, suppose that the support in the middle is suddenly removed, taking care not to disturb the string in any way in the process. That is to say, we  neither put in any more energy in the system nor take out of it, in the process of removing the middle support.

Once the support is thus removed, the waves from the left-half can now travel to the right-half, get reflected from the right end-support, travel all the way to the left end-support, get reflected there, etc. Thus, they will travel back and forth, in both the directions.

Modeled as a two-point BV/IC problem, assume that the system settles down into a steadily repeating pattern of some kind of standing waves.

The question now is:

What would be the pattern of the standing waves formed in the system at a time $t_F \gg t_1$?

The theory suggests that there is no unique answer!:

Since the support in the middle was exactly at the midpoint, removing it has the effect of suddenly doubling the length for the string.

Now, simple maths of the normal modes tells you that the string can vibrate in the fundamental mode for the entire length, which means: the system should show standing waves of the frequency $\nu_F = \nu_H/2$.

However, there also are other, theoretically conceivable, answers.

For instance, it is also possible that the system gets settled into the first higher-harmonic mode. In the very first higher-harmonic mode, it will maintain the same frequency as earlier, i.e., $\nu_F = \nu_H$, but being an isolated system, it has to conserve its energy, and so, in this higher harmonic mode, it must vibrate with a lower max. amplitude $A_F < A_H$. Thermodynamically speaking, since the energy is conserved also in such a mode, it also should certainly be possible.

In fact, you can take the argument further, and say that any one or all of the higher harmonics (potentially an infinity of them) would be possible. After all, the system does not have to maintain a constant frequency or a constant max. amplitude; it only has to maintain the same energy.

OK. That was the idealized model and its maths. Now let’s turn to reality.

Relevant empirical observations show that only a certain answer gets selected:

What do you actually observe in reality for systems that come close enough to the above mentioned idealized description? Let’s take a range of examples to get an idea of what kind of a show the real world puts up….

Consider, say, a violinist’s performance. He can continuously alter the length of the vibrations with his finger, and thereby produce a continuous spectrum of frequencies. However, at any instant, for any given length for the vibrating part, the most dominant of all such frequencies is, actually, only the fundamental mode for that length.

A real violin does not come very close to our idealized example above. A flute is better, because its spectrum happens to be the purest among all musical instruments. What do we mean by a “pure” tone here? It means this: When a flutist plays a certain tone, say the middle “saa” (i.e. the middle “C”), the sound actually produced by the instrument does not significantly carry any higher harmonics. That is to say, when a flutist plays the middle  “saa,” unlike the other musical instruments, the flute does not inadvertently go on to produce also the “saa”s from any of the higher octaves. Its energy remains very strongly concentrated in only a single tone, here, the middle “saa”. Thus, it is said to be a “pure” tone; it is not “contaminated” by any of the higher harmonics. (As to the lower harmonics for a given length, well, they are ruled out because of the basic physics and maths.)

Now, if you take a flute of a variable length (something like a trumpet) and try very suddenly doubling the length of the vibrating air column, you will find that instead of producing a fainter sound of the same middle “saa”, the flute instead produces the next lower “saa”. (If you want, you can try it out more systematically in the laboratory by taking a telescopic assembly of cylinders and a tuning fork.)

Of course, really speaking, despite its pure tones, even the flute does not come close enough to our idealized description above. For instance, notice that in our idealized description, energy is put into the system only once, at $t_R$, and never again. On the other hand, in playing a violin or a flute we are continuously pumping in some energy; the system is also continuously dissipating its energy to its environment via the sound waves produced in the air. A flute, thus, is an open system; it is not an isolated system. Yet, despite the additional complexity introduced because of an open system, and therefore, perhaps, a greater chance of being drawn into higher harmonic(s), in reality, a variable length flute is always observed to “select” only the fundamental harmonic for a given length.

How about an actual guitar? Same thing. In fact, the guitar comes closest to our idealized description. And if you try out plucking the string once and then, after a while, suddenly removing the finger from a fret, you will find that the guitar too “prefers” to immediately settle down rather in the fundamental harmonic for the new length. (Take an electric guitar so that even as the sound turns fainter and still fainter due to damping, you could still easily make out the change in the dominant tone.)

OK. Enough of empirical observations. Back to the connection of these observations with the theory of physics (and maths).

The question:

Thermodynamically, an infinity of tones are perfectly possible. Maths tells you that these infinity of tones are nothing but the set of the higher harmonics (and nothing else). Yet, in reality, only one tone gets selected. What gives?

What is the missing physics which makes the system get settled into one and only one option—indeed an extreme option—out of an infinity of them of which are, energetically speaking, equally possible?

Update on 18 June 2017:

Here is a statement of the problem in certain essential mathematical terms. See the three figures below:

The initial state of the string is what the following figure (Case 1) depicts. The max. amplitude is 1.0. Though the quiescent part looks longer than half the length, it’s just an illusion of perception.:

Case 1: Fundamental tone for the half length, extended over a half-length

The following figure (Case 2) is the mathematical idealization of the state in which an actual guitar string tends to settle in. Note that the max. amplitude is greater (it’s $\sqrt{2}$) so  as to have the energy of this state the same as that of Case 1.

Case 2: Fundamental tone for the full length, extended over the full length

The following figure (Case 3) depicts what mathematically is also possible for the final system state. However, it’s not observed with actual guitars. Note, here, the frequency is half of that in the Case 1, and the wavelength is doubled. The max. amplitude for this state is less than 1.0 (it’s $\dfrac{1}{\sqrt{2}}$) so as to have this state too carry exactly the same energy as in Case 1.

Case 3: The first overtone for the full length, extended over the full length

Thus, the problem, in short is:

The transition observed in reality is: $T1:$ Case 1 $\rightarrow$ Case 2.

However, the transition $T2:$ Case 1 $\rightarrow$ Case 3 also is possible by the mathematics of standing waves and thermodynamics (or more basically, by that bedrock on which all modern physics rests, viz., the calculus of variations). Yet, it is not observed.

Why does only $T1$ occur? why not $T2$? or even a linear combination of both? That’s the problem, in essence.

While attempting to answer it, also consider this : Can an isolated system like the one depicted in the Case 1 at all undergo a transition of modes?

Enjoy!

Update on 18th June 2017 is over.

That was the classical mechanics problem I said I happened to think of, recently. (And it was the one which took me away from the program of answering the E&R questions.)

Find it interesting? Want to give it a try?

If you do give it a try and if you reach an answer that seems satisfactory to you, then please do drop me a line. We can then cross-check our notes.

And of course, if you find this problem (or something similar) already solved somewhere, then my request to you would be stronger: do let me know about the reference!

In the meanwhile, I will try to go back to (or at least towards) completing the task of answering the E&R questions. [I do, however, also plan to post a slightly edited version of this post at iMechanica.]

Update History:

07 June 2017: Published on this blog

8 June 2017, 12:25 PM, IST: Added the figure and the section headings.

8 June 2017, 15:30 hrs, IST: Added the link to the brief version posted at iMechanica.

18 June 2017, 12:10 hrs, IST: Added the diagrams depicting the mathematical abstraction of the problem.

A Song I Like:

(Marathi) “olyaa saanj veli…”
Music: Avinash-Vishwajeet
Singers: Swapnil Bandodkar, Bela Shende
Lyrics: Ashwini Shende

# See, how hard I am trying to become an Approved (Full) Professor of Mechanical Engineering in SPPU?—3

I was looking for a certain book on heat transfer which I had (as usual) misplaced somewhere, and while searching for that book at home, I accidentally ran into another book I had—the one on Classical Mechanics by Rana and Joag [^].

After dusting this book a bit, I spent some time in one typical way, viz. by going over some fond memories associated with a suddenly re-found book…. The memories of how enthusiastic I once was when I had bought that book; how I had decided to finish that book right within weeks of buying it several years ago; the number of times I might have picked it up, and soon later on, kept it back aside somewhere, etc.  …

Yes, that’s right. I have not yet managed to finish this book. Why, I have not even managed to begin reading this book the way it should be read—with a paper and pencil at hand to work through the equations and the problems. That was the reason why, I now felt a bit guilty. … It just so happened that it was just the other day (or so) when I was happily mentioning the Poisson brackets on Prof. Scott Aaronson’s blog, at this thread [^]. … To remove (at least some part of) my sense of guilt, I then decided to browse at least through this part (viz., Poisson’s brackets) in this book. … Then, reading a little through this chapter, I decided to browse through the preceding chapters from the Lagrangian mechanics on which it depends, and then, in general, also on the calculus of variations.

It was at this point that I suddenly happened to remember the reason why I had never been able to finish (even the portions relevant to engineering from) this book.

The thing was, the explanation of the $\delta$—the delta of the variational calculus.

The explanation of what the $\delta$ basically means, I had found right back then (many, many years ago), was not satisfactorily given in this book. The book did talk of all those things like the holonomic constraints vs. the nonholonomic constraints, the functionals, integration by parts, etc. etc. etc. But without ever really telling me, in a forth-right and explicit manner, what the hell this $\delta$ was basically supposed to mean! How this $\delta y$ was different from the finite changes ($\Delta y$) and the infinitesimal changes ($\text{d}y$) of the usual calculus, for instance. In terms of its physical meaning, that is. (Hell, this book was supposed to be on physics, wasn’t it?)

Here, I of course fully realize that describing Rana and Joag’s book as “unsatisfactory” is making a rather bold statement, a very courageous one, in fact. This book is extraordinarily well-written. And yet, there I was, many, many years ago, trying to understand the delta, and not getting anywhere, not even with this book in my hand. (OK, a confession. The current copy which I have is not all that old. My old copy is gone by now (i.e., permanently misplaced or so), and so, the current copy is the one which I had bought once again, in 2009. As to my old copy, I think, I had bought it sometime in the mid-1990s.)

It was many years later, guess some time while teaching FEM to the undergraduates in Mumbai, that the concept had finally become clear enough to me. Most especially, while I was going through P. Seshu’s and J. N. Reddy’s books. [Reflected Glory Alert! Professor P. Seshu was my class-mate for a few courses at IIT Madras!] However, even then, even at that time, I remember, I still had this odd feeling that the physical meaning was still not clear to me—not as as clear as it should be. The matter eventually became “fully” clear to me only later on, while musing about the differences between the perspective of Thermodynamics on the one hand and that of Heat Transfer on the other. That was some time last year, while teaching Thermodynamics to the PG students here in Pune.

Thermodynamics deals with systems at equilibria, primarily. Yes, its methods can be extended to handle also the non-equilibrium situations. However, even then, the basis of the approach summarily lies only in the equilibrium states. Heat Transfer, on the other hand, necessarily deals with the non-equilibrium situations. Remove the temperature gradient, and there is no more heat left to speak of. There does remain the thermal energy (as a form of the internal energy), but not heat. (Remember, heat is the thermal energy in transit that appears on a system boundary.) Heat transfer necessarily requires an absence of thermal equilibrium. … Anyway, it was while teaching thermodynamics last year, and only incidentally pondering about its differences from heat transfer, that the idea of the variations (of Cov) had finally become (conceptually) clear to me. (No, CoV does not necessarily deal only with the equilibrium states; it’s just that it was while thinking about the equilibrium vs. the transient that the matter about CoV had suddenly “clicked” to me.)

In this post, let me now note down something on the concept of the variation, i.e., towards understanding the physical meaning of the symbol $\delta$.

Please note, I have made an inline update on 26th December 2016. It makes the presentation of the calculus of variations a bit less dumbed down. The updated portion is clearly marked as such, in the text.

The Problem Description:

The concept of variations is abstract. We would be better off considering a simple, concrete, physical situation first, and only then try to understand the meaning of this abstract concept.

Accordingly, consider a certain idealized system. See its schematic diagram below:

There is a long, rigid cylinder made from some transparent material like glass. The left hand-side end of the cylinder is hermetically sealed with a rigid seal. At the other end of the cylinder, there is a friction-less piston which can be driven by some external means.

Further, there also are a couple of thin, circular, piston-like disks ($D_1$ and $D_2$) placed inside the cylinder, at some $x_1$ and $x_2$ positions along its length. These disks thus divide the cylindrical cavity into three distinct compartments. The disks are assumed to be impermeable, and fitting snugly, they in general permit no movement of gas across their plane. However, they also are assumed to be able to move without any friction.

Initially, all the three compartments are filled with a compressible fluid to the same pressure in each compartment, say 1 atm. Since all the three compartments are at the same pressure, the disks stay stationary.

Then, suppose that the piston on the extreme right end is moved, say from position $P_1$ to $P_2$. The final position $P_2$ may be to the left or to the right of the initial position $P_1$; it doesn’t matter. For the current description, however, let’s suppose that the position $P_2$ is to the left of $P_1$. The effect of the piston movement thus is to increase the pressure inside the system.

The problem is to determine the nature of the resulting displacements that the two disks undergo as measured from their respective initial positions.

There are essentially two entirely different paradigms for conducting an analysis of this problem.

The first paradigm is based on an approach that was put to use so successfully by Newton. Usually, it is called the paradigm of vector analysis.

In this paradigm, we focus on the fact that the forced displacement of the piston with time, $x(t)$, may be described using some function of time that is defined over the interval lying between two instants $t_i$ and $t_f$.

For example, suppose the function is:
$x(t) = x_0 + v t$,
where $v$ is a constant. In other words, the motion of the piston is steady, with a constant velocity, between the initial and final instants. Since the velocity is constant, there is no acceleration over the open interval $(t_i, t_f)$.

However, notice that before the instant $t_i$, the piston velocity was zero. Then, the velocity suddenly became a finite (constant) value. Therefore, if you extend the interval to include the end-instants as well, i.e., if you consider the semi-closed interval $[t_i, t_f)$, then there is an acceleration at the instant $t_i$. Similarly, since the piston comes to a position of rest at $t = t_f$, there also is another acceleration, equal in magnitude and opposite in direction, which appears at the instant $t_f$.

The existence of these two instantaneous accelerations implies that jerks or pressure waves are sent through the system. We may model them as vector quantities, as impulses. [Side Exercise: Work out what happens if we consider only the open interval $(t_i, t_f)$.]

We can now apply Newton’s 3 laws, based on the idea that shock-waves must have begun at the piston at the instant $t = t_i$. They must have got transmitted through the gas kept under pressure, and they must have affected the disk $D_1$ lying closest to the piston, thereby setting this disk into motion. This motion must have passed through the gas in the middle compartment of the system as another pulse in the pressure (generated at the disk $D_1$), thereby setting also the disk $D_2$ in a state of motion a little while later. Finally, the pulse must have got bounced off the seal on the left hand side, and in turn, come back to affect the motion of the disk $D_2$, and then of the disk $D_1$. Continuing their travels to and fro, the pulses, and hence the disks, would thus be put in a back and forth motion.

After a while, these transients would move forth and back, superpose, and some of their constituent frequencies would get cancelled out, leaving only those frequencies operative such that the three compartments are put under some kind of stationary states.

In case the gas is not ideal, there would be damping anyway, and after a sufficiently long while, the disks would move through such small displacements that we could easily ignore the ever-decreasing displacements in a limiting argument.

Thus, assume that, after an elapse of a sufficiently long time, the disks become stationary. Of course, their new positions are not the same as their original positions.

The problem thus can be modeled as basically a transient one. The state of the new equilibrium state is thus primarily seen as an effect or an end-result of a couple of transient processes which occur in the forward and backward directions. The equilibrium is seen as not a primarily existing state, but as a result of two equal and opposite transient causes.

Notice that throughout this process, Newton’s laws can be applied directly. The nature of the analysis is such that the quantities in question—viz. the displacements of the disks—always are real, i.e., they correspond to what actually is supposed to exist in the reality out there.

The (values of) displacements are real in the sense that the mathematical analysis procedure itself involves only those (values of) displacements which can actually occur in reality. The analysis does not concern itself with some other displacements that might have been possible but don’t actually occur. The analysis begins with the forced displacement condition, translates it into pressure waves, which in turn are used in order to derive the predicted displacements in the gas in the system, at each instant. Thus, at any arbitrary instant of time $t > t_i$ (in fact, the analysis here runs for times $t \gg t_f$), the analysis remains concerned only with those displacements that are actually taking place at that instant.

The Method of Calculus of Variations:

The second paradigm follows the energetics program. This program was initiated by Newton himself as well as by Leibnitz. However, it was pursued vigorously not by Newton but rather by Leibnitz, and then by a series of gifted mathematicians-physicists: the Bernoulli brothers, Euler, Lagrange, Hamilton, and others. This paradigm is essentially based on the calculus of variations. The idea here is something like the following.

We do not care for a local description at all. Thus, we do not analyze the situation in terms of the local pressure pulses, their momenta/forces, etc. All that we focus on are just two sets of quantities: the initial positions of the disks, and their final positions.

For instance, focus on the disk $D_1$. It initially is at the position $x_{1_i}$. It is found, after a long elapse of time (i.e., at the next equilibrium state), to have moved to $x_{1_f}$. The question is: how to relate this change in $x_1$ on the one hand, to the displacement that the piston itself undergoes from $P_{x_i}$ to $P_{x_f}$.

To analyze this question, the energetics program (i.e., the calculus of variations) adopts a seemingly strange methodology.

It begins by saying that there is nothing unique to the specific value of the position $x_{1_f}$ as assumed by the disk $D_1$. The disk could have come to a halt at any other (nearby) position, e.g., at some other point $x_{1_1}$, or $x_{1_2}$, or $x_{1_3}$, … etc. In fact, since there are an infinity of points lying in a finite segment of line, there could have been an infinity of positions where the disk could have come to a rest, when the new equilibrium was reached.

Of course, in reality, the disk $D_1$ comes to a halt at none of these other positions; it comes to a halt only at $x_{1_f}$.

Yet, the theory says, we need to be “all-inclusive,” in a way. We need not, just for the aforementioned reason, deny a place in our analysis to these other positions. The analysis must include all such possible positions—even if they be purely hypothetical, imaginary, or unreal. What we do in the analysis, this paradigm says, is to initially include these merely hypothetical, unrealistic positions too on exactly the same footing as that enjoyed by that one position which is realistic, which is given by $x_{1_f}$.

Thus, we take a set of all possible positions for each disk. Then, for each such a position, we calculate the “impact” it would make on the energy of the system taken as a whole.

The energy of the system can be additively decomposed into the energies carried by each of its sub-parts. Thus, focusing on disk $D_1$, for each one of its possible (hypothetical) final position, we should calculate the energies carried by both its adjacent compartments. Since a change in $D_1$‘s position does not affect the compartment 3, we need not include it. However, for the disk $D_1$, we do need to include the energies carried by both the compartments 1 and 2. Similarly, for each of the possible positions occupied by the disk $D_2$, it should include the energies of the compartments 2 and 3, but not of 1.

At this point, to bring simplicity (and thereby better) clarity to this entire procedure, let us further assume that the possible positions of each disk forms a finite set. For instance, each disk can occupy only one of the positions that is some $-5, -4, -3, -2, -1, 0, +1, +2, +3, +4$ or $+5$ distance-units away from its initial position. Thus, a disk is not allowed to come to a rest at, say, $2.3$ units; it must do so either at $2$ or at $3$ units. (We will thus perform the initial analysis in terms of only the integer positions, and only later on extend it to any real-valued positions.) (If you are a mechanical engineering student, suggest a suitable mechanism that can ensure only integer relative displacements.)

The change in energy $E$ of a compartment is given by
$\Delta E = P A \Delta x$,
where $P$ is the pressure, $A$ is the cross-sectional area of the cylinder, and $\Delta x$ is the change in the length of the compartment.

Now, observe that the energy of the middle compartment depends on the relative distance between the two disks lying on its sides. Yet, for the same reason, the energy of the middle compartment does depend on both these positions. Hence, we must take a Cartesian product of the relative displacements undergone by both the disks, and only then calculate the system energy for each such a permutation (i.e. the ordered pair) of their positions. Let us go over the details of the Cartesian product.

The Cartesian product of the two positions may be stated as a row-by-row listing of ordered pairs of the relative positions of $D_1$ and $D_2$, e.g., as follows: the ordered pair $(-5, +2)$ means that the disk $D_1$ is $5$ units to the left of its initial position, and the disk $D_2$ is $+2$ units to the right of its initial position. Since each of the two positions forming an ordered pair can range over any of the above-mentioned $11$ number of different values, there are, in all, $11 \times 11 = 121$ number of such possible ordered pairs in the Cartesian product.

For each one of these $121$ different pairs, we use the above-given formula to determine what the energy of each compartment is like. Then, we add the three energies (of the three compartments) together to get the value of the energy of the system as a whole.

In short, we get a set of $121$ possible values for the energy of the system.

You must have noticed that we have admitted every possible permutation into analysis—all the $121$ number of them.

Of course, out of all these $121$ number of permutations of positions, it should turn out that $120$ number of them have to be discarded because they would be merely hypothetical, i.e. unreal. That, in turn, is because, the relative positions of the disks contained in one and only one ordered pair would actually correspond to the final, equilibrium position. After all, if you conduct this experiment in reality, you would always get a very definite pair of the disk-positions, and it this same pair of relative positions that would be observed every time you conducted the experiment (for the same piston displacement). Real experiments are reproducible, and give rise to the same, unique result. (Even if the system were to be probabilistic, it would have to give rise to an exactly identical probability distribution function.) It can’t be this result today and that result tomorrow, or this result in this lab and that result in some other lab. That simply isn’t science.

Thus, out of all those $121$ different ordered-pairs, one and only one ordered-pair would actually correspond to reality; the rest all would be merely hypothetical.

The question now is, which particular pair corresponds to reality, and which ones are unreal. How to tell the real from the unreal. That is the question.

Here, the variational principle says that the pair of relative positions that actually occurs in reality carries a certain definite, distinguishing attribute.

The system-energy calculated for this pair (of relative displacements) happens to carry the lowest magnitude from among all possible $121$ number of pairs. In other words, any hypothetical or unreal pair has a higher amount of system energy associated with it. (If two pairs give rise to the same lowest value, both would be equally likely to occur. However, that is not what provably happens in the current example, so let us leave this kind of a “degeneracy” aside for the purposes of this post.)

(The update on 26 December 2016 begins here:)

Actually, the description  given in the immediately preceding paragraph was a bit too dumbed down. The variational principle is more subtle than that. Explaining it makes this post even longer, but let me give it a shot anyway, at least today.

To follow the actual idea of the variational principle (in a not dumbed-down manner), the procedure you have to follow is this.

First, make a table of all possible relative-position pairs, and their associated energies. The table has the following columns: a relative-position pair, the associated energy $E$ as calculated above, and one more column which for the time being would be empty. The table may look something like what the following (partial) listing shows:

(0,0) -> say, 115 Joules
(-1,0) -> say, 101 Joules
(-2,0) -> say, 110 Joules

(2,2) -> say, 102 Joules
(2,3) -> say, 100 Joules
(2,4) -> say, 101 Joules
(2,5) -> say, 120 Joules

(5,0) -> say, 135 Joules

(5,5) -> say 117 Joules.

Having created this table (of $121$ rows), you then pick each row one by and one, and for the picked up $n$-th row, you ask a question: What all other row(s) from this table have their relative distance pairs such that these pairs lie closest to the relative distance pair of this given row. Let me illustrate this question with a concrete example. Consider the row which has the relative-distance pair given as (2,3). Then, the relative distance pairs closest to this one would be obtained by adding or subtracting a distance of 1 to each in the pair. Thus, the relative distance pairs closest to this one would be: (3,3), (1,3), (2,4), and (2,2). So, you have to pick up those rows which have these four entries in the relative-distance pairs column. Each of these four pairs represents a variation $\delta$ on the chosen state, viz. the state (2,3).

In symbolic terms, suppose for the $n$-th row being considered, the rows closest to it in terms of the differences in their relative distance pairs, are the $a$-th, $b$-th, $c$-th and $d$-th rows. (Notice that the rows which are closest to a given row in this sense, would not necessarily be found listed just above or below that given row, because the scheme followed while creating the list or the vector that is the table would not necessarily honor the closest-lying criterion (which necessarily involves two numbers)—not at least for all rows in the table.

OK. Then, in the next step, you find the differences in the energies of the $n$-th row from each of these closest rows, viz., the $a$-th, $b$-th, $c$-th and $c$-th rows. That is to say, you find the absolute magnitudes of the energy differences. Let us denote these magnitudes as: $\delta E_{na} = |E_n - E_a|$$\delta E_{nb} = |E_n - E_b|$$\delta E_{nc} = |E_n - E_c|$ and $\delta E_{nd} = |E_n - E_d|$.  Suppose the minimum among these values is $\delta E_{nc}$. So, against the $n$-th row, in the last column of the table, you write the value $\delta E_{nc}$.

Having done this exercise separately for each row in the table, you then ask: Which row has the smallest entry in the last column (the one for $\delta E$), and you pick that up. That is the distinguished (or the physically occurring) state.

In other words, the variational principle asks you to select not the row with the lowest absolute value of energy, but that row which shows the smallest difference of energy from one of its closest neighbours—and these closest neighbours are to be selected according to the differences in each number appearing in the relative-distance pair, and not according to the vertical place of rows in the tabular listing. (It so turns out that in this example, the row thus selected following both criteria—lowest energy as well as lowest variation in energy—are identical, though it would not necessarily always be the case. In short, we can’t always get away with the first, too dumbed down, version.)

Thus, the variational principle is about that change in the relative positions for which the corresponding change in the energy vanishes (or has the minimum possible absolute magnitude, in case the positions form a discretely varying, finite set).

(The update on 26th December 2016 gets over here.)

And, it turns out that this approach, too, is indeed able to perfectly predict the final disk-positions—precisely as they actually are observed in reality.

If you allow a continuum of positions (instead of the discrete set of only the $11$ number of different final positions for one disk, or $121$ number of ordered pairs), then instead of taking a Cartesian product of positions, what you have to do is take into account a tensor product of the position functions. The maths involved is a little more advanced, but the underlying algebraic structure—and the predictive principle which is fundamentally involved in the procedure—remains essentially the same. This principle—the variational principle—says:

Among all possible variations in the system configurations, that system configuration corresponds to reality which has the least variation in energy associated with it.

(This is a very rough statement, but it will do for this post and for a general audience. In particular, we don’t look into the issues of what constitute the kinematically admissible constraints, why the configurations must satisfy the field boundary conditions, the idea of the stationarity vs. of a minimum or a maximum, i.e., the issue of convexity-vs.-concavity, etc. The purpose of this post—and our example here—are both simple enough that we need not get into the whole she-bang of the variational theory as such.)

Notice that in this second paradigm, (i) we did not restrict the analysis to only those quantities that are actually taking place in reality; we also included a host (possibly an infinity) of purely hypothetical combinations of quantities too; (ii) we worked with energy, a scalar quantity, rather than with momentum, a vector quantity; and finally, (iii) in the variational method, we didn’t bother about the local details. We took into account the displacements of the disks, but not any displacement at any other point, say in the gas. We did not look into presence or absence of a pulse at one point in the gas as contrasted from any other point in it. In short, we did not discuss the details local to the system either in space or in time. We did not follow the system evolution, at all—not at least in a detailed, local way. If we were to do that, we would be concerned about what happens in the system at the instants and at spatial points other than the initial and final disk positions. Instead, we looked only at a global property—viz. the energy—whether at the sub-system level of the individual compartments, or at the level of the overall system.

The Two Paradigms Contrasted from Each Other:

If we were to follow Newton’s method, it would be impossible—impossible in principle—to be able to predict the final disk positions unless all their motions over all the intermediate transient dynamics (occurring over each moment of time and at each place of the system) were not be traced. Newton’s (or vectorial) method would require us to follow all the details of the entire evolution of all parts of the system at each point on its evolution path. In the variational approach, the latter is not of any primary concern.

Yet, in following the energetics program, we are able to predict the final disk positions. We are able to do that without worrying about what all happened before the equilibrium gets established. We remain concerned only with certain global quantities (here, system-energy) at each of the hypothetical positions.

The upside of the energetics program, as just noted, is that we don’t have to look into every detail at every stage of the entire transient dynamics.

Its downside is that we are able to talk only of the differences between certain isolated (hypothetical) configurations or states. The formalism is unable to say anything at all about any of the intermediate states—even if these do actually occur in reality. This is a very, very important point to keep in mind.

The Question:

Now, the question with which we began this post. Namely, what does the delta of the variational calculus mean?

Referring to the above discussion, note that the delta of the variational calculus is, here, nothing but a change in the position-pair, and also the corresponding change in the energy.

Thus, in the above example, the difference of the state (2,3) from the other close states such as (3,3), (1,3), (2,4), and (2,2) represents a variation in the system configuration (or state), and for each such a variation in the system configuration (or state), there is a corresponding variation in the energy $\delta E_{ni}$ of the system. That is what the delta refers to, in this example.

Now, with all this discussion and clarification, would it be possible for you to clearly state what the physical meaning of the delta is? To what precisely does the concept refer? How does the variation in energy $\delta E$ differ from both the finite changes ($\Delta E$) as well as the infinitesimal changes ($\text{d}E$) of the usual calculus?

Note, the question is conceptual in nature. And, no, not a single one of the very best books on classical mechanics manages to give a very succinct and accurate answer to it. Not even Rana and Joag (or Goldstein, or Feynman, or…)

I will give my answer in my next post, next year. I will also try to apply it to a couple of more interesting (and somewhat more complicated) physical situations—one from engineering sciences, and another from quantum mechanics!

In the meanwhile, think about it—the delta—the concept itself, its (conceptual) meaning. (If you already know the calculus of variations, note that in my above write-up, I have already supplied the answer, in a way. You just have to think a bit about it, that’s all!)

An Important Note: Do bring this post to the notice of the Officially Approved Full Professors of Mechanical Engineering in SPPU, and the SPPU authorities. I would like to know if the former would be able to state the meaning—at least now that I have already given the necessary context in such great detail.

Ditto, to the Officially Approved Full Professors of Mechanical Engineering at COEP, esp. D. W. Pande, and others like them.

After all, this topic—Lagrangian mechanics—is at the core of Mechanical Engineering, even they would agree. In fact, it comes from a subject that is not taught to the metallurgical engineers, viz., the topic of Theory of Machines. But it is taught to the Mechanical Engineers. That’s why, they should be able to crack it, in no time.

(Let me continue to be honest. I do not expect them to be able to crack it. But I do wish to know if they are able at least to give a try that is good enough!)

Even though I am jobless (and also nearly bank balance-less, and also cashless), what the hell! …

…Season’s greetings and best wishes for a happy new year!

A Song I Like:

[With jobless-ness and all, my mood isn’t likely to stay this upbeat, but anyway, while it lasts, listen to this song… And, yes, this song is like, it’s like, slightly more than 60 years old!]

(Hindi) “yeh raat bhigee bhigee”
Music: Shankar-Jaikishan
Singers: Manna De and Lata Mangeshkar
Lyrics: Shailendra

[E&OE]

/

# Analyzing the Eco-Cooler, part 1

OK, that was ample time for you to have hit your fluid mech/heat transfer/thermo books, and to have it verified whether the 5 deg. C drop was believable or not… You must have made your notes, too, no?…  So, in this post, let’s cross-check our notes.

On my part, I will first present the simplest (and the most approximate) model, and also give you a simple Python script to play with, to see what kind of predictions this model makes. Then, we will go on considering more and more complicated but still approximate “engineering” models that hopefully become more and more realistic. We will cross-check their predictions too. We may eventually find that a full-fledged CFD analysis is called for. However, I will save that—I mean doing a full-fledged CFD analysis—for another day. (I in fact plan to write a paper on this problem, using CFD. (…Some day…).)

The reason we follow this method—from the simplest and crudest models to the more complicated and better ones—is because for problems related to fluid mechanics, it is this method which works best!

I mean to say, the full-fledged Navier-Stokes equations are too complicated to solve for, even when they are applied to the simplest of practically encountered geometries—e.g., the flow of air through the Eco-Cooler bottle. Since the NS equations cannot be solved exactly, the traditional engineering models (which are based on analytical or semi-analytical solutions) fall short, and then, a CFD analysis is called for.

But the fact of the matter also is, even CFD itself is only an approximate technique. CFD solutions sometimes even happen to carry more numerical artifacts than real physics. We therefore cannot approach CFD blindly. We ourselves have to have some good idea in the first place of what the desired solution should look like. We should have this idea right before we even think of setting up a CFD model/simulation. The traditional engineering models provide precisely these insights.

Yes, that’s right.

The traditional engineering models actually are more approximate than CFD. Yet, since they are also simpler than CFD, and since they explicitly carry conceptual connections with the major fluid mechanical phenomena in a more direct manner, they also make it easier to gain insights about both the nature of the problem, and the nature of the expected solution. No similar insights can be had by directly using CFD, for several reasons. The CFD theory itself is too complicated, and the CFD practice involves too many different analysis options. In the jungle of all those parameters, iterations, and convergence requirements, CFD happens to loose the directness of the conceptual connections with the basic analytical theory—with the fluid dynamical phenomena.

That’s why we first deal with the simplest engineering models, even though they are known to be approximate—and therefore, they are easily capable of giving us wrong results. But this way, we can build insights. Building insights is an art, and the process progresses slowly.

As I said, we will follow an iterative scheme of model building. In each phase or iteration of the model building activity, we will actually be applying the same set of principles: the conservation of mass, momentum, and energy. However, in going from a model-building phase to the next, we will aim to incorporate an increasing level of complexity or sophistication—and accuracy, hopefully.

Actually, in the fluid dynamics theory, all the three conservation equations come coupled to each other. You cannot solve for conservation of only the mass, or the momentum, or the energy, by neglecting any of the other two. However, for this particular problem (of the Eco-Cooler), for various reasons (which you will come to appreciate slowly), it so turns out that we can get away considering the mass, momentum and energy equations in a decoupled manner, and in the order stated: first mass, then momentum, and then energy. (It’s no accident that text-books spell out these three principles in this order. Many fluids-related phenomena with which we are well familiar through our direct experience of the world are such that for solving problems involving these phenomena, this order happens to be the best one to follow.)

So, with that big introduction, let’s now get going calculating—even though we will not shut up even while performing those calculations.

Model-Building Phase I: The Simplest Possible Model:

Geometry:

Consider the plastic-bottles used in the Eco-Cooler; they all lie horizontally. Consider one of these bottles. A tube has been obtained by cutting off the base of the bottle. Let the base-plane be identified by the subscript 1 and the neck-plane by 2. See the figure below:

Air flows from the base-plane (1) to the neck-plane (2).

Conservation of mass:

The simplest possible expression for mass conservation, applied to the bottle geometry, would be the continuity equation, given below:
$A_1 U_1 = A_2 U_2$
where $A_1$ and $A_2$ are cross-sectional areas of the bottle, and $U_1$ and $U_2$ are wind velocities, at the base and the neck, respectively. Rearranging for $U_2$, we get:
$U_2 = \dfrac{A_1}{A_2} U_1$             …(1)

We can use this simple an equation for mass conservation only if the flow is incompressible. To determine if our flow is incompressible or not, we have to calculate the Mach number for the flow. To do that, we have to first know the expected wind velocities.

Referring to the Wiki article on the Beaufort scale [^], we may make an assumption that the inlet speed can go up to about 60 kmph. Actually, the wind-speeds covered by the Beaufort scale go much higher (in excess of 118 kmph). However, practically speaking, the only times such high wind-speeds (gales etc.) occur in India is when rains also accompany them. The rains bring down the ambient temperature anyway, thereby obviating the need for any form of a cooler. Thus, we have to consider only the lower range of speeds.

Assuming the speed of sound in air to be about 340 m/s, we find that the Mach number (for the range of the winds we consider) goes up to about 0.65. Now, for $\text{Ma} < 0.33$, the flow is sub-sonic, and can be regarded as incompressible. For $0.33 < \text{Ma} < 1.0$, the flow is trans-sonic, meaning, the changes in pressures do not adjust “instantaneously” everywhere in the flow, and so, it is increasingly not possible to even idealize the flow as incompressible.

Therefore, in the interest of simplicity, for our first solution cut, we choose to consider only the wind-speeds up to 100 m/s, i.e. 28 kmph, so that the incompressibility assumption can be justified. Making this assumption about the highest possible wind-speed, we are then free to use the simplest form of mass conservation equation given above as Eq. (1).

For a typical one liter bottle, the base diameter is 7.5 cm, and the neck diameter is 2 cm (both referring to IDs i.e. inner diameters). (I measured them myself!) So, the area ratio $\frac{A_1}{A_2}$ turns out to be about 14.

Thus, the wind accelerates inside the bottle; the outlet velocity is about 14 times the inlet velocity!

This looks like a remarkable bit of acceleration to happen over just some 20 cm of length. (The bottle is cut somewhere in the middle.) More on it, later.

Conservation of momentum:

The simplest possible equation to use for momentum conservation is the steady-state Bernoulli’s equation:
$\dfrac{P_1}{\rho} + \dfrac{U_1^2}{2} = \dfrac{P_2}{\rho} + \dfrac{U_2^2}{2}$        …(2)
where we have ignored the potential head term ($gz$) because the tube is horizontal as well as symmetrical about its central horizontal axis (and because the air is so thin that its weight can be easily neglected here).

Carefully note the assumptions behind Eq (2). It holds only for a steady-state and laminar flow, and only after neglecting viscosity.

Since we are in a hurry, we will assume them all, and proceed!

Rearranging Eq (2) for $P_2$, we obtain:
$P_2 = P_1 + \dfrac{\rho}{2} \left( U_1^2 - U_2^2 \right)$

Conservation of energy:

We basically need the equation of energy conservation only in order to calculate the temperature at the neck ($T_2$), from a knowledge of: (i) the temperature at the base, $T_1$, and (ii) the pressures $P_1$ and $P_2$.

In the last line, I said “from.” This usage implies that there already is an assumption I made, viz., that the energy equation can be decoupled from the momentum equation. How reasonable is this assumption? It seems pretty OK. Think: can air flowing through a 7.5 cm or a 2.0 cm diameter tube at under 100 m/s get heated up to a significant fraction of 5 deg. C, over a length of just a foot or less? Not likely. Can heating up the neck region cause the air flow to come a halt, say because it helps build up a sufficient amount of pressure? Not even remotely likely. So, we may get away by decoupling the two.

The simplest equation to compute $T_2$ from the other three quantities would be: the ideal gas law, given as:
$\dfrac{P_1 V_1}{T_1} = \dfrac{P_2 V_2}{T_2}$.

We have already found that the flow can be considered incompressible (at least up to 28 kmph of wind-speeds). Hence, the volume of each fluid part remains constant, i.e., $V_1 = V_2$. (Note, in this equation, we have to consider the volume of a fluid element or a part, not the volume for a unit axial length at a given cross-section of the bottle.) For constant volume, the ideal gas law reduces to:
$\dfrac{P_1}{T_1} = \dfrac{P_2}{T_2}$,
from which we can conclude:
$T_2 = P_2 \dfrac{T_1}{P_1}$.

But we have to know whether the assumption of the ideal gas itself can be used in our problem—for the real air—or not. For doing that, we have to know the critical pressure and temperature of air. Cengel (“Thermodynamics: An Engineering Approach,” 8th SI units edition) lists them (Appendix A1) as 132.5 K, and 3.77 MPa. Using these values, the reduced temperature and the reduced pressure of air turn out to be $T_{1_R} = (30 + 273.15)/132.5 \approx 2.29$, and $P_{1_R} = (101.32\times10^3)/(3.77\times 10^6) \approx 0.027$. Further, the expected drops in the pressure and temperature would be just small fractions of the inlet values. Hence, the reduced quantities for the outlet also would not differ significantly from those for the inlet. Referring to the chart and the remarks on p. 139 in Cengel, it seems like we can get away using the ideal gas law.

Note, we are using the ideal gas law not as an approximation to the energy equation, but in place of it, simply because looking at the variables, we noticed that $T_2$ had to be determined from the other three variables, and this set of variables reminded us of the ideal gas law! Then, referring to the reduced pressure and temperature, the ideal gas approximation seemed to be OK. In short, we have not directly considered the energy conservation principle at all. We may subsequently have an occasion to revisit this issue.

Assumed data:

Now, let us make assumptions about the data to be used for our calculations.

Suppose that the ambient temperature is 30 deg C, and the Eco-Cooler is kept at the mean sea level (MSL), say by the sea-side (rather than somewhere on a slope going down into the Death Valley [^]). Now, seated at a sea-side, the evaporative cooler isn’t going to be feasible because of humidity, and that’s the reason why we are at all considering using the Eco-Cooler. Alright. So to wrap up this point, we have to use data values at the MSL.

Suppose that we can use the ambient MSL atmospheric pressure for the inlet of the bottle; it would then mean that $P_1 = 101330$ Pa. (Note, this is an assumption; like with many other assumptions, we may have occasion to revisit it later on.) The air density at MSL and at 30 deg. C may be taken to be $\rho = 1.169$ kg/m^3. (Minor changes to this value turn out to have minimal impact on the predictions, so it’s OK to use, even if we might have made a mistake in looking it up hurriedly.)

For wind-speeds, let’s assume that the speed at the inlet (i.e., at the base of the bottle, which is exposed to the outside) is the same as the ambient wind-speed. (Again, this is an assumption; we may have the occasion to revisit it later on.)

Since the wind-speeds vary, and since the pressure drop (and hence the temperature drop) obviously depends on the inlet wind-speed, we will have to repeat our calculations for each wind-speed, again and again.

Referring again to the Wiki article for the Beaufort scale [^], to have representative wind-speed values, we choose to take the averages of the lower and upper wind speeds which together define the range for each wind-grade on the Beaufort scale. Thus, our $U_1$ could be one of: 0.5, 3.0, 8.5, 15.5, 24.0, 33.5, 44.0, all in kmph.

We are now ready to do our calculations. To recap: First, we calculate $U_2$ from $U_1$ using the continuity (mass conservation) equation and the known area ratio (which is about 14). Then we substitute the data in the re-arranged Bernoulli’s equation (which brings in its own assumptions) and obtain $P_2$, the pressure at the outlet (i.e. the neck of the bottle). From the ideal gas law (being used in place of the energy equation proper), we then calculate $T_2$.

Python script:

We use a Python script only because the calculations for $T_2$ have to be repeated again and again for different wind-speeds. Anyway, here is the Python script:

'''
This Python script calculates the expected
outlet temperature for a bottle in the Eco-Cooler.
See the relevant blog posts by Ajit R. Jadhav.
All units are in SI.
'''

d1 = 7.5 # ID at the inlet (i.e. at the base), in cm
d2 = 2.0 # ID at the outlet (i.e. at the neck), in cm
AR = (d1*d1)/(d2*d2) # Area ratio for the bottle

T1 = 30.0 # Ambient temp., in deg. C
T1 = T1 + 273.15 # Conversion from deg. C to K
rho = 1.169 # Density of air, in kg/m^3
P1 = 101330 # Ambient pressure, in Pa

# The Beaufort Scale wind speeds in kmph, and their text descriptions
bsa = [0.5, 3.0, 8.5, 15.5, 24.0, 33.5, 44.0]
bsta = ["Calm","Light air", "Light breeze", "Gentle breeze", "Moderate breeze", "Fresh breeze", "Strong breeze"]

# Calculate the expected temperatures for various wind-speeds
for i in range(7):
s1 = bsa[i] # wind-speed, in kmph
sText = bsta[i] # wind-speed description

U1 = s1 / 3.6 # conversion from kmph to m/s
U2 = U1*AR

# Calculate P2 using Bernoulli's equation
P2 = P1 + rho*(U1*U1 - U2*U2)/2.0

# Calculate T2 using the ideal gas law
T2 = T1*P2/P1
TC = T2 - 273.15 # conversion from K to deg. C
print( "%20s %6.1f %6.2lf %10.0f %6.1f" % (sText, s1, U1, P2, TC) )



The program written using Python is so simple, that you can very easily modify it, say to report any additional calculations that were made along the way.

Output data:

Here is the output data I get. The columns are: wind description in words, wind speed in kmph, wind speed in m/s, outlet pressure, and outlet temperature:

    Wind Description        U_1        P2, Pa   T2, deg. C
kmph  m/s
Calm    0.5   0.14     101328   30.0
Light air    3.0   0.83     101250   29.8
Light breeze    8.5   2.36     100689   28.1
Gentle breeze   15.5   4.31      99198   23.6
Moderate breeze   24.0   6.67      96219   14.7
Fresh breeze   33.5   9.31      91372    0.2
Strong breeze   44.0  12.22      84151  -21.4



Interpretation of results:

Our model predicts that when the breeze is strong (but less strong than when the tube is held near the window of a car that is traveling on an expressway), we should get ice formation at the neck of the bottle—in fact, it should be a super-cooled ice!

From your knowledge of what happens on the Indian sea-side, do you expect a mere half-foot tube of variable diameter, to begin to have ice-formation at its neck?

Obviously, the model we dreamt up has gone wrong somewhere. …

… But precisely where? … We have made so many assumptions…. Which of these assumptions are likely to have impacted our analysis the most? In what all places should we bring in the corrections to our model?

I have already given a lot of explicit notes—not just hints—while writing down the analysis above. So, go through the entire post once again, now pausing especially at the assumptions, and think how we may go on to improve our model.

And as always, sure drop me a line if you think I am going wrong somewhere.

Alright, enjoy!

Ummm… Since I have given you Python scripts to play with, guess the usual section on a song I like has become redundant. So, let me mention the “other” song (re. my last post) when I finish this series of posts—which will take one, or at the most two more posts).

[I may come back and cross-check the latex entries in this post, grammar, etc., later on today itself, when I may perhaps edit this post just a bit, but not much. Done on 2016.10.02 itself.]

[E&OE]

# A few remarks on the Eco-Cooler

While generally browsing ISHRAE[^]’s Web site after a long while today, I ran into this coverage of the so-called Eco-Cooler [^] in their News Section.

… My earlier coverage of another creative usage of the used plastic bottles was here: [^] (see the “farm ponds” section in that post).

Anyway, coming back to the Eco-Cooler, a simple Google search on the inventor’s name (“Ashis Paul”) will give you quite a few links, e.g. here [^] and here [^]. A sketchy story as to how Paul ended up inventing the cooler is mentioned here [^].

The idea is so simple that you just have to wonder why no one else thought of it before!

Apart from the cultural reasons (people in this part of the world arguably don’t always try to tackle their life’s problems creatively; they arguably just sit idle and whine and complain) the other reason, I think, is that to a learned engineer (and I will call myself that), it would be difficult to think that the cooling effect obtained this way could be significant—the claim is a drop of up to 5 degrees Celcius (i.e. 9 degrees Fahrenheit) in the room temperature!

… I don’t know why, but somehow, at least on the face of it, a claim of this big a temperature drop does seem unbelievable, at least initially.

Anyway, here are a few things you could pursue, especially if you are a student of mechanical engineering:

• First, name (or hit your text books and find out) the principle that explains the cooling effect.
• Then, assume suitable values for the air flow, and using the appropriate thermodynamic/psychrometric charts and property tables, determine whether the inventor’s claim is acceptable. (I have not done this cross-check myself before writing this post; I just assumed that someone at ISHRAE must have done it!)
• Now check out the DIY YouTube videos on this invention. If interested, think of building an Eco-Cooler and measuring its performance yourself. (And if you do that, and if you are from Pune or a nearby place, then do drop me a line. I would love to come over and check it out myself.) Alternatively, think of doing a “simple” CFD analysis and compute the estimated temperature drop. [… And to think how people keep asking me where I get all my student-project ideas from!]

Next are a few notings (assuming that the cooling effect is indeed big enough) to help you put it all in the right context, and then also some pointers as to how you could try and modify (and even optimize) the existing design.

• Bamboo Curtains: First, try to put it in some context: People in India often use bamboo “chaTai”s or mats [^] as window covers/curtains. (Also the khus curtains.) Some of these “chaTai”s do carry regularly spaced holes. Do such mats (or even Venetian blinds) give rise to any cooling effect? Can they? Why or why not?
• Flow Pattern: Using ink blobs or other tracers in a flow of water, visualize the geometry of the flow going into a hole, and ask yourself: Is a bottle surface at all necessary? Why? When?
• Shape and Size: Would you get a better effect if you modify the dimensions of the bottles used in the Eco-Cooler? Is the size of the water bottle optimal? How about the shape?
• Mounting: What if you mount the bottles on the board not at the neck but at the base? Would it be more stable? Would it be more convenient because nothing goes protruding outside the room? Many questions below assume mouting on the base, such that the bottles come protruding inside the room. Let me call it the Internally Protrduding Design (IPD for short), as compared to the Externally Protruding Design (EPD, which is shown in the original photographs and videos).
• Materials: How about changing the material? What if you use clay for the tube?
• Evaporative Cooling: Assume IPD. Would keeping the clay tubes wet help enhance the cooling? You could keep them wet via a simple system of water from an overhead tank running over or percolating through the thickness of the clay tubes. For this purpose, arrange the base circles in a hexagonal lattice arrangement (rather than the simple square lattice they show in the original sketches and videos). In any case, compare the cooling obtained using the dry Eco-Cooler vs. that using the desert cooler. Then, compare it with the wet Eco-Cooler. To do that, first find out the natural cooling limitations of the desert cooler. (Something like this was a unit test question I had asked my ME (Heat Power) students last year.) Where would the wet Eco-Cooler be more effective—in the humid coastal areas (e.g. in Mumbai), or in the dry-and-hot areas (like in the plains or Delhi)?
• Cooling Achieved: Estimate the size of the biggest room that may be effectively cooled using EPD. Repeat for IPD. Also find out (by CFD analysis or by experiment) the locations where the cooling would be effective enough to bring (at least a bit of) comfort to a human being.
• Forced Circulation: What if you use forced air circulation with IPD? Would it lead to any better cooling? Don’t guess! Bring out your charts, tables and calculators once again, assume suitable values for fans, and provide a quantitative estimate. Then, also figure out if a forced air circulation could be economical enough.
• Enhanced Natural Circulation: (Assume both designs in turn.) Think if you could possibly enhance the natural air circulation by using some simple cardboard flaps erected on the outside of the room. (Do a quick-and-simple CFD analysis if you wish.)
• Radiation: How much of the temperature drop can be attributed to the obvious reduction in the radiative heating alone?
• Internal Reflection: Is the total internal reflection an important factor here? Would using clay tubes (of varying cross section) reduce the glare due to total internal reflection?
• Noise Generation: Does the arrangement emit sound (as in a patch of bamboo trees)?
• Aesthetics: Assume IPD: Think of how the cooler design may be used creatively for aesthetic enhancements of the room interiors in a middle-class apartment or bungalow. (The cooler doesn’t have to be used only in the slums!) Could ready-made panels of standard sizes be made in clay or alternative materials (e.g. cheap ceramics) just as cost-effectively? Would painting the bottles help?
• Reverberations: Assume IPD: Refer to technical acoustics. Can you reduce the sound reverberations if you use such shapes near walls? Could plastic bottles be at all effective in this respect? How about the clay tubes? Would the existence of the holes modify the sound-dampening effect due to the protruding tubes? Would they introduce unwanted modulations? Estimate the range of sound-frequencies (or of musical tones) that stand to get impacted (for the better or for the worse) due to the presence of the Eco-Cooler.

Enjoy…

A Song I Like:

(Hindi) “too laalee hai savere waalee, gagan rang de tu mere man kaa…”
Music: Sapan-Jagmohan
Singers: Kishore Kumar, Asha Bhosale
Lyrics: Indivar Naqsh Lyallpuri [^]

[BTW, this song reminds me of another song which has a similar tune. (I don’t know music well enough to make out “raaga”s. In fact, I often cannot even make out tones! I can only compare the tones in a hand-waving sort of a manner, that’s all! … It’s just that sometimes I happen to notice some similarities.) See if you can guess it—the other song. I will tell you the answer in my next post.]

[I have a habit of coming back and modifying my post a bit even after publication. But guess, at least for this post, there really isn’t anything left to add or modify.  Actually, I did modify! [sigh!] I clarified the two designs and even added the names for them. I even changed the title a bit!!…  Anyway, bye for now, and take care…]

[E&OE]