Unruled Notebook

Entries from August 2006

The Koch Curve and Visual Resolution

August 28, 2006 · 13 Comments

The Koch Curve or the von Koch snowflake was discovered by Helge von Koch (1870-1924) in 1904. It is a closed fractal curve of infinite length within a finite region of space, enclosing a finite area. The construction of such a curve and some of its unique properties is explained in this post, with a connection to porous media.

Take an equilateral triangle as shown below in Figure 1, with sides of, say, 1 unit each. This is the “zeroth” iteration of the Koch curve.

KochCurveIteration0

KochCurveIteration1

Figure 1: Iteration zero (and) Figure 2: Iteration one

For the first iteration we split each side of the equilateral triangle into 3 equal parts, with the middle 1/3rd being replaced essentially by another smaller equilateral triangle made of side of 1/3 unit. The resulting figure is shown below.

What we have done in the first iteration is to replace each side made of 1 unit stick by four, 1/3 unit sticks, arranged such that their end points remain fixed between the original 1 unit length of iteration zero. This procedure automatically fixes the angles of the protruding equilateral triangle in Figure 2, and thus increases the original area confined by the equilateral triangle of Figure 1, to that of Figure 2.

By following the previous procedure, in the second iteration, we could now generate Figure 3 out of Figure 2, with each of the sides of Figure 2 now being replaced in their middle thirds with even smaller equilateral triangles of sides made of 1/9 unit sticks.

KochCurveIteration2

KochCurveIteration3

Figure 3: Iteration two (and) Figure 4: Iteration three

Results of iteration three, four and five are shown in Figure 4, 5 and 6 respectively.

KochCurveIteration4

KochCurveIteration5

Figure 5: Iteration four (and) Figure 6: Iteration five

The procedure could be repeated many times, to reach the von Koch Snowflake, when the iteration tends to infinity. There is an applet available at [http://www.efg2.com/lab], for you to try out more iterations and other curves, starting from other basic geometrical figures.

Intuitively we could see that this resulting figure would have “infinite” length for its sides (if we could possibly draw it) while enclosing a “finite area” as shown in the above figures. How much would be the area when we actually do the infinite iterations?

To answer this let us put this procedure in a mathematical footing. It only involves simple algebra.

The derivation for calculating the enclosed area done here follows the explanation given in When Least is Best by Paul J. Nahin. Similar explanations are available in the Wikipedia page for the Koch Curve.

Let after n iterations (n > or = 0)

  • Nn = number of sides
  • ln = length of each side
  • Ln = length of perimeter = Nn x ln

For instance, if l0 = 1, N0 = 3, then L0 = 3. Since we increase in every iteration the number of sides by a factor of four (compare Figure 1 and Figure 2) and since N0 = 3, then it means

Nn = 3 x 4n, where n = 0, 1, 2, ……

Further, for each iteration, the length of a side decreases by a factor of 3. for instance we used a stick of length unit 1 to form the side of Figure 1, while in Figure 2 each side of Figure 1 is replaced by 4 sticks of length unit 1/3 of the original. Since l0 = 1, we could write then,

l_n = 1\cdot\left(\frac{1}{3}\right)^n = \frac{1}{3^n}, n=0,1,2,\cdots

The perimeter after the n’th iteration becomes

L_n = N_nl_n = 3 \cdot 4^n\cdot \frac{1}{3^n} = 3 \cdot (\frac{4}{3})^n

which, in other words, means

lim _{n \rightarrow \infty }[L_n] = lim _{n \rightarrow \infty } \left[ 3 \cdot (\frac{4}{3})^n \right] = \infty

So, the length of this closed curve does actually goes to infinity. Now, for the area it encloses.

We know from our high school trigonometry that the area of a triangle can be calculated using the Heron’s formula

\sqrt {s(s-l_n)(s-l_n)(s-l_n)} = \sqrt { \frac {3}{2}l_n \times \frac{1}{2}l_n \times \frac{1}{2}l_n \times \frac{1}{2}l_n } = \frac {\sqrt {3}}{4}(l_n)^2

where ln is the length of a side, with s = 1/2 x (ln + ln + ln) = 3/2 x ln.

Using the above formula, for Figure 1, the zeroth iteration of the Koch curve, we could write

A_0 = \frac{\sqrt{3}}{4}

By comparing Figure 1 and Figure 2 we could see that we are increasing the area by adding 3 x 4 n-1 equilateral triangles (a triangle for each side) with ln = 1/3n. For example, with n = 1, we increase from A0 to A1, by adding three equilateral triangles, each with l1 = 1/3.The area of each of these triangles is obviously,

\frac{\sqrt{3}}{4}\left(\frac{1}{3^n}\right)^2 = \frac{\sqrt{3}}{4}\cdot\frac{1}{9^n}

and so for the figure after n iterations

A_n = A_{n-1} + 3 \times 4^{n-1} \times \frac {\sqrt {3}}{4} \times \frac {1}{9^n}

= A_{n-1} + A_0 \times \frac{1}{3} \times (\frac {4}{9})^{n-1}

For clarity, if we write explicitly for the first few n values

n = 1: A_1 = A_0 + A_0 \times \frac{1}{3}\left(\frac{4}{9}\right) ^0 = A_0\Big[1+\frac{1}{3}\left(\frac{4}{9}\right) ^0\Big]

n = 2: A_2 = A_1 + A_0 \times \frac{1}{3}\left(\frac{4}{9}\right) ^1 = A_0\Big[1+\frac{1}{3}\left(\frac{4}{9}\right) ^0 + \frac{1}{3}\left(\frac{4}{9}\right)\Big]

n = 3: A_3 = A_2 + A_0 \times \frac{1}{3}\left(\frac{4}{9}\right) ^2 = A_0\Big[1+\frac{1}{3}\left(\frac{4}{9}\right) ^0 + \frac{1}{3}\left(\frac{4}{9}\right) ^1 + \frac{1}{3}\left(\frac{4}{9}\right) ^2 \Big]

Therefore, in general we could write

\lim_{n\to\infty} A_n = A_0 \Big[1+\frac{1}{3}\left(\frac{4}{9}\right) ^0 + \frac{1}{3}\left(\frac{4}{9}\right) ^1 + \frac{1}{3}\left(\frac{4}{9}\right) ^2 + \cdots \Big]

= A_0 \Big[1 + \frac{1}{3} \Big( 1 + \frac{4}{9} + \left(\frac{4}{9}\right) ^2 + \cdots \Big) \Big]

The expression inside the braces of the above equation can be seen to be in a geometric progression, which then could be simplified as

\lim_{n\to\infty} A_n = A_0 \Big[1+\frac{1}{3} \cdot \frac{1}{1 - \frac {4}{9}} \Big] = \frac {8}{5} A_0

So, the area enclosed by the closed curve of infinite length is actually only 60 percent more than that of the original area of the equilateral triangle we started in Figure 1. A remarkable property indeed.

Now to give all of this a porous medium perspective, we could think of each successive figure from 1 through 6 to be of finer and finer “resolution level”, depicting however, the same porous medium. The ‘gray color region’ inside of each figure being one constituent and the ‘outside’ of this figure being the other constituent (of course, bounded by a outer boundary, which is not seen in these pictures), with the Koch curve forming the “identifiable interface” between these two constituents.

The visual resolution going from Figure 1 to 2, the next level of the Koch curve ‘porous medium’, is 3. Because, each original length of the earlier iteration is divided into four parts of each 1/3 of the original length. So, indirectly, the visual resolution is represented by the interface length in a porous medium.

We could see now why visual resolution is important to define a porous medium, as we kept saying the previous posts on this topic. Even the interface length is increasing in the above example of a porous medium, if we resolve things finer and finer, an effect which is bound to influence interface phenomena, like heat transfer across the constituents of a porous medium.

Categories: Maths
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Composite Heat Sinks for Cooling Electronics

August 20, 2006 · 10 Comments

Higher volumetric heat generation is anticipated when miniaturization of electronics is done in typical engineering applications. Recent days, this value of heat flux released from electronics is as much as 1 MW/m^2 (100 W/cm^2).

For reliable functioning of the electronics, the generated heat has to be dissipated quickly and efficiently such that the electronics are maintained below a set point temperature (SPT, usually the junction temperature of the transistors, ~ 90 to 120 C) during their time of operation.

In this regard, heat sinks using Phase Change Materials (PCM) such as high carbon paraffin find extensive applications in the thermal management of portable electronic devices, owing to the PCM property of absorbing a large quantum of heat at a constant temperature while melting.

However, the low conductivity of the PCM is a hindrance by the creation of hot spots, which is overcome by a Composite Heat Sink (CHS) comprising a judicious distribution of high thermal conductivity Base Material (BM) along with the PCM.

The thermal design of such a CHS, constructed using a vertical array of �fins� (or Elemental CHS, ECHS) of similar shape and size, made of PCM and BM, for its performance improvement has also been studied in detail by many researchers in the recent decade [Prof. Suresh Garimella at Purdue Uty., Prof. Cristina Amon at Carnegie Mellon Uty., Prof. Pradip Dutta and Prof. Srinivasan at the IISc]. One such simple design is shown in Figure 1 below.

chsfigure1.jpg

Figure 1

For a CHS constructed with the design of BM fins placed vertically inside a PCM bath (Fig. 1a), operating under a given SPT (i.e. keeping the electronics that need to be cooled), under-usage of the latent heat of the PCM because of incomplete melting is a more critical CHS issue than the complete melting of PCM before the CHS reaches the SPT. The former results in a poorly performing CHS while the latter merely reduces the CHS potency for heat transfer enhancement (from PCM latent heat to sensible heat) in time.Of course, this maximization of the operation of CHS can be resolved, as was done in [Akhilesh et al., 2005] – I am one of the co-authors. For a chosen PCM quantity, a “critical upper-bound dimension” (d_c and B_c) for the ECHS, as shown in Figure 2b, can be determined by analysis. Constructing a CHS (shown in Figure 2a) with a finite number of ECHS (shown in Figure 2b) made of the critical dimensions (d = d_c and B = B_c) ensure complete melting of all of the PCM exactly when the SPT is reached. This design gives an increased time of operation for the CHS.

In other words, in a CHS design of Figure 1, exhausting all of the PCM latent heat storage before SPT is reached, should yield maximum time of operation of the CHS.

chsfigure2.jpg

Figure 2

By the way, the above CHS idea is not far-fetched. See for instance [ ESLI Products: PCM Heat sinks ]. This product, shown in the right here, is a typical PCM based CHS that uses light weight brush like metal fibers as interconnecting media pervading the PCM matrix, different from the simple parallel ECHS arrangement shown in Figure 1.

chsfigure3.jpg

According to the ESLI website,

ESLI’s paraffin based heat sinks can be designed to operate at or near 5, 18, 28, 37, 44, 55, or 61 °C. Non-paraffin based designs are also available for cryogenic or high temperature applications. ESLI’s PCM composite heat sinks have been space flight tested on the Space Shuttle (STS-95) aboard the CRYOTSU experiment module, and will fly aboard NASA’s Vegetation Canopy LIDAR experiment in 2000.

Another product developed by NASA to cool on-board space batteries in rockets, utilizes carbon fibers as the metal BM, a variant from the above CHS design of Figure 1 and 2.

For an illustration of the applicability of the CHS explained in Fig. 1 and 2, consider a cooling application that requires a thermal dissipation of 62.5 W and the electronics to be maintained below T_SET = 90 C through a typical CHS area of 5cm x 5cm. For this case then the heat flux would be q” = 25 kW/m^2 and by choosing a CHS with D = 5 cm (left to right length, in Figure 1) and height A = 5 cm (see Fig. 2a), the number of critical dimensioned ECHS (Fig. 2b) made of 50 percent of PCM that should ‘fill’ the CHS (Fig. 2a) can be calculated from analysis. This turns out to be d_c = B_c = 0.004 m (see Fig. 2b) and the number of such ECHS that would fill the CHS is equal to 4. For such a design the CHS would operate successfully for about 453 seconds. After this, the CHS has to be ‘cooled’ to release all of the stored energy in the PCM, before charging it again.

The CHS design in Figure 1 and 2, or something similar, is a required one for cooling the on-board electronics in the space vehicles launched by ISRO, the Indian Space Research Organization. Typically, this means, the CHS has to operate successfully (taking maximum energy and not allowing the electronics to reach SPT) for about 20 to 30 minutes (one way).

Categories: Research Notes · Thermal Sciences
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Alphabet Recital and Creativity

August 20, 2006 · 9 Comments

A Sunday evening query, otherwise fit to be told as an anecdote (only) in the Readers Digest.

My daughter surprised me yesterday, when she recited the English alphabets.

It started as A, B, C, D….went up to O and suddenly went like Q, P, T, S, R, U, V, W,…until Z.

So?

Well, it set me thinking why should I have “taught” her the ABCD in THAT particular ‘conventional’ order and act surprised now? What is SO methodical and creative about THAT conventional ordering of the English alphabets, other than the fact that I learnt it that way and didn’t bother to give a damn about why it should be taught and remembered that way alone?

Of course, the above query applies equally well to other languages also.

As far as I could observe, my daughter didn’t miss (forget) any alphabets. She could recall all of it, but not in a conventional order.

Should I now ‘train’ her to follow ‘convention’, for no logical reason, OR should I leave her alone? Any thoughts?

Categories: Micro Muse
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Convection Carnot Engine

August 18, 2006 · 5 Comments

Free or Natural Convection has been explained in detail in Free and Paid Convection, Free Convection For Dummies and Free Convection and the Rayleigh Number. The natural convection phenomenon explained can be viewed as a Heat Engine – a device that performs the conversion of heat energy to mechanical work – say, a car engine – by exploiting the temperature gradient between a hot “source” and a cold “sink”.Let us explain how the above stated analogy is possible using the figure below.

convectioncarnotengine.jpg

Figure 1

The processes that constitute the natural convection ‘cycle’ (a closed set of processes that changes the thermodynamic properties of a system like pressure temperature, only to bring it back to values with which it started) inside an enclosure are as follows.

  • Process 1 . Here the fluid packet considered to experience the force imbalance is composed of mass say dm. The actual movement of this packet is because it receives heat energy at the bottom from a heat source isothermally because of which it immediately expands. In effect, it receives heat energy by undergoing an isothermal expansion process.
  • Process 2 . With this energy, it increases its volume and as seen earlier experiences a force imbalance resulting in the displacement, which causes further expansion as, the packet raises. This results in an adiabatic expansion process in which the packet moving up can perform some work.
  • Process 3 . As it goes up the packet loses the remaining heat energy to the surrounding at the top to maintain equilibrium. This it does as an isothermal compression process as it cools and contracts at the top.
  • Process 4 . Finally this cooled packet is what we get circulated back to the bottom, if we have to account for the ‘lost’ mass in the bottom by mass conservation principle. This happens through an adiabatic compression process, as the packet further contracts as it comes down.

The fluid packet executes a cycle comprising heating – expansion – cooling – compression the processes that constitute a Carnot Heat Engine Cycle (an idea proposed by Nicolas Leonard Sadi Carnot in 1824, which we use even today as a reference maximum efficiency cycle, for analyzing any heat engine cycle).

Existence of a Work producing potential in the Natural Convection flow inside enclosures can be made evident by inserting a propeller (made of a light material) across the flow path of the fluid packet. The situation can also be easily visualized in the laboratory with the help of aluminum powder strewn in water. In reality, the Work Output of this cycle, unfortunately, is sufficient only to accelerate the fluid packet against the viscous drag (the fluid brake, which opposes the motion) the packet experiences on the way. This has been explained earlier in Free Convection and the Rayleigh Number.

That is all with the idea.

Now, if you are comfortable with some equations, we could place this idea in better quantitative light.

The ‘buoyancy force’ generated in the convecting system above (see Free Convection For Dummies for an explanation) is the chief cause for the ‘work producing potential’. As an order of magnitude, the ‘energy’ gained as a result of this buoyancy force – which is equivalent to the ‘work done’ of the convecting system neglecting viscous and thermal diffusion – is given by

W sim (rho U Delta T cdot g cdot alpha) cdots (1)

where $lates rho $ is the density and alpha   is the thermal diffusivity of the convection fluid, U is the velocity of the convection wheel, Delta T = (T_1 - T_2) is the temperature difference across the enclosure as defined in figure 1 and g is the acceleration due to gravity.

The Heat energy added to the system is given, again as an order of magnitude, by

Q_1 sim (rho cdot c_P cdot U cdot Delta T ) / L cdots (2)

where c_P is the specific heat of the convecting fluid at constant pressure

The ‘efficiency‘ of the Convection Carnot cycle can be defined as the ratio of how much useful work we derived from the rotating convection wheel (given in Eq. 1) and how much heat energy we ‘spent’ to achieve this (given in Eq. 2). This ratio is given as

frac{W}{Q_1} sim frac{gLalpha}{c_P} cdots (3)

The notation on the LHS of equation (3) corresponds to conventional thermodynamic usage (see figure 1).Typical to the convection system described in figure 1, the W << Q1 and Q1 ~ Q2, which results in the low efficiency. The fluid packet received some energy in the bottom. Subsequently, this energy will be dispersed (depending on the strength of c_P) from it. Before this happens, the fluid moves up because of buoyancy. This displacement can lift a weight (do work). However, the displacement has to overcome the viscous resistance and the thermal diffusion along the way for which some of the energy is used. The remaining energy in the packet can be realized as Work Output.

It is now evident that the convection situation described above (and in our earlier posts) obeys the First Law of Thermodynamics.

However, the kinetic energy (after overcoming the fluid brake and thermal diffusion) in the packet is almost a small perturbation in the First Law of Thermodynamics. When Convection is modeled as a Carnot engine, it should be completely reversible.

From the Second Law of Thermodynamics we infer the change in Entropy for the cycle is zero. In reality, the viscous dissipation and conduction irreversibility generates enough Entropy to ‘restrict’ the Work potential of the Convective heat engine. These considerations are embedded in a ‘dimensionless group’ called the Rayleigh Number.

A recent research paper by Prof. V. A. F. Costa [1], explores the role of viscous dissipation in natural convection engine very nicely. I have also written an article on the above idea [2], but without considering viscous dissipation. I would be happy to send both the papers in pdf format, for the interested readers.

Some references

  • On natural convection in enclosures filled with fluid-saturated porous media including viscous dissipation, V. A. F. Costa, International Journal of Heat and Mass Transfer, Volume 49, Issues 13-14, , July 2006, Pages 2215-2226 [abstract]
  • A. Narasimhan, Convective Carnot engine, Phys. Educ. 35 (2000) (3), pp. 178-181, [see Abstract-INSPEC for summary]
  • Physical Fluid Dynamics – D. J. Tritton, (1988), Oxford Science Pub.
  • Convection Heat Transfer , 3rd ed. – A. Bejan, (2004), John Wiley & Sons
  • Convection – M. G. Velarde & C. Normand, (1980), Scientific American
  • Rayleigh Benard Convection : Structures and Dynamics – A. V. Getling, (1998), World Scientific

Categories: Research Notes · Thermal Sciences
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The Scian Melt – Edition Twenty

August 15, 2006 · 17 Comments

Welcome to the Twentieth Edition of The Scian Melt, a science blog carnival originated by Prashant Mullick of Kyun and nurtured by Selvakumar at The Scientific Indian. Contributions and tip-offs didn’t exactly swamp me off my feet, but were just enough and I searched for some more, worthy of this carnival.

This edition coincides with the move of Selva’s The Scientific Indian into the ScienceBlogs compendium. We wish him more blogging success.

We start the carnival with two posts that mark this significant leap [notice I don't use the much abused "quantum" leap]

The rice mills of our desi-villages are served by the concept in the first post above, while the windmills of our mind are served by the second.

With that science-appetite growing strong, move on to wonder about the flavors of neutrinos (they come in three) in

If you liked your neutrino flavors there, try this one for a contrast on,

and while you feel spiced up by the above post, munch these desi-chips and tortillas of mixed flavors as well

If that dose of high fa(c)t science junk food, is not enough, you can try cooking it all by yourself and pour hot oil onto your feet for a change, to know how it feels. Wait, somebody was already

for you, so you don’t have to repeat it. Unless you are like me, who believes in nobody and had burnt a finger testing if fire actually is hot…

Let this all digest. If you are worried about your weight because of what you have eaten so far, you could try the

If on the other hand, like the Romans, if you want immediate relief, try this purgatory of

which is sure to cause you relief. Alternately, you may stir it all up with

or some Indian pot served in

If you already feel like puking, don’t do it at my blog. There is always a

to do that, for you to be diagnosed of you know what. If you don’t know, read the above post (male audience: did you check out the link in the last sentence in the above post?).

And let me be the fair host and spare you of my cuisine in this carnival. Its all over my blog, anyway.

Hope you enjoyed what we could Melt for you, the 20th time. Visit again, at another blog (check here to know where) on another day, with another menu and another host to entertain you…

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Categories: Carnivals