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User:Egm6322.s09.xyz/proj draft

=Mass Transport via Molecular Diffusion for Fuel Cell Application=

Introduction to Fuel Cell Technology

Fuel cells are a clean energy source that holds great promise for displacing conventional energy systems toward a cleaner, more efficient future. Similar to a battery, fuel cells produce DC current from an electrochemical reaction that takes place between hydrogen and oxygen via a polymer membrane electrolyte.

The basic structure of a PEM (polymer electrolyte membrane) fuel cell is given in Figure 1. It consists of a polymer electrolyte membrane sandwiched between two gas diffusion layers, known as the anode and the cathode. The electrochemical reaction takes place with the aid of two catalyst layers, located at the anode/electrolyte and cathode/electrolyte interfaces, respectively. The chemical reactions for a PEM fuel cell are given as follows :

Anode RXN: $$H_2 \Rightarrow 2H^+ + 2e^-$$

Cathode RXN: $$\frac{1}{2}O_2 + 2H^+ + 2e^- \Rightarrow 2H_20$$

Overall RXN: $$ H_2 + \frac{1}{2}O_2 \Rightarrow H_20$$

As shown in the above equations, fuel cells require both hydrogen and oxygen to enable the reactions at the catalyst layers. The product of the electrochemical reaction is simply water (and heat).

In a fuel cell system, the reactant (fuel & oxygen) flows for the fuel cell are closely monitored. Depending on the performance set point, information is needed about the required flow rates, the amount of storage needed to hold the reactants, as well as the amount of storage needed to hold the reaction product water.

Mass transport of reactants through the anode/cathode layers to the catalyst layers (reaction zones) is governed by molecular (mass) diffusion. Consideration of mass diffusion is presented here in the form of 1) a detailed derivation of the governing partial differential equation using a control volume approach, 2) a discussion about how mass diffusion is similar to the heat conduction, and, 3) an illustrative example using gas diffusion through the electrode layers of a PEM fuel cell.

It is hoped that reader will gain a deeper understanding of the transport processes within a fuel cell as well as an appreciation of the mathematics that govern this real-world physical process.

Derivation of Mass Diffusion Partial Differential Equation

The foundation of the molecular mass diffusion is that the flux, F, of a substance is related to the change in concentration, C, over the change is distance, x, via a constant of proportionality known as the diffusion coefficient, D. An expression of this is given as follows:

 * $$F = -D \frac{\partial C}{\partial x}$$

Note that the flux is given as the mass transfer of a substance per unit time per unit area.

Consider a control volume of an isotropic medium as shown in the Fig. 1. Isotropic materials have the same properties in all orientations. The control volume features a rectangular box that is parallel to the coordinate axis. The lengths for the box are given as:

 * $$AA' = BB' = DD' = CC' = 2dx$$

 * $$AD = BC = A'D' = B'C' = 2dy$$

 * $$AB = DC = A'B' = D'C' = 2dz$$

The point P = P(x,y,z) represents the center of the control volume.

If the ABCD and A'B'C'D' faces are perpendicular to the x-axis, then the flux of a substance entering the ABCD face is given as:

 * $$4dydz \left( F_x - \frac{\partial F_x}{\partial x} dx\right)$$

Respectively, the flux exiting the c.v. from the A'B'C'D' face is given as:

 * $$4dydz \left( F_x + \frac{\partial F_x}{\partial x} dx\right)$$

The rate of increase of the species between the ABCD and A'B'C'D' faces is given as:

 * $$-8dxdydz\frac{\partial F_x}{\partial x}$$

The other faces can be treated the same way, therefore:

 * $$-8dxdydz\frac{\partial F_y}{\partial y}$$, and $$-8dxdydz\frac{\partial F_z}{\partial z}$$

Lastly, the increase in the rate of the diffusing substance is given as:

 * $$8dxdydz\frac{\partial C}{\partial t}$$

The mass conservation principle that governs unsteady diffusion can be stated as :

(net rate of diffusion of species A into medium) = (rate of increase of species A in medium)

Following this principle, then the following expression results:

$$\frac{\partial C}{\partial t} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} = 0$$

Substituting the above expression for the flux, F, as a function of the concentration, yields:

$$\frac{\partial C}{\partial t} = D \left( \frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2} + \frac{\partial^2 C}{\partial z^2} \right)$$

In this equation, the diffusion coefficient is constant. For the 1D case, the equation reduces to:

$$\frac{\partial C}{\partial t} = D \left( \frac{\partial^2 C}{\partial x^2} \right)$$

This 1D formulation for mass transport via diffusion is the governing partial differential equation that will be explored deeper for the fuel cell application example that follows below.

source: J. Crank, The Mathematics of Diffusion, Oxford University Press, New York, 1956.

Diffusion Thoery: Historical Background ,

The 1D mass diffusion equation given above was first proposed by Adolf Eugene Fick in 1855. He based his derivation on the work of Jean Baptiste Fourier who created the landmark studies of heat conduction. The mass diffusion equation is analogous to the heat conduction equation which is of the form:

$$\alpha \frac{\partial^2 T}{\partial x^2} = \frac{\partial T}{\partial t}$$

where T is the temperature, the constant of proportionality $$\alpha$$ is the thermal diffusivity.

An Illustrative Example: Gas Diffusion through electrode layer of a fuel cell

This section will attempt to utilize the mathematical theory presented above to model a real-world application of hydrogen gas diffusion through the anode layer of a PEM fuel cell.

Figure 3 shows a schematic diagram of the anode layer along with the respective coordinate system that will be used in the calculations. The anode layer is treated as a 1D slad of thickness L. The boundary conditions are such that there is a known concentration, C1, of hydrogen on the bulk flow side of the anode (x = 0). The concentration, C2, represents the hydrogen concentration at the anode/catalyst interface (x = L). The anode layer has an initial concentration equal to some unknown function given by f(x).

The following is a derivation of the hydrogen gas concentration, C(x,t), as a function of space and time given the boundary conditions listed here.

Problem Statement (summary)
$$\frac{\partial^2 C}{\partial x^2} = \frac{1}{D} \frac{\partial C}{\partial t}$$

(initial condition)

IC: $$C(x,t=0) = f(x)$$, at $$t = 0$$

(boundary conditions)

BC1: $$C(x = 0,t) = C_1$$

BC2: $$C(x = L,t) = C_2$$

Splitting a Non-Homogenous Problem into a Steady-state and Homogeneous Problem
The problem as posed is non-homogeneous and must be split into two simpler problems in order to obtain an analytical solution. The first problem will be a steady-state problem that results in a simple 2nd order ODE with respect to the space variable, x. The second problem will be a homogeneous problem that results in a PDE with only the initial condition as a non-homogeneity.

Details of the two sub-problems are given below.

Steady-state Solution
$$\frac{\partial^2 C_ss}{\partial x^2} = 0$$

BC1: $$C_{ss}(x = 0) = C_1$$

BC2: $$C_{ss}(x = L) = C_2$$

Integrating once, yields:

$$\frac{\partial C_{ss}}{\partial x} = A$$ where A is an integration constant

A second integration yields yields the auxillary solution for the steady-state condition:

$$C_{ss}(x) = Ax + B$$

The constants of integration, A & B, can be determined by applying the boundary conditions as follows:

$$\blacktriangleright C_{ss}(x = 0) = C_1$$

 * $$C_{ss} = C_2 = A(0) + B$$

 * $$\therefore B = C_1$$

$$\blacktriangleright C_{ss}(x = L) = C_2$$

 * $$C_{ss} = C_2 = A(L) + C_1$$

 * $$\therefore A = \frac{C_2 - C_1}{L}$$

The steady-state solution is then

$$C_{ss}(x)= \left[ \frac{C_2 - C_1}{L} \right] x + C_1$$

Homogeneous Solution using Separation of Variables
$$\frac{\partial^2 C_h}{\partial x^2} = \frac{1}{D} \frac{\partial C_h}{\partial t}$$

IC: $$C_h(x, t = 0) = f(x) - C_s(x) = f^*(x)$$

BC1: $$C_h(x = 0, t >0) = 0$$

BC2: $$C_h(x = L, t >0) = 0$$

Separation of Variables
Using separation of variables method, the concentration can be expressed as the product of two functions that each have a single dependent variable:

$$C_h(x,t) = X(x)\Gamma(t)$$

Substituting this expression into the homogeneous PDE, yields:

$$\frac{X''}{X} = \frac{1}{D\Gamma} \frac{d\Gamma}{dt} = -\lambda^2$$

The greek lambda in the above equation represents a constant value. The right hand side and left hand side of the the expression are both functions of single dependent variables (x and t, respectively). This can only be valid if they equal a constant. The sign of the constant lambda term has been chosen as negative in order to force the boundary value problem in the x-direction (finite).

Auxiliary solutions for each variable (x & t) are determined as follows:

$$\blacktriangleright $$Space variable, X

 * $$X''+\lambda^2 X = 0$$
 * The solution of this equation is of the form:$$X(x) = G cos(\lambda x) + H sin(\lambda x)$$
 * Applying the boundary condition at x=0, yields:
 * $$G(1) + H (0) = 0$$, therefore $$G = 0$$

$$\therefore X(x) = H sin(\lambda x)$$
 * Applying the second boundary condition at x = L
 * $$0 = H sin(\lambda x)$$

$$\therefore \lambda_n = \frac{n\pi}{L}$$ where n = 0,1,2,...

$$\blacktriangleright $$Time variable, t
 * $$\frac{d\Gamma}{\Gamma} = -\lambda^2dt$$
 * $$ln\Gamma = -\lambda^2 D t + E^*$$ where E is a constant of integration

$$\therefore \Gamma(t) = Eexp(-\lambda^2 D t)$$

The homogeneous solution is given by merging both solutions for the X and t variables and summing over all possible solutions for $$\lambda_n$$. The homogenous solution is:

$$C_h(x,t) = \sum_{n=1}^\infty H sin(\lambda_n x) \left[ exp(-\lambda_n^2 D t)\right]$$

Principle of Orthogonality
The first step in finding the coefficient H is to apply the initial condition to the auxiliary homogeneous solution which states:

$$C_h(x, t=0) = f^*(x)$$

$$\therefore f^*(x) = \sum_{n=1}^\infty H sin(\lambda_n x) \cdot (1) = \sum_{n=1}^\infty H sin(\lambda_n x) $$

Observing the orthogonality of the sin function, both sides of this equation can be operated by the following expression:

$$\int_{0}^{L}sin(\lambda_n x)dx$$

Solving for the coefficient, H, yields:

$$H = \frac{\int_{0}^{L}f^*(x)sin(\lambda_n x)dx}{\int_{0}^{L}\left[sin(\lambda_n x) \right]^2 dx}$$

where$$f^*(x) = f(x) - C_{ss}(x)$$ ,or, $$f^*(x) = f(x) - \left[ \frac{C_2 - C_1}{L} \right] x - C_1$$

The values for $$H$$and $$f^*$$ complete the homogeneous solution given above.

Final Solution
The final solution is found by summing the steady-state and the homogeneous solutions

$$C(x,t) = C_{ss}(x) + C_h(x,t)$$ $$\therefore C(x,t)= \left[ \frac{C_2 - C_1}{L} \right] x + C_1 + \sum_{n=1}^\infty H sin(\lambda_n x) \left[ exp(-\lambda_n^2 D t)\right]$$

Substituting the respective values for $$H$$ is given above (see Principle of Orthogonality) as well as $$f^*$$, then grouping like terms yields :

$$C(x,t) = C_1 + (C_2 - C_1) \frac{x}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)sin(\lambda_n x)\int_{0}^{L} \left[ f(x) - C_1 - (C_2-C_1) \frac{x}{L}\right]$$

The final solution is found by integrating this expression, resulting in the following equation :

$$C(x,t) = C_1 + (C_2 - C_1) \frac{x}{L} + \frac{2}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)sin(\lambda_n x)\int_{0}^{L} \left[ f(x) sin(\lambda_n x) dx\right] + \frac{2}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)\frac{1}{\lambda_n}sin(\lambda_n x) \left[ T_2 cos(n\pi) - T_1\right]$$

Simulation using MathCAD

The final solution to the given problem was simulated using MathCAD. The following parameters were used:

Electrode diffusion layer thickness: L = 0.005;

Diffusion Coefficient: D = 2*10^(-9);

Bulk Flow Channel Concentration: C1 = 100;

Catalyst Layer Concentration: C2 = 0;

Initial Concentration: f = 0;

Figure 4 shows a data plot of the resulting hydrogen concentration profiles throughout the thickness of the anode layer for various time elapse conditions based on the boundary (and initial) conditions listed above. It illustrates just how the hydrogen gas migrates through the anode thickness over time. As shown in the the data plot, the hydrogen concentration converges to a linear profile at large time scales.

Conclusion
Fuel cells hold great promise for a clean energy future. Fuel cells consist of multiple layers including anode/cathode, electrolyte, and catalyst layers. Mass transport within the various layers of a fuel cell is governed by Fick's equation for mass diffusion which is derived using a control volume approach for an isotropic medium.

Fick's law is analogous to the heat diffusion equation given by Fourier. Both process of mass transport and heat conduction rely on random molecular interactions. Respectively, the mathematical formulation and derivation is similar between the two physical phenomena.

A real-world application of Fickian diffusion is the gas transport of hydrogen fuel from the bulk flow stream through the anode diffusion layer to the catalyst reaction cite at the anode/electrolyte interface. Though a relatively simple case was presented here, it should be noted that mass transport for fuel cells is a very complex topic. Levels of model complexity increases with consideration of the multi-phase liquid/gas reactant streams, non-constant diffusion parameters, changing material properties from layer to layer, chemical reactions (chemical kinetics), and the interaction of reactants with the material layers.

It is a truly exciting topic!

title: User:Egm6322.s09.xyz/proj draft


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Talk:Linux/Distributions

Alrighty then
I am working with Debian, Ubuntu and Tiny Core Linux these days. As I get further along, my intent is to share what I learn as I go. I'm wondering how many Wikiversity participants are running Linux and in what flavors. • CQ 04:14, 30 April 2009 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Linux/Distributions


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User:CarrieBerg/The Sorcerer's Daughter

Chapter 1: The Castle on the Edge of Forever

In the most remote corner of the kingdom of Ruziya, beyond the barren wastes that none visit willingly, in the shadows of the fearsome Onzag mountains, there lies the castle of Malikon the Heartless.

It lies as far in the northwest corner of Ruziya as possible: just a step or two more and the castle would no longer be on land, but would be engulfed by the icy waters of the Barzen-Glassae Sea.

To get to the castle was considered a near impossible feat in and of itself. For it was as if the very land of Ruziya wished to keep the wizard’s fortress free from prying eyes. And, given the powers of Malikon, it was entirely possible that he had made the land do so.

After leaving the joys of civilized lands, anyone who wanted to reach Malikon had to first pass through the Lost Plains of Adaran. The plains themselves were not lost. No, long had they been marked on maps as a place to be avoided. Something terrible had happened there in the past, and no one wished to run into any of the lingering effects. For the effects were said to be worse than the original terror which had occurred. The plains were supposed to be devoid of all natural life. No one liked to think about what unnatural life might have sprung up over the centuries. Sane travelers avoided the place entirely. But there were always the few crazy fools who tried, and found that the tales were true: they could not readily or easily travel through that cursed place.

If, by luck or chance, travelers successfully made it through the plains, a second challenge awaited them. For the Lost Plains of Adaran turned into hills the color of dried blood. Hot and dry, they too offered nothing to those who would travel through them. However, the hills were not cursed as the plains, so life did occur. Plants grew. Though most of them poisonous, and quite a few were deadly. Horrible poisonous lizards dwell in the vales between the hills, living on the poisonous plants or the occasional unwary traveler. There were even a few streams; all with water as red as the hills. And where the hills were shadowed by the Onzag Mountains, creatures even more horrible than the lizards were said to make their home.

Finally, the travelers would come to the last challenge; the Onzag mountains. Black as shadow, jagged and harsh, they offered as much to a traveler as the lands before them: nothing. They were formed from massive slabs of basalt, pushed up over the centuries into form harsh points, as of yet hardly worn away by time. Where ancient rivers used to flow, the normally black rock was stained red as blood. The few rivers that still flowed out of the ancient peaks were a murky blood-red. It became obvious where the hills got their coloring. It was impossible to climb the mountains: the jagged basalt would cut through leather, cloth and flesh. There was only one harsh path through them. And at the path’s end lies the castle of Malikon.

The castle itself is a testament to the power the wizard Malikon, for it is carved from a single piece of black basalt. It is, if possible, even more forbidding than the land around it. There are no gaps or seems, no windows or doors. Just harsh angles and edges and walls that absorb the light, making the castle seem like a shadow even during the light of day. All except for the northernmost tower. For the north tower is made out of a single glistening ruby. It glows softly each dawn and dusk, but hangs like a bloody sword above the castle during the rest of the day. It is the first part of the castle anyone coming from the mountain pass can see, winking like a blood-shot eye in a shadowed face.

If that is not enough to inspire awe, the castle also seems to hang in the air, balanced on the edge where the mountains become sheer basalt cliffs. Again, the wizard’s might can readily be seen, for it seems that Malikon’s will alone keeps his fortress from collapsing into the dark waters below. The castle rests precariously at a crazy angle between the sea and the sky, with towers pointing not straight up, but to the west, as if they would follow the sun and throw themselves into the sea. Or perhaps, over the years, the castle has been trying to join with its reflection.

In the ocean at the base of the cliff there is a perfect reflection of the castle. Too perfect. It is not a real reflection, but a whole other castle. Most people who visit Malikon notice only that the reflection is where it should be. They never look closely to see that the reflection doesn’t ripple with the waves, but lies untouched beneath them. There are a few who look a little closer; but they dismiss what they see, thinking it is another enchantment to awe the unwary. Or perhaps a mirage to lead them to their death chasing fantasy castles in the icy sea. It would not be the first time Malikon had done such a thing. And so they do no look long, not wanting to be snared if it is a deadly illusion.

But if they were to look closer, they would see that this castle is not made to look like blood and shadows. It is a vision of bubbles and sunlight, dancing beneath the stormy waves. Instead of black basalt and crimson ruby, the walls are made of glistening pearl and creamy alabaster, golden helidor and pale aquamarine, sparkling emerald and watery beryl. During the afternoon, when it is fully illuminated with sunlight, it dazzles and delights, a magical marvel. But during the rest of the day and night, when it is hidden in the shadow of the cliffs, the castle blends into the colors of the ocean and looks like the pale reflection it is thought to be. None suspect the truth: that it is not so much a reflection as it is the original. Malikon’s castle on the cliffs is this castle’s shadowy reflection.

Yet every so often, about once or twice a century, someone realizes that buried in the rumors about the Castle on the Edge of Forever are grains of truth. That person takes the time to learn all the stories by heart, to extract from those tiny grains the greater truth. Then the person painstakingly finds their way across plains and hills and mountains, to the winding path that leads down the cliff. What they find next is never spoken of, for so far none of those dreamers have ever returned. But perhaps this time it would be different.

.:.:.:.:.:.

Celandine walked through the halls of the castle. With each step they shifted color. The emerald walls were now bright green, now dark, now bright again. Once, the motion of the sunlight streaming through the ocean had made her dizzy, but that had been a long time ago. The castle had long ceased to affect Celandine in that way. In fact, the castle no longer held any wonder for her at all. For while it was beautifully crafted, and exquisitely designed; it was also her prison. And after so many years as a prisoner, it held no marvels for Celandine anymore.

She had not always been a prisoner. She remembered a time when sunlight did not stream greenly through the water. When the sky was a different shade of blue. When merfolk did not stop by and chat through the windows. She could barely remember what a breeze felt like. And the only way she remembered clouds was by looking at them in one of the many magic mirrors that lay about the castle. There were times when she thought she would do anything to leave. And yet here she stayed. Locked in a castle of scintillating colors beneath the icy waves.

.:.:.:.:.:.

Here I'm trying to capture Patricia McKillip's gorgeous writing style. She writes what could be considered modern folklore, fantasy stories with roots in folklore but are entirely unique tales. Her descriptions are wonderful - and I have a feeling I'm not quite doing them justice with my pale imitation.

title: User:CarrieBerg/The Sorcerer's Daughter


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User:CarrieBerg/Number Games

Four, eight, five, six. More and more. Count down, count around, count adds ten, over and over again. Then count once for me and then twice more times nine, seven digits counted multiply. Recount thrice, minus, divide, until none are left. Nil, zero, zip.

.:.:.:.:.:.

I chose a rather unique limitation, or so I'd like to believe. Can you guess what it is? Yes, it has to do with numbers. Let me give you a hint. One above or below a number surround each number. Do you need another hint? 4, 5, 4, 3....

title: User:CarrieBerg/Number Games


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User:CarrieBerg/STANDS FOR

______ stands for...

Something special. Something spectacular. Foreign, strange, necessarily so. Otherwise the day, dull and dreary and far too ordinary, obtains no splendiferousness. No sorrow or anguish allowed around now, for today's too special for that. Dauntless and daring, shiny, tiny and delicate. And amidst the familiar, seemingly so out of the norm, so ostentatious, that no one tries to deny the fascination they feel. Staring openly, they dance around, daring others to deny the thing resting there. That darling organism that so transforms the day.

(_____ stands for: a single flower.)

.:.:.:.:.:.

The title says it all - it sets the limits of this piece, one letter at a time.

title: User:CarrieBerg/STANDS FOR


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User:Yeoman/2009/W18/4

Thursday, 30 April 2009 - Debian laptop up and running

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Debian

This is a Wikiversity Learning by doing unit that encourages participants to Learn Linux. If you are new to Linux, find an older discarded machine and try it out. People throw away perfectly good computers all the time just because they are virus-infected or no longer run the latest and greatest non-free OS.

Hardware
Debian runs on all sorts of computers from tiny laptops to giant servers. It's easy to install on both modern machines and presumably "obsolete" older ones. "Each new release of Debian generally supports a larger number of computer architectures."

See the documentation for the stable release

Installation
Ryan Murray of Debian operates the build daemon for the i386 release at "Lenny". Using the debian-50x-i386-netinst.iso works fine on something like a Dell® laptop or an older Compaq® desktop PC. The Network install method is good because it allows you to get up and running quickly, provided you have a reasonably fast connection and a dedicated hard disk. Accepting the defaults works for the typical user. The net-install CD detects hardware quickly and thoroughly and presents you with reasonable options.

NOTE: Installing Linux begins by creating a bootable disk unless you have one or can purchase one.

Choices
It's important to understand the community-driven nature of Debian. Debian's default answer for participating in their "Configuring popularity contest" is  but it's a good idea to change that default to. (opinion, of course)

Basic Commands
Knowing your Debian version : lsb_release -a No LSB modules are available. Distributor ID:	Debian Description:	Debian GNU/Linux 9.4 (stretch) Release:	9.4 Codename:	stretch

title: Debian


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User:Egm6322.s09.Three.nav/Project

 See my comments below. Egm6322.s09 14:29, 1 May 2009 (UTC)

About me
I am a doctoral candidate enrolled in the Mechanical & Aerospace Engineering Department at the University of Florida, Gainesville. I work in the Computational Plasma Dynamics Laboratory & Testing Facility under the supervision of Dr.Subrata Roy.

Plasma Physics

On earth, we know and have learnt to live with only three states of matter. However Sir William Crookes identified Plasma, the fourth state in 1879. It is by far the most common form of matter. It is present in the stars and in the tenuous space between them. It is present, though invisible, in almost 99.9% of the visible universe. Ideally plasma consists of neutrals, electrons and ions. However commercial plasma is ionized gas. In order to separate electrons and ions from atoms, energy is needed and this energy can be in any form, thermal, electric etc. Most commonly, a gas is passed through a pair of electrodes and an electric field. This electric field provides the energy needed to ionize the gas into atoms and ions. Plasma has helped create a better understanding of the world we live in. It provides us the prospect of abundant energy. It provides alternatives to surface cleaning, waste removal, in the medical industry and many more.

My research focuses on Dielectric Barrier Discharges(DBDs) & their applications. Commerically produced plasma is classified into two types- Volume Plasma (wherein the plasma is produced in the discharge space between two electrodes) & Surface Plasma (wherein both electrodes are embedded on the same surface). DBDs are a kind of surface plasma. They are generated between two electrodes with a dielectric barrier in between them. The electrodes are separated by a gap of a few millimetres. Typically DBDs are operated at voltages of 1-100 kV and frequencies of 50Hz-1MHz. When electric field is sufficiently high to cause breakdown of the discharge gas, a large number of micro-discharges can be observed emanating from the electrodes.

Current Research Focus- Plasma Sterilization Plasma Sterilization is a process in which low-temperature plasma is utilized to kill micro-organisms. Our research focus is on using Dielectric Barrier Discharges for this purpose. Plasma is produced usually by passing a gas through an electric field and ionizing it. Dielectric Barrier Discharges are a special case, in which two electrodes are embedded into the top and bottom of a dielectric surface (an insulator). An electric field is produced between these two electrodes. When a gas is passed through this electric field, it is ionized, producing ions, electrons, neutrals and UV photons. All of these species collide with the micro-organism, reacting with its essential elements and forming harmless compounds. The micro-organism, when devoid of its essential elements becomes harmless and is thus inactivated. More information about preliminary experiments on this subject is given here.

Plasma Physics, for the most part,deals with a lot of non-linear partial differential equations. The governing equations defining the behaviour of electrons, ions and neutrals inside a plasma are often expressed in differential forms and resemble closely the governing equations for fluid mechanics. Some of the differential equations are 1st order while some others are 2nd order but inevitably problems of non-linearity of PDEs are experienced. For a good explanation of the basic principles of Plasma Physics, refer to this site :Fundamentals of Plasma Physics

Abstract
One of the oldest problems that has sparked a lot of research in plasma physics is the plasma-wall problem. The sheath is a region of electric field that exists between the plasma and the boundary into which it runs. The basic problem of plasma running into a wall needs to be understood in order to better visualize some of the other aspects of plasma physics. In this paper, we attempt to understand the processes happening in the sheath region. There exist a set of governing equations, akin to the governing equations of fluid dynamics that outline the physical processes running in the plasma. These equations are discretized using a simple, first order finite difference scheme and then solved using the Newton-Raphson Method. The boundary conditions applied however are those pertaining to the sheath region existing between plasma and a solid wall. The objective in conducting this analysis is to investigate the accuracy of the numerical scheme used. If the numerical scheme is accurate, the electron density at the wall should go to zero. Further a probability density function is constructed using electron density data generated from the solver code.

Introduction
To completely describe the state of plasma, we would need to write down all the particle locations and velocities and describe the electromagnetic field in the plasma. But considering the zillions of particles, this would be an impossible task! In order to make our life easier, we construct models which simplify the processes taking place in plasma.

Models are of two types:
 * Fluid Model- This describes plasmas in terms of smoothed quantities like density and averaged velocity around each position. A general description is the two-fluid picture, where ions and electrons are described separately. Fluid models usually model the particles using a Maxwell-Boltzmann Distribution. However they are usually unable to capture accurately finer structures such as beams or double layers.
 * Kinetic Model- While being more accurate than the fluid model, it is usually more computationally intensive. It describes the particle velocity distribution function at each point in the plasma and therefore does not need to assume a distribution function like the fluid model. The more commonly used Particle-In-Cell (PIC) technique includes kinetic information by following the trajectories of a large number of individual particles

Understanding plasma sheaths is perhaps one of the oldest problems in plasma physics. A plasma sheath is the localized electric field that separates plasma from a material boundary. It confines the more mobile species (electrons) in the plasma and accelerates the less mobile species (ions) out of the plasma and toward the walls.This means that in the typical case, where the electrons are more mobile than the positively charged ions, the electric field in the sheath points towards the boundary.This behaviour can be further influenced by external processes such as variations in the applied electric field or the imposition of an external magnetic field.

Modeling of the behaviour of species in the Sheath Region
Many numerical models have been developed to describe Sheaths. The simplest visualization of this process is the behaviour of electrons and ions in the sheath region. Consider a planar slab of collisionless plasma, modeled in 1D space, using a fluid approach and assuming a Boltzmannian electron distribution. The governing equations for such a model are given below:

Continuity Equation for Ion flux:
 * $$ \frac{\partial (n_{i}v_{i})}{\partial x}= zn_{e}$$ ---(1)

Continuity Equation for Electron flux:
 * $$ \frac{\partial (n_{e}v_{e})}{\partial x}= zn_{e}$$ ---(2)

Momentum of ions:
 * $$ \frac{\partial (Mn_{i}v_{i}^{2})}{\partial x}+en_{i}\frac{\partial(\psi)}{\partial{x}}= 0 $$ ---(3)

Momentum of electrons:
 * $$ \frac{\partial (mn_{e}v_{e}^{2}+kn_{e}T_{e})}{\partial x}-en_{e}\frac{\partial(\psi)}{\partial{x}}= 0$$(4)

Poisson Equation:
 * $$ \epsilon_{0}\frac{\partial^2 \psi}{\partial x^2}= -e(n_{i}-n_{e})$$ --(5)

where k is the Boltzmann Constant, m is the electron mass, M is the ion mass, Te is the electron temperature,ve is the electron velocity, viis the ion velocity, $$\psi$$ is the electric potential, z is the frequency of direct ionization, e is the electron charge, $$\epsilon_{0}$$ is the permittivity of free space, niis the ion density and ne is the electron density.

More on the Poisson Equation The Poisson's Equation given by $$\nabla^{2}\phi= E$$ gives way to the Laplace's Equation for vanishing 'E'. The Poisson's Equation is one of the cornerstones of Electrostatics. The derivation of Poisson's Equation from basic concepts of electrostatics is a fascinating exercise and is given below:

 * Gauss' Law of Electricity (in a differential control volume) $$\Rightarrow \nabla.D= \rho_{f}$$

where D= electric field displacement, $$\rho_{f}$$ describes the external charge density

 * For a linear, isotropic and homogenous medium, D= $$\epsilon E$$

where $$\epsilon$$ is the permittivity of medium and E is the electric field.

 * Substituting the above equation into Gauss' Law $$\Rightarrow \nabla.E= \frac {\rho_{f}}{\epsilon}$$

 * Faraday's Law of Induction(for a constant magnetic field of strength B)$$\Rightarrow \nabla \times E= -\frac{\partial B}{\partial t}$$

When the curl of a vector is zero, it can be expressed as the gradient of a scalar. Express
 * $$\Rightarrow E= -\nabla \phi$$, where $$\phi$$ is defined as electric potential(scalar)

$$\Rightarrow \nabla^{2}\phi= -\frac {\rho_{f}}{\epsilon}$$

Defining Charge $$\rho_{f}$$ as the product of the charge of a single electron multiplied by the number density of electrons inside the plasma, we get the form shown in (5).

More details on these equations and their non-dimensionalization can be obtained from the paper by Sternberg and Godyak. For our purpose, we take the normalized versions of Equations (1)-(5) used in the paper by

 * $$\frac{\partial{(y_{i}V_{i}})}{\partial{x}}= y_{e}= exp(-\phi)$$--(5)
 * $$ V_{i}\frac{\partial{V_{i}}}{\partial{x}}-E+V_{i}\frac {y_{e}}{y_{i}}=0$$---(6)
 * $$ \frac{\partial{E}}{\partial{x}}- \kappa^{2}(y_{i}-y_{e})= 0$$ --(7)
 * $$ E= \frac{\partial{\phi}}{\partial{x}}$$(8)

where $$y_{i}$$ is the normalized ion density, $$y_{e}$$ is the normalized electron density, $$V_{i}$$ is the normalized ion velocity, E is the electric field strength and $$\phi$$ is the electric potential.

Here the variables solved for will be $$y_{i}, V_{i}$$ and E or $$\phi$$ (E can be calculated from $$\phi$$ from (8)). Knowing $$y_{i}$$, $$y_{e}$$ can be calculated (from (5)).

 It is useful for the general readers to identify the unknowns to be solved for by numerical methods as described further below. Egm6322.s09 17:57, 1 May 2009 (UTC) Unknowns identified here Egm6322.s09.Three.nav 13:15, 4 May 2009 (UTC)

Boundary Conditions
It is assumed with sufficiently high accuracy that at the center of the plasma ni= ne. This is termed a quasi-neutral plasma.

Debye Length & Quasi-neutrality of Plasmas The characteristic length scale in a plasma is the electron Debye length, $$\lambda_{DE}$$. It is typically the distance scale over which significant charge-densities can exist.

However across a plasma of length L>>>$$\lambda_{DE}$$,

Eqn.(5) $$\Rightarrow \nabla^{2}\phi= -\frac{e(zn_{i}-n_{e})}{\epsilon}$$

We generally expect that $$\phi \leq T_{e}= \frac{e n_{e}\lambda_{DE}^{2}}{\epsilon}$$

$$\Rightarrow \frac {|zn_{i}-n_{e}|}{n_{e}} \leq \frac {\lambda_{DE}^{2}}{L^2}$$

For L>>>$$\lambda_{DE}$$, $$\frac {\lambda_{DE}^{2}}{L^2} \leq 1$$

$$\Rightarrow \frac {|zn_{i}-n_{e}|}{n_{e}}\leq 1$$

$$\Rightarrow |zn_{i}-n_{e}| \leq n_{e}$$

$$\Rightarrow zn_{i}= n_{e}$$

Which defines a quasi neutral plasma. This assumption helps simplify the solution of the macro-scopic equations listed from (1)-(5) greatly.

The boundary conditions at the plasma center are specified as follows:

$$ n_{i}(0)= n_{0}  = n_{e}(0) $$

$$ v_{i}(0)= 0= v_{e}(0)$$

$$\psi(0)= 0= \frac{\partial{\psi}}{\partial{x}}(0)$$

The normalized boundary conditions are then going to be as follows:

$$y_{e}(x=0)=0$$ - Defining the electron density at the start of the plasma

$$\phi(x=0)=0$$- Defining the electric potential at this location

$$y_{i}(x=0)=1$$- Defining the ion density at the start of the plasma

$$V_{i}(x=0)=0$$- Defining the ion-velocity at the start of the plasma.

Remember: 'Start of the plasma' implies the starting point, at which there is no plasma generated. Obviously, here the gas is still not fully ionized, wherein ideally, the ion density, electron density and ion velocity are zero.

Solution through Numerical Schemes
A finite difference scheme is applied to discretize the first derivatives in Equations(5)-(7). Equation(8) is just needed to calculate the electric potential $$\phi$$ Finite difference schemes are usually applied for complex, non-linear partial differential equations, wherein the solution space is divided into n number of grid points and the required value at each grid point is calculated. For the single derivative, there are typically three types of schemes: Forward difference, Backward difference and Central Difference. Each of these have a different order of accuracy and are applied depending on how accurate the numerical scheme applied has to be.

Forward Difference$$\Rightarrow \frac{\partial {A}}{\partial x}= \frac {A_{n+1}-A_{n}}{\delta x}$$

Backward Difference$$\Rightarrow \frac{\partial {A}}{\partial x}= \frac {A_{n}-A_{n-1}}{\delta x}$$

Central Difference $$\Rightarrow \frac{\partial {A}}{\partial x}= \frac {A_{n+1}-A_{n-1}}{2\delta x}$$

where An+1 is the value of A(the quantity being calculated) at the n+1th grid point and An-1 is the value of A at the n-1th grid point.

The Central Difference Scheme usually has an order of accuracy higher than the other two schemes but this does not necessarily mean that using this scheme might give us the most accurate solution. The nature of the PDE being solved should also be taken into account. This is where we define terms like 'stability' and 'consistency' of a finite difference scheme. For more information, the book by Anderson is a very informative book to read. The basics of fundamental numerical schemes as well as more complex ones are outlined along with illustrative applications to common problems in Fluid Mechanics like the Burger's Equation.

Newton Raphson Method

Consider a typical problem that gives N functional relations to be zeroed involving variables xi, i=1,2....N.

f(xi)= 0, i=1,2....N

Each of these functions can be expanded into a Taylor Series i.e $$ f(x_{i}+\delta x_{i})= f(x_{i})+ \delta x_{j}\frac {\partial{f_{i}}}{\partial{x_{j}}}+O(\delta x^2)$$

Assuming $$f \to 0$$, we get
 * $$ \Sigma \alpha_{ij} \delta x_{ij}= \beta_{ij} \ where \ \beta_{ij}= -f_{i} $$

$$ \alpha_{ij} $$ is known as the Jacobian. Here Equations 5-7 are the equations that are discretized and contain three unknowns. So $$\alpha$$ is going to be a 3x3 matrix and $$\beta$$ will be a 3x1 matrix. $$\delta x_{ij}$$ is then solved for iteratively in MATLAB.

 * Hence once Equations (5)-(7) are discretized using forward difference, they reduce to a set of non-linear equations in which the unknowns are yi, Vi and E of the form

 * Defining Vi= u, yi= y and ye= ne (for simplicity), Eqns.(5)-(7) when discretized reduce to the following equations

 * $$\frac {y_{n}(u_{n+1}-u_{n})+u_{n}(y_{n+1}-y_{n})}{\delta x}= ne = exp(-\phi)$$

 * $$\frac {u_{n}(u_{n+1}-u_{n})-E+\frac {u_{n}ne}{y_{n}}}{\delta x}= 0$$

 * $$\frac {\phi_{n+1}- 2\phi_{n}+\phi_{n-1}}{\delta x^2}= \kappa^2(y_{n}-ne)$$

where the subscript 'n+1' implies the (n+1)th grid point, 'n' implies the nth grid point and 'n-1' implies the (n-1)th grid point.

This set of non-linear equations is solved using the Newton-Raphson Method,described above.

 You can also linearize the PDEs in Eqs.(5)-(7); the linearized PDEs would look close to an elliptic PDE, particularly after you replace the linearized Eq.(6) into Eq.(7). Egm6322.s09 17:57, 1 May 2009 (UTC)

Hence we get a matrix of type Ax= B where A= $$\alpha_{ij}$$ and B= $$\beta_{ij}$$ as described above. Once A and B are calculated, x= A-1B. This process is repeated for several iterations in order to calculate the necessary plasma parameters. Once these are known, ye can be calculated using Equation(5).
 * Now, once the electron density distributions and ion-density distributions are calculated, how do we estimate what happens down the line, say after like 100 iterations? Does the electron density in the Sheath Region remain zero always, as it should? Does the ion-density increase or decrease? This check achieves two purposes: The accuracy of the numerical scheme is verified and the sheath behaviour after a long time is understood. This check is performed with the help of a probability density analysis. A MATLAB code was written to simulate the entire process described above. This code furthe takes the electron-density data generated, divides it into 100 blocks and constructs a probability density function. This is repeated for every iteration so that we know the nature of the pdf after about 100 iterations.

Results
 Remember to put a license to each image that you uploaded to prevent their removal due to a lack of a license. Egm6322.s09 17:57, 1 May 2009 (UTC)

Licenses inserted for all images Egm6322.s09.Three.nav 13:46, 4 May 2009 (UTC)

Fig(1) shows the published results from the paper by Roy et.al(2004) In this paper, equations (1-7) to (1-10) are solved using a second-order time accurate, implicit finite element method using a computational domain extending from x=0 to the wall. Fig. 1(a) is a plot of ion density versus position x. Fig. 1(b) is the plot of ion velocity as a function of position, x. Fig. 1(c) is the plot of electric potential as a function of x, while Fig. 1(d) represents the plot of electric field versus x. As expected, since the more mobile species is confined within the plasma and the less mobile species is accelerated out, it is seen accordingly that at xw~0.71, the ion velocity increases. Since electrons are being confined, one would expect electric field at wall to rise (more charge being accumulated). This is seen in Fig. 1(c) and consequently in Fig. 1(d).

In Fig.(2), the plasma parameter distributions obtained from the MATLAB code used are given. In fig. 2(a), as can be seen, ion density prediction matches the trend as predicted in Fig.1(a) i,e ion density decreases in the direction of the wall. This is expected. Fig. 2(b), which shows the plot of ion velocity versus x also seems to match the trend predicted in 1(b). It makes senses too because one would expect ion velocity to increases as sheath approaches. Fig.2(c) makes perfect sense because electron density decreases to zero as wall is approached. The value of x at which ne goes to zero is a little earlier than expected (at x=0.2). Also to be noted is the important fact that ion density at wall does not go to zero while electron density does go to zero, which simply confirms the fact the electrons stay behind in the plasma while ions accelerate towards the wall. Fig. 2(d) is a plot of the potential versus x, which also matches the plot shown in fig.1(c). This tells us that the code is predicting the behaviour of the species in the sheath correctly to some extent. The linear trend present in fig. 2(a), 2(b), 2(d) is of slight concern as we are looking for more of an exponential trend. This could be because of an error in the code or due to slight errors introduced by the numerical scheme used. Note that the results in fig.(1) are the result of a finite element code while the results in fig. (2) are those computed from a finite difference scheme. Finite element schemes are much more robust and stable schemes and hence what needs to be done here is take a higher order finite difference scheme and apply it to the problem at hand.

Fig. 3 shows the pdf analysis for the ion density. Ion density data is taken from the code, divided into 100 blocks and a pdf is constructed. Fig.3(a) shows this pdf as a bar diagram. Further the convergence of this pdf is investigated by calculating moments as high as E[[x7]. Fig. 4 shows the results of a similar analyis for the electron density data. There are a couple of important points to note in these two figures. Fig.3(a) shows that the ion density does not go to zero at the wall. Fig. 4(a) shows that there is a 4% probability of electron density going to zero at the wall. Fig. 3(b) and Fig. 4(b) confirm convergence of the pdf for both electron density and ion density data. Note also that the electron density is steady after about 40 iterations while the ion density is steady after about 20 iterations

Conclusion & Future Work
The governing equations for the plasma wall problem were solved using a finite difference scheme to discretize them. The electron and ion density are calculated using a MATLAB code. The computed distributions of ion and electron density exhibit the trend shown in the published results as shown in Fig.(1). The linear trend may be due to accuracy errors in solver code. However the ion and electron density plots are very similar to ion and electron behaviour in sheath region. Also the PDF analysis shown in Fig. (3) and (4) for electron and ion density data respectively shows that there is convergence after 100 iterations, proving that the results predicted by the code are accurate. Also we see that there is a 4% probability of the electron density going to zero. This value might further reduce, provided the accuracy errors in the code are resolved.

Future work involves using more complex, higher order numerical schemes to discretize the governing equations. Also a simplistic case of the plasma-wall phenomenon was examined here. More complex phenomenon can be introduced into this simulation in order to further understand this phenomenon.

Since understanding Plasma Physics requires a thorough understanding of PDES, their types, solution methods used in solving them, this project was a good review of the PDE principles that were discussed in class. Principles, discussed in class, such as the Laplace's Equation and the significance of the 'gradient' operator, 'curl' and so on were good tools in providing a better understanding of the theory behind the vast subject of Plasma Physics.

Signature
Egm6322.s09.Three.nav 20:40, 30 April 2009 (UTC)

title: User:Egm6322.s09.Three.nav/Project


text: User:Yukari~enwikiversity

title: User:Yukari~enwikiversity


text:
User:Egm6322.s09.bit.la/hw1

= General non-Linear PDE's =

N independent variables

 * {xi}={x1,...,xn}

if expand the index "i"=1,...,n

Example

1D case with time variable (1D time independent problem)

(x,t)=(x1,x2)

x1=x  spatial variable

x2=t  temporal variable

3D time independent problem

(x,y,z,t)=(x1,...,x4)

Unknown Functions: u1,u2,...

e.g. Navier-Stokes in 3D

ui({xj})  i=1,2,

ui={u1,u2,u3}  j=1,2,3,4 ui={u1,u2,u3}$$\rightarrow$$ velocity field (x,y,z)

Restrict to 1 unknown function u
n independent variable {xi}  i=1,...,n

mth partial derivatie: $$\frac{\partial u^m}{\partial x_{i1} ... \partial x_{im}}$$

{i1,...,im} subset of m index among n possible index

i1,...,im=1,...,n

=Nonlinear PDE's=

n independent variables {xi} 1 Unknown function u
$$F=\left(\left\{xi \right\},\left\{\frac{\partial u}{\partial x_{i}} \right\},\left\{\frac{\partial u^2}{\partial x_{i} \partial x_{j}} \right\},... \right)=0$$

$$\left\{xi \right\}$$ - n arguments

$$\left\{\frac{\partial u}{\partial x_{i}} \right\}$$ - n arguments (components of grad. u)

$$\left\{\frac{\partial u^2}{\partial x_{i} \partial x_{j}} \right\}$$ - $$\frac{n\left(n+1 \right)}{2}$$ arguments (assuming $$\left\{\frac{\partial u^2}{\partial i \partial j} \right\}=\left\{\frac{\partial u^2}{\partial i \partial j} \right\}$$

i=1,...,n

Hij:= $$\frac{\partial u^2}{\partial x_{i} \partial x_{j}}$$

Hnxn:=[Hij]nxn and HT=H

$$\begin{vmatrix} U_{11}  U_{12} U_{13}\\ U_{21} U_{22} U_{23}\\ U_{31} U_{32} U_{33} \end{vmatrix}$$

if you just the elements above the main diagonal - $$\frac{n^{2}-n}{2}$$

if you just the elements above the main diagonal (including the diagonal) - $$\frac{n\left(n+1 \right)}{2}$$

2 Independent variables, 1 unknown function
$$F((x,y),u(u_{x},u_{y}),(u_{xx},u_{xy},u_{yy}),...)=0$$

Example

2uxx+3uyy=ax2+bx

uxx+uyy=0  Laplace Equation

div(grad u)

$$\triangledown. \triangledown u = \triangledown ^{2} u$$

5(uxx2)+7(uyy)3=0

Linearity with respect to u (unknown function)
"We must welcome the future, remembering that soon it will be the past; and we must respect the past, remembering that it was once all that was humanly possible" George Santayana

$$\mathcal{L}$$: an operator $$\mathcal{L}$$ (u) is linear with respect to u if

$$\mathcal{L}$$($$\alpha $$u + $$\beta$$v)= $$\alpha $$ $$\mathcal{L}$$(u)+$$\beta $$ $$\mathcal{L}$$(v)

u:$$\Omega$$ $$\rightarrow $$ $$\mathbb{R}$$

(x,y) $$\rightarrow$$ u(x,y)

$$\mathbb{R}$$=set of real numbers

(x,y) $$\epsilon$$ $$\Omega$$

$$\Omega$$ is domain of u, $$\mathbb{R}$$ is range of u,

$$u\left(\Omega \right)$$ is image of $$\Omega$$ under mapping $$u$$,

$$v: \Omega \rightarrow \mathbb{R}$$ just anather function like $$u$$,

$$\alpha ,\beta \epsilon \mathbb{R}$$ (2 real numbers)

Example:

Egm6322.s09.bit.sahin 21:02, 24 April 2009 (UTC)

$$2u_{xx}+3u_{yy}-7x^{2}-x=0$$

$$F\left(x,y,u,u_{x},u_{y}... \right)=0$$

Therefore,

$$\mathcal{L}\left(\cdot \right)=2{\frac{\partial ^{2}\left(\cdot  \right)}{\partial x^2}}{}+3{\frac{\partial ^{2}\left(\cdot  \right)}{\partial y^2}}{}-7x^{2}-x$$ and

$$\mathcal{L}\left(u \right)=2u_{xx}+3u_{yy}-7x^{2}-x$$

$$\mathcal{L}\left(v \right)=2v_{xx}+3v_{yy}-7x^{2}-x$$

let, $$f\left(x \right)=-7x^{2}-x$$

$$\mathcal{L}\left(\alpha u+\beta v \right)=2\left(\alpha u+\beta v \right)_{xx}+3\left(\alpha u+\beta v \right)_{yy}+f\left(x \right)=\alpha \left[2u_{xx}+3u_{yy} \right]+\beta \left[2v_{xx}+3v_{yy} \right]+f\left(x \right)$$

and

$$\alpha \mathcal{L}\left(u \right)+\beta \mathcal{L}\left(v \right)=\alpha \left[2u_{xx}+3u_{yy}+f\left(x \right) \right]+\beta \left[2u_{xx}+3u_{yy}+f\left(x \right) \right]$$

so,

$$\mathcal{L}\left(\alpha u+\beta v \right)\neq\alpha \mathcal{L}\left(u \right)+\beta \mathcal{L}\left(v \right)$$

It is clear that the partial differential equation is non-linear (Figure 3).

If we define,

$$\mathcal{L}\left(\cdot \right)=2\left(\cdot  \right)_{xx}+3\left(\cdot  \right)_{yy}$$

then cleary

$$\mathcal{L}\left(\alpha u+\beta v \right)=\alpha \mathcal{L}\left(u \right)+\beta \mathcal{L}\left(v \right)$$

this partial differential equation is linear (Figure 3).

= Order of PDE =

Egm6322.s09.bit.sahin 21:03, 24 April 2009 (UTC)

The order of a differential equation is the order of the highest derivative that appears in the equation. Followings are both first order PDEs. First one is a linear first order PDE, whereas second one is non-linear.

$$ 5u_{x}-7u_{y}=0$$

$$6u_{x}^{3}+2u_{y}^{2}+u^{1/2}+x^{2}+siny=0$$

An example for the second order PDE is,

$$div\left(grad\ u \right)+f\left(x,y \right)=0$$

This equation can also be written in cartesian coordinates as:

$$u_{xx}+u_{yy}+f\left(x,y \right)=0$$

Another example for 2nd order example is:

$$div\left(grad\ \kappa \cdot u \right)+f\left(x,y \right)=0$$

Here $$\kappa$$ is known as conductivity tensor (2nd order tensor).

$$\kappa =\kappa _{ij}e_{i}\otimes e_{j}$$

Here $$e_{i}$$ shows the unit vectors. Cartesian coordinate system and corresponding unit vectors can be seen in Figure 4.

$$\kappa \cdot gradu=\left(\kappa _{ij} e_{i}\otimes e_{j}\right)\cdot \left(\frac{\partial u}{\partial x_{k}}e_{k} \right)=\kappa _{ij}\frac{\partial u}{\partial x_{k}}\left(e_{i}\otimes e_{j} \right)e_{k}$$

$$=\kappa _{ij}\frac{\partial u}{\partial x_{k}}e_{i}\left(e_{j}\cdot e_{k} \right)$$

Since $$e_{j}\cdot e_{k}=\delta _{jk} $$

$$\kappa \cdot grad u=\kappa _{ij}\frac{\partial u}{\partial x_{j}}e_{i}=: v_{i}e_{i}$$

$$v:=\kappa \cdot gradu$$

$$divv=\frac{\partial }{\partial x_{i}}\kappa _{ij}\frac{\partial u}{\partial x_{j}}$$

The conductivity tensor is a second order tensor and obeys the usual transformation rule when the coordinate system rotated. Moreover it is usually assumed that $$\kappa$$ is symmetric.

$$\left[\kappa _{ij} \right]=\begin{bmatrix} \kappa _{11} & \kappa _{12}\\ \kappa _{21}& \kappa _{22} \end{bmatrix}$$, $$\kappa _{ij}=\kappa _{ji}$$ or $$\kappa ^{T}=\kappa $$

If $$\kappa$$ is constant, the equation is a linear 2nd order PDE.

If $$\kappa=\kappa(x,y)$$, the equation is a linear 2nd order PDE.

If $$\kappa=\kappa(x,y,u)$$, the equation is a quasi-linear 2nd order PDE.

=Home work Solutions=

Further Examples of Linear PDE
{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Examples of Linear and Non Linear PDE

 * A simple example of a linear PDE is the 2-D Laplace Equation

$$L\left( \phi\right)=\frac{\partial^2\left( \phi\right) }{\partial^2 x}+\frac{\partial^2\left( \phi\right) }{\partial^2 y}$$

Let,

$$L\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Therefore, in the present problem, we have,

$$L\left( \right)=\frac{\partial^2\left( \right) }{\partial^2 x}+\frac{\partial^2\left( \right) }{\partial^2 y}$$

$$\Rightarrow L\left(\alpha \phi \right)=\frac{\partial^2\left(\alpha \phi \right) }{\partial^2 x}+\frac{\partial^2\left(\alpha \phi \right) }{\partial^2 y}$$

Where, $$\phi=\phi\left(x,y\right)$$

$$\Rightarrow L\left(\alpha \phi \right)=\alpha L\left(\phi\right)$$

$$\Rightarrow Linear$$

This also is an example to illustrate the homogeneous property of the linear PDE

 * Another example for linear PDE

$$\frac{\partial\left(\ u\right)}{\partial t}+\frac{\partial\left(\ u\right)}{\partial x}=0$$

Let,

$$L\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Therefore, in the present problem, we have,

$$L\left( \right)=\frac{\partial\left( \right) }{\partial t}+\frac{\partial\left( \right) }{\partial x}$$

$$L\left(\alpha u+\beta v \right)=\frac{\partial\left(\alpha u+\beta v \right) }{\partial t}+\frac{\partial\left(\alpha u+\beta v \right) }{\partial x}$$

$$\Rightarrow L\left( \alpha u+\beta v \right)=\alpha\frac{\partial\left(\ u\right) }{\partial t}+\beta\frac{\partial\left(\ v\right) }{\partial t}+\alpha\frac{\partial\left(\ u\right) }{\partial x}+\beta\frac{\partial\left(\ v\right) }{\partial t}$$

$$\Rightarrow L\left(\alpha u+\beta v \right)=\alpha\left\{\frac{\partial\left(\ u \right) }{\partial t}+\frac{\partial\left(\ u \right) }{\partial x} \right\}+\beta\left\{\frac{\partial\left(\ v\right) }{\partial t}+\frac{\partial\left(\ v\right) }{\partial x} \right\}$$

$$\Rightarrow L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

This also is an example to illustrate the additivity property of the linear PDE

Further Examples of non linear PDE

 * The inviscid Burgers equation

$$\frac{\partial\left(\ u\right)}{\partial t}+u\frac{\partial\left(\ u\right)}{\partial x}=0$$

Let,

$$L\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Therefore, in the present problem, we have,

$$L\left( \right)=\frac{\partial\left( \right) }{\partial t}+\left(\right)\frac{\partial\left( \right) }{\partial x}$$

$$L\left(\alpha u+\beta v \right)=\frac{\partial\left(\alpha u+\beta v \right) }{\partial t}+\left(\alpha u+\beta v \right)\frac{\partial\left(\alpha u+\beta v \right) }{\partial x}$$

$$\Rightarrow L\left( \alpha u+\beta v \right)=\alpha\frac{\partial\left(\ u\right) }{\partial t}+\beta\frac{\partial\left(\ v\right) }{\partial t}+\alpha^2 u\frac{\partial\left(\ u\right) }{\partial x}+\beta^2 v\frac{\partial\left(\ v\right) }{\partial x}+\alpha\beta\left\{u\frac{\partial\left(\ v\right) }{\partial x}+ v\frac{\partial\left(\ u\right) }{\partial x}\right\}$$

$$\Rightarrow L\left(\alpha u+\beta v\right)\neq\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

 * Another example for non linear PDE

$$L\left( u\right)=\frac{\partial^2\left( u\right) }{\partial^2 t}+\frac{\partial^2\left( u\right) }{\partial^2 x}-u^2$$

Let,

$$L\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Therefore, in the present problem, we have,

$$L\left( \right)=\frac{\partial^2\left( \right) }{\partial^2 t}+\frac{\partial^2\left( \right) }{\partial^2 x}-\left(\right)^2$$

From the definition of operator $$L\left(\right)$$,

$$L\left(\alpha u+\beta v \right)=\frac{\partial^2\left(\alpha u+\beta v \right) }{\partial^2 t}+\frac{\partial^2\left(\alpha u+\beta v \right) }{\partial^2 x}-\left(\alpha u+\beta v \right)^2$$

Clearly,

$$\Rightarrow L\left(\alpha u+\beta v\right)\neq\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Egm6322.s09.bit.gk 20:27, 24 April 2009 (UTC)
 * }

Problem 5
128.227.51.230 22:29, 26 January 2009 (UTC) Egm6322.s09.bit.sahin 22:31, 26 January 2009 (UTC) Egm6322.s09.bit.la 22:34, 26 January 2009 (UTC) Egm6322.s09.bit.gk 22:37, 26 January 2009 (UTC)

title: User:Egm6322.s09.bit.la/hw1


text:
Wikiversity talk:Community projects

When was this updated?
Time for a change, something new! IDangerMouse 15:44, 27 January 2011 (UTC)

for being on the side bar.
for being on the side bar. this seems a bit sparse. could this be updated to be more useful, helping, and interesting? bless up. Michael Ten (discuss • contribs) 04:47, 4 January 2024 (UTC)

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User:Graeme E. Smith/Modularity Revisited

Modularity Revisited Three Phases of Modularity in the Cerebral Cortex Graeme E. Smith, GreySmith Institute of Advanced Studies  http://en.wikiversity.org/wiki/Portal:GreySmith_Institute   http://en.wikiversity.org/wiki/User:GreySmith_Institute  grysmith@telus.net In his book The Modularity of Mind Jerry Fodor introduced the idea of Modules in the Cerebral cortex that acted like databases specialized for a particular domain. His conception of these Modules was quite strict, and suggested a innate knowledge about the subject matter of the domain. Annette Karmiloff-Smith, in her book Beyond Modularity: A developmental perspective on Cognitive Science rebutted the innate nature of the Modules, clearly showing that some of the data in the modules was generated over the early life of the individual during developmental stages. Since then, there has been considerable discussion about some of the other limits that Fodor placed on the modules. In this Article I explore the nature of the Attention System, and posit a theory of three separate modular Phases that are all to be found within the Cerebral Cortex.

Modularity in the mind, is a seductive concept, that seems to favor the implementation of Functional Consciousness, by dividing the functionality of the mind, into byte sized pieces that can be analyzed and transferred to a computer once they are fully understood. The originator of this concept is Jerry A. Fodor, who wrote about it in The Modularity of Mind: an Essay on Faculty Psychology now in its 11th edition. He believed that such modules were started with innate knowledge, were relatively lightweight having shallow processing, were not selected by attention but competed for output, were defined by data-encapsulization and so on. Proponents of Massive Modularity chafe under the restrictions he placed on his modules since they consider even high-end processing to operate in the same manner   While I am not a proponent of Massive Modularity, I find that I must also chafe under the restriction that the modules are not selected by attention. In my attention model, I have room for three phases of modularity one of which is passively selected by attention, and one of which is actively selected for by attention. Fodor's Modularity only captures the first phase of modules in this theory, because of the limitations he places on the definition of modules   It is interesting that my reason for deciding that Fodor's rules of modularity don't fit, came partly because of Fodor's own work in a later book The Mind doesn't work That Way!: the scope and limits of Computational Psychology in which he discusses the limitations of Computational approaches to phenomenal psychology and the Computational Theory of Mind, as defined by Alan Turing. Personally I take more David Chalmers point of view in his article A Computational Foundation for the Study of Cognition in which he indicates that computation is not limited to Alan Turings concept of "Truth Preserving Functions" today, but includes many other forms of mathematical calculation because it is a general term. At the time that Fodor wrote his book on the limits to Computational Psychology, scientists were just beginning to deal with the need to convert from a Phenomenal form of storage such as neural networks might create, to an explicitly addressable form of storage, such as might be needed to isolate specific memories. It is because of the work of  David LaBerge, that I have determined the nature of the process needed to make such a conversion. However in doing so, I have learned of caveats on the nature of the element being converted, that cause me to hew closer to Qualia Theory than I would normally have considered because of my computer background. The problem as Jerry Fodor pointed out, in his 2000 book, is that neural networks are phenomenal in their output, and as such, you cannot separate out specific memories from a pure neural network model. in fact, if David Marr, was correct, the natural mode of the Cerebral Neocortex is a content addressable memory. Marrs comments, and my own thought experiments quickly indicated to me, that the output of such a memory had to be a redundant field of data, rather than a differentiable memory as such. The only way to narrow the field, it seemed is to reduce the number of outputs by suppression at the pyramidal Neuron level, or filter the outputs in some manner so that only outputs with a specific tag characteristic are allowed. Because neural networks are non-local in their storage, prediction of the location of a memory within the memory matrix is problematic. However Jerry Fodor in his 2001 book, suggested that somehow it was thought that the column architecture of the brain might have something to do with locating memories. In fact the theory of Neural Groups, is that groups of a hundred or so neurons get together in a cluster to promote a single neuron as the output for the whole group. The Center Surround Theory suggests that in fact the 6th layer pyramidal neuron, acts to localize these neural groups by suppressing neurons that are not as strongly activated, and promoting the centroid of the set as being the strongest neuron in the neural group. Evidence shows that the neuron being promoted to fire, often changes over time, suggesting that individual neurons might be overcome with fatigue, but the neural group can continue to output a value.((indent|8}}In Neural Darwinism: The Theory of Neural Group Selection Dr. Edelman suggests that neural groups are interchangeable. This means that at the neural group level, no common mapping can be found for even individuals of the same species. However in Self-Organizing Maps, Teuvo Kohonen discusses attempts to map motor signals to arm movements that have successfully used Neural Probe Arrays, and Artificial Neural Network SOM's to detect and map which neural groups activated which arm muscles, well enough for a monkey to guide a robotic arm in order to give himself a treat. The main requirement seems to be that the SOM has to learn the characteristic activation patterns for a particular individual monkey, before it can do the job. This interchangeability of neural groups means that there is no species specific mapping of neural Groups that could be used to access a specific memory, the organism has to learn its own unique pattern of Neural Groups in order to access memories. This in turn means that the conversion from content addressable memory to demand memory, is not simple or straight forward like it would be in a computer where we would have to implement the content addressable memory on top of an already place-code addressable memory, since each memory element in a computer has its own unique address. In essence this means that even if we wanted to, we cannot demand a memory from the Cerebral Cortex content addressable mode memory, until we have done some mapping. this is a tricky concept for some, because it seems so obvious that all we need to do is activate a specific column and we will get a specific memory. There is no doubt that such is true, because brain surgeons during open brain surgery, have shown that anytime a stimulus happens on a column, a specific memory is triggered. Columns invoke specific memories. The problem is in knowing which column to activate. This is where Dr. Laberges Apical Dendrites from his Attentional Control Brief and Prolonged article come into play. they are a way of activating a sub-unit of a Column, he calls a Mini-column. Mini-columns in turn cause the pre-activation of content addressable neurons in Layer 2/3 of the cerebral neocortex, and these in turn, activate their local neural groups. In other words, pre-activation by a Cortico-Thalamic Connection to the 5th layer pyramidal neurons that are the hub of the mini-column, causes the content addressable memory to output a field of data. We don't know what exactly that data will be, at this point, but we have a method of addressing the content addressable memory, at a level below the neural group level. Furthermore we know that this method should be capable of addressing specific memories, if only we knew which were which. Obviously to make a demand memory work, we need an index to the mini-columns so we know which columns to activate for which memories. If we could memorize the specific mini-column addresses for a particular field of data, we could simulate that field of data, by pre-activation of the selected mini-column addresses. Now I am going to stick my neck out, and say that the main problem in mapping a field of data to a particular set of mini-column addresses is to isolate the specific field of data. Once you have done that, you can simply compare each mini-column address with the field of data and if they are both active, you have found a mini-column address that is part of the specific field of data. The problem here is simply that neural networks do not lend themselves to isolation of specific memories so that the field of data you must work with, is whatever is left after you shut off everything else at the 2/3 layer pyramidal Neurons and filter for some tag element. I call this highly redundant field of data a quale, because it is phenomenally implicit memory and there is no fine control over what you get out of it. The problem I suggest is not how you end up with such a rich form of data, but how you filter out any significant portion of it, or make sense of all the redundant data that comes with it. This inverts the philosophical quandry of Qualia, which is a good thing. By inversion the problem becomes how do you make sense out of a Quale, which is a lot more approachable than asking what the color red feels like. Once you have mapped a clump of mini-column addresses to a specific Quale (Redundant Field of Data), The problem of how to make sense of it, still rears it's ugly head. If you rehearse the whole clump, you are going to get something a lot like the original quale. Now if Jerry Fodor's Module theory is at all correct, the output of the quale includes the outputs of all the modules that have been applied automatically to the data, unless somehow they were suppressed by suppression of the individual pyrimidal Neurons in layer 2/3. In David Marr's "A theory for the Cerebral Neocortex" he notes that a certain type of neuron in the 4th layer called a basket neuron arbors on the soma of the Pyramidal Neuron. At the time he wrote his theory, it was considered derigor that neurons were doing numerial calculation on the data. Since then we have found that there is too much chaos in the connections for such a prosaic function to work, so I can suggest that instead of being a division step, it is likely that the basket neurons are soma shunts capable of turning off the pyramidal neurons without affecting their inputs. If this is true we have a mechanism for shutting off portions of the quale, we just need a method of activating these neurons   In David LaBerges "Attention, Awareness and the Triangular Circuit, he draws a circuit of three areas of the brain that are interconnected. We have already heard how the Thalamus might be involved in addressing the mini-columns, We already have an idea of how the cerebral cortex connects the mini-column address to the content addressable memory, so the missing element seems to be the third area of the brain that is involved in LaBerges Triangle, the Prefrontal Cortex. The Prefrontal Cortex, connects to the cerebral neocortex and seems according to Dr. Laberge to have a suppression mechanism that is used to select with.    One of the problems that scientists have with Dr. Laberges theory is that he holds it out as the theory of attention, yet it fails to capture nearly half the functions that are thought to belong to attention. In other words it is not a complete model, but it is, a useful model, especially for understanding explicit memory. Explict memory, in my lexicon, is that memory that can be addressed using place-code addressing involving the Apical Dendrites of the 5th layer pyramidal neurons activated by cerebro-thalamic connections. One of the problems with Dr. LaBerges theory, is that it doesn't explain how the selection is acoomplished or what the effect on the datafield is. One of the things that Dr. LaBerges Triangular Theory of attention fails to deal with, is the binding of cross-modal sensory data. Evidence of the encapsulation of the Fodorian Modules, seems to suggest that sensory data, is processed in a stove-pipe fashion where each sensory modality is processed separately. Dr. Edelman in The Remembered Present: A biological Theory of Consciousness notes that the binding effect probably has to do with the creation of Functional Clusters, where clusters of neurons distributed across the cortex, resonate in synchrony at about 40 hertz. These Gamma Sychronized Oscillations (GSOs) seem to link across modal stove-pipes which would suggest a binding effect. In Wider than the Sky: The Phenomenal Gift of Consciousness Dr. Edelman notes a similar oscillation in the Thalamus, suggesting that perhaps either the thalamus is influencing the Cerebral Cortex GSO's or the Cerebral Cortex is influencing the Thalamus GSO's. Because the Thalamus is part of the Reticular Activating System, which has been implicated in distributing brain waves, it makes sense that the thalamus would be influencing the Cerebral Cortex. This of course does not explain how the Synchrony is coordinated since the same signals must pass through networks of different lengths to reach different areas of the cerebral cortex, nor does it explain what function the GSO's might have other than binding cross modal data. But if we look at our model two possible roles might be found for GSO's, they might, tag data, or filter it. A hypothesis that I have been looking at, is that the GSO's tag data so that different functional clusters can be separated, by filtering only for a particular GSO frequency. If so, it is probably the role of the Anterior Cingulate Cortex, in the Vento-lateral PreFrontal Cortex, that uses the GSO frequency to suppress all but a single GSO. This tagged and filtered functional cluster, then defines the Quale that I have already discussed. This would not work exactly like a band pass filter, in that any signal that synchronized with a particular frequency reference signal would be passed, rather than passing only those frequencies carried on a particular band. There are two problems with this hypothesis, first there is no frequency loss associated with the signal, like might be expected in a radio-like band-pass filter, and secondly, it has been shown that there is no separation in signals between the foreground and background at this level. If we expected to have the foreground separated from the background at this level, we would have quite rightly doubted the role of the Anterior Cingulate Cortex, in filtering the Quale. However, the nature of the processing before this level, seems more to suggest that what we are filtering for, is a consistent Point of View, rather than any detail in the output. Further, we don't necessarily expect a radio type band-pass filter, if only because the nature of the signal is more of a tag added to the 2/3 Layer activations, rather than a modulation of the signals from those Layers. In actual fact, we would expect in this model that the pre-activation of the mini-columns with a frequency, would simply add a resonation to the already existing data frequencies, which is more like the effect that has been observed. However this model also works for the matching role for determining a clump of mini-column addresses, If the mini-column connection resonates at the same speed as the Quale, it must be part of the Quale. And as I like to say, the functional cluster creates a quale that is information rich, but organizationally poor. The problem with this type of signal is that it has no meaning, since the information cannot be easily factored out, until we have an index. But how to build such an index, Aye there's the rub. If we rehearse our clump of mini-column addresses we get something similar to the original quale, but because the network has cnanged in the meantime, we don't get the exact same quale, we get a second quale that resembles the first, but includes new information. In fact this is exactly the nature of implicit memory, is that memories get added information, when they are rehearsed. In The Remembered Present Dr. Edleman called this re-categorical memory. To get some meaning out of the Quale, we need to reduce it to smaller clumps of mini-column addresses. One way we might do this, is if we edit the set of mini-column addresses before we rehearse the quale. At first we would have to do this at random, if only because there is no significance to the individual mini-columns at first. One way we could do this, is to suppress some of the mini-columns. Dr. LaBerge, has noted a connection between the PreFrontal Cortex, and the Thalamus, and other researchers have suggested that it flows through the striate cortex, and a special element in that cortex called the Nucleus Accumbens. Then it flows into a sort of inverter sheaf around the thalamus called the Nucleus Reticularis Thalmi. What this suggests is that the PreFrontal Cortex can influence which mini-column addresses are pre-activated by actively suppressing a subset of them. All we would need is some inference engine that could sense whether or not a particular sub-clump arrangement of mini-column addresses were more or less significant than the Original Quale, and we would be on the way to mapping significant areas of the Cerebral Cortex. Now where the original Fodorian Modules would be limited to the unselected qualar output, and what they can analyze from that, a secondary set of Modules could be monitoring the Qualar output of these sub-clump arrangements, and analyze them for even more information. Further, one of the things we would want to do was to map the significant sub-clump arrangements so that we can find them again in a similar quale. this seems to be a role that is made for the striate cortex, where the prefrontal cortex signals and the Thalamic Signals are both available. I have recently heard of connections from the subiculum, the hippocampus and the entorhinal cortex, and possibly the parahippocampus all also dumping into this area, and have hypothesised that their roles involve similar types of connections. Of course the science does not reach that far into this area of the brain yet, if only because the tools we have for studying it are so expensive. As we learn more about the brain, we will be able to confirm or deny many different hypothesis and of course this is just one. The last modular elements I would like to discuss, are those that are recruited by the Dorso-Lateral PFC. This area of the cortex, hss the ability to selectively activate areas of the cerebral cortex that are not usually active, An example of this is the Fusiform Facial Area, an area on the Fusiform Gyrus, that lights up when something that might be a face is detected. Another example is a motor neuron that is only recruited when you want to point your finger at something. These Modules are actively selected by the Attention system, and recruited to do a specific action or function with a specific piece of data. Like the Fodorian Modules these other two layers of modules I have described, are relatively shallow and encapsulated However both of these two arrays of modules are to some extent directed by the attention system, the second array, by selection of sub-clump arrangements of mini-column addresses, and the third array by specific selection via the Attention System. In fact the third type, because it is process oriented, is done relative to the data produced by the second type and the sub-clump arrangements. This forms a strange type of attention I call Complicit attention because it depends on another form of attention for selection of its data.((indent|8}} In fact, I suspect that Complicit Attention defines a value-function tuple that in aggregate defines a processing language of some sort. It is therefore the nature of this third type of module, that defines the content of high order processing. While I cannot speak to the functions that might be practical what would be needed in order to define such a value-function tuple was two generators that generated the clump address for first the data element and then the Function, to form the value-function tuple. While I do not know personally of the existence of these generators, I am assured that short term memory is dependent on centers both in the frontal lobes and in the hippocampal area. Either center can be removed or silenced chemically with only a short term loss of short term memory, while the brain relearns to use the other center to accomodate. However if both centers are lost or silenced, short term memory is also lost. While I can't reference the actual documents in which these experiments were written, I thank Dr. Clayton Dickson of the UofA in Edmonton Alberta for bringing them to my attention.


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User:Egm6322.s09.xyz/final project

 See my comments below. Egm6322.s09 14:29, 1 May 2009 (UTC)

=Mass Transport via Molecular Diffusion for Fuel Cell Application=

Introduction to Fuel Cell Technology

Fuel cells are a clean energy source that holds great promise for displacing conventional energy systems toward a cleaner, more efficient future. Similar to a battery, fuel cells produce DC current from an electrochemical reaction that takes place between hydrogen and oxygen via a polymer membrane electrolyte.

The basic structure of a PEM (polymer electrolyte membrane) fuel cell is given in Figure 1. It consists of a polymer electrolyte membrane sandwiched between two gas diffusion layers, known as the anode and the cathode. The electrochemical reaction takes place with the aid of two catalyst layers, located at the anode/electrolyte and cathode/electrolyte interfaces, respectively. The chemical reactions for a PEM fuel cell are given as follows :

Anode RXN: $$H_2 \Rightarrow 2H^+ + 2e^-$$

Cathode RXN: $$\frac{1}{2}O_2 + 2H^+ + 2e^- \Rightarrow 2H_20$$

Overall RXN: $$ H_2 + \frac{1}{2}O_2 \Rightarrow H_20$$

As shown in the above equations, fuel cells require both hydrogen and oxygen to enable the reactions at the catalyst layers. The product of the electrochemical reaction is simply water (and heat).

In a fuel cell system, the reactant (fuel & oxygen) flows for the fuel cell are closely monitored. Depending on the performance set point, information is needed about the required flow rates, the amount of storage needed to hold the reactants, as well as the amount of storage needed to hold the reaction product water.

Mass transport of reactants through the anode/cathode layers to the catalyst layers (reaction zones) is governed by molecular (mass) diffusion. Consideration of mass diffusion is presented here in the form of 1) a detailed derivation of the governing partial differential equation using a control volume approach, 2) a discussion about how mass diffusion is similar to the heat conduction, and, 3) an illustrative example using gas diffusion through the electrode layers of a PEM fuel cell.

It is hoped that reader will gain a deeper understanding of the transport processes within a fuel cell as well as an appreciation of the mathematics that govern this real-world physical process.

Derivation of Mass Diffusion Partial Differential Equation

The foundation of the molecular mass diffusion is that the flux, F, of a substance is related to the change in concentration, C, over the change is distance, x, via a constant of proportionality known as the diffusion coefficient, D. An expression of this is given as follows:

$$F = -D \frac{\partial C}{\partial x}$$

Note that the flux is given as the mass transfer of a substance per unit time per unit area.

Consider a control volume of an isotropic medium as shown in the Fig. 1. Isotropic materials have the same properties in all orientations. The control volume features a rectangular box that is parallel to the coordinate axis. The lengths for the box are given as:

$$AA' = BB' = DD' = CC' = 2dx$$

$$AD = BC = A'D' = B'C' = 2dy$$

$$AB = DC = A'B' = D'C' = 2dz$$

The point P = P(x,y,z) represents the center of the control volume.

If the ABCD and A'B'C'D' faces are perpendicular to the x-axis, then the flux of a substance entering the ABCD face is given as:

$$4dydz \left( F_x - \frac{\partial F_x}{\partial x} dx\right)$$

Respectively, the flux exiting the c.v. from the A'B'C'D' face is given as:

$$4dydz \left( F_x + \frac{\partial F_x}{\partial x} dx\right)$$

The rate of increase of the species between the ABCD and A'B'C'D' faces is given as:

$$-8dxdydz\frac{\partial F_x}{\partial x}$$

The other faces can be treated the same way, therefore:

$$-8dxdydz\frac{\partial F_y}{\partial y}$$, and $$-8dxdydz\frac{\partial F_z}{\partial z}$$

Lastly, the increase in the rate of the diffusing substance is given as:

$$8dxdydz\frac{\partial C}{\partial t}$$

The mass conservation principle that governs unsteady diffusion can be stated as :

(net rate of diffusion of species A into medium) = (rate of increase of species A in medium)

Following this principle, then the following expression results:

$$\frac{\partial C}{\partial t} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} = 0$$

Substituting the above expression for the flux, F, as a function of the concentration, yields:

$$\frac{\partial C}{\partial t} = D \left( \frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2} + \frac{\partial^2 C}{\partial z^2} \right)$$

In this equation, the diffusion coefficient is constant. For the 1D case, the equation reduces to:

$$\frac{\partial C}{\partial t} = D \left( \frac{\partial^2 C}{\partial x^2} \right)$$

This 1D formulation for mass transport via diffusion is the governing partial differential equation that will be explored deeper for the fuel cell application example that follows below.

source: J. Crank, The Mathematics of Diffusion, Oxford University Press, New York, 1956.

Heat Conduction Analogy ,

The 1D mass diffusion equation given above was first proposed by Adolf Eugene Fick, , in 1855. He based his derivation on the work of Jean Baptiste Fourier ,, who created the landmark studies of heat conduction in 1822. The mass diffusion equation is analogous to the heat conduction equation which is of the form:

$$\alpha \frac{\partial^2 T}{\partial x^2} = \frac{\partial T}{\partial t}$$

where T is the temperature, the constant of proportionality $$\alpha$$ is the thermal diffusivity.

An Illustrative Example: Gas Diffusion through electrode layer of a fuel cell

This section will attempt to utilize the mathematical theory presented above to model a real-world application of hydrogen gas diffusion through the anode layer of a PEM fuel cell.

Figure 3 shows a schematic diagram of the anode layer along with the respective coordinate system that will be used in the calculations. The anode layer is treated as a 1D slad of thickness L. The boundary conditions are such that there is a known concentration, C1, of hydrogen on the bulk flow side of the anode (x = 0). The concentration, C2, represents the hydrogen concentration at the anode/catalyst interface (x = L). The anode layer has an initial concentration equal to some unknown function given by f(x).

The following is a derivation of the hydrogen gas concentration, C(x,t), as a function of space and time given the boundary conditions listed here.

Problem Statement (summary)
$$\frac{\partial^2 C}{\partial x^2} = \frac{1}{D} \frac{\partial C}{\partial t}$$

(initial condition)

IC: $$C(x,t=0) = f(x)$$, at $$t = 0$$

(boundary conditions)

BC1: $$C(x = 0,t) = C_1$$

BC2: $$C(x = L,t) = C_2$$

 Clearly, the above PDE has an easy classification... Egm6322.s09 19:11, 1 May 2009 (UTC)

Splitting a Non-Homogenous Problem into a Steady-state and Homogeneous Problem
The problem as posed is non-homogeneous and must be split into two simpler problems in order to obtain an analytical solution. The first problem will be a steady-state problem that results in a simple 2nd order ODE with respect to the space variable, x. The second problem will be a homogeneous problem that results in a PDE with only the initial condition as a non-homogeneity.

Details of the two sub-problems are given below.

Steady-state Solution
$$\frac{\partial^2 C_ss}{\partial x^2} = 0$$

BC1: $$C_{ss}(x = 0) = C_1$$

BC2: $$C_{ss}(x = L) = C_2$$

Integrating once, yields:

$$\frac{\partial C_{ss}}{\partial x} = A$$ where A is an integration constant

A second integration yields yields the auxillary solution for the steady-state condition:

$$C_{ss}(x) = Ax + B$$

The constants of integration, A & B, can be determined by applying the boundary conditions as follows:

$$\blacktriangleright C_{ss}(x = 0) = C_1$$

 * $$C_{ss} = C_2 = A(0) + B$$

 * $$\therefore B = C_1$$

$$\blacktriangleright C_{ss}(x = L) = C_2$$

 * $$C_{ss} = C_2 = A(L) + C_1$$

 * $$\therefore A = \frac{C_2 - C_1}{L}$$

The steady-state solution is then

$$C_{ss}(x)= \left[ \frac{C_2 - C_1}{L} \right] x + C_1$$

Homogeneous Solution using Separation of Variables
$$\frac{\partial^2 C_h}{\partial x^2} = \frac{1}{D} \frac{\partial C_h}{\partial t}$$

IC: $$C_h(x, t = 0) = f(x) - C_s(x) = f^*(x)$$

BC1: $$C_h(x = 0, t >0) = 0$$

BC2: $$C_h(x = L, t >0) = 0$$

Separation of Variables
Using separation of variables method, the concentration can be expressed as the product of two functions that each have a single dependent variable:

$$C_h(x,t) = X(x)\Gamma(t)$$

Substituting this expression into the homogeneous PDE, yields:

$$\frac{X''}{X} = \frac{1}{D\Gamma} \frac{d\Gamma}{dt} = -\lambda^2$$

The greek lambda in the above equation represents a constant value. The right hand side and left hand side of the the expression are both functions of single dependent variables (x and t, respectively). This can only be valid if they equal a constant. The sign of the constant lambda term has been chosen as negative in order to force the boundary value problem in the x-direction (finite).

Auxiliary solutions for each variable (x & t) are determined as follows:

$$\blacktriangleright $$Space variable, X

 * $$X''+\lambda^2 X = 0$$
 * The solution of this equation is of the form:$$X(x) = G cos(\lambda x) + H sin(\lambda x)$$
 * Applying the boundary condition at x=0, yields:
 * $$G(1) + H (0) = 0$$, therefore $$G = 0$$

$$\therefore X(x) = H sin(\lambda x)$$
 * Applying the second boundary condition at x = L
 * $$0 = H sin(\lambda x)$$

$$\therefore \lambda_n = \frac{n\pi}{L}$$ where n = 0,1,2,...

$$\blacktriangleright $$Time variable, t
 * $$\frac{d\Gamma}{\Gamma} = -\lambda^2dt$$
 * $$ln\Gamma = -\lambda^2 D t + E^*$$ where E is a constant of integration

$$\therefore \Gamma(t) = Eexp(-\lambda^2 D t)$$

The homogeneous solution is given by merging both solutions for the X and t variables and summing over all possible solutions for $$\lambda_n$$. The homogenous solution is:

$$C_h(x,t) = \sum_{n=1}^\infty H sin(\lambda_n x) \left[ exp(-\lambda_n^2 D t)\right]$$

Principle of Orthogonality
The first step in finding the coefficient H is to apply the initial condition to the auxiliary homogeneous solution which states:

$$C_h(x, t=0) = f^*(x)$$

$$\therefore f^*(x) = \sum_{n=1}^\infty H sin(\lambda_n x) \cdot (1) = \sum_{n=1}^\infty H sin(\lambda_n x) $$

Observing the orthogonality of the sin function, both sides of this equation can be operated by the following expression:

$$\int_{0}^{L}sin(\lambda_n x)dx$$

Solving for the coefficient, H, yields:

$$H = \frac{\int_{0}^{L}f^*(x)sin(\lambda_n x)dx}{\int_{0}^{L}\left[sin(\lambda_n x) \right]^2 dx}$$

where$$f^*(x) = f(x) - C_{ss}(x)$$ ,or, $$f^*(x) = f(x) - \left[ \frac{C_2 - C_1}{L} \right] x - C_1$$

The values for $$H$$and $$f^*$$ complete the homogeneous solution given above.

Final Solution
The final solution is found by summing the steady-state and the homogeneous solutions

$$C(x,t) = C_{ss}(x) + C_h(x,t)$$ $$\therefore C(x,t)= \left[ \frac{C_2 - C_1}{L} \right] x + C_1 + \sum_{n=1}^\infty H sin(\lambda_n x) \left[ exp(-\lambda_n^2 D t)\right]$$

Substituting the respective values for $$H$$ is given above (see Principle of Orthogonality) as well as $$f^*$$, then grouping like terms yields :

$$C(x,t) = C_1 + (C_2 - C_1) \frac{x}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)sin(\lambda_n x)\int_{0}^{L} \left[ f(x) - C_1 - (C_2-C_1) \frac{x}{L}\right]$$

The final solution is found by integrating this expression, resulting in the following equation :

$$C(x,t) = C_1 + (C_2 - C_1) \frac{x}{L} + \frac{2}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)sin(\lambda_n x)\int_{0}^{L} \left[ f(x) sin(\lambda_n x) dx\right] + \frac{2}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)\frac{1}{\lambda_n}sin(\lambda_n x) \left[ T_2 cos(n\pi) - T_1\right]$$

Simulation using MathCAD

The final solution to the given problem was simulated using MathCAD. The following parameters were used:

Electrode diffusion layer thickness: L = 0.005;

Diffusion Coefficient: D = 2*10^(-9);

Bulk Flow Channel Concentration: C1 = 100;

Catalyst Layer Concentration: C2 = 0;

Initial Concentration: f = 0;

Figure 4 shows a data plot of the resulting hydrogen concentration profiles throughout the thickness of the anode layer for various time elapse conditions based on the boundary (and initial) conditions listed above. It illustrates just how the hydrogen gas migrates through the anode thickness over time. As shown in the the data plot, the hydrogen concentration converges to a linear profile at large time scales.

 Why was the initial-condition curve (purple dot-dash line) not smooth? There was a kink. Egm6322.s09 19:11, 1 May 2009 (UTC)

Conclusion
Fuel cells hold great promise for a clean energy future. Fuel cells consist of multiple layers including anode/cathode, electrolyte, and catalyst layers. Mass transport within the various layers of a fuel cell is governed by Fick's equation for mass diffusion which is derived using a control volume approach for an isotropic medium.

Fick's law is analogous to the heat diffusion equation given by Fourier. Both process of mass transport and heat conduction rely on random molecular interactions. Respectively, the mathematical formulation and derivation is similar between the two physical phenomena.

A real-world application of Fickian diffusion is the gas transport of hydrogen fuel from the bulk flow stream through the anode diffusion layer to the catalyst reaction cite at the anode/electrolyte interface. Though a relatively simple case was presented here, it should be noted that mass transport for fuel cells is a very complex topic. The level of model complexity increases with consideration of the multi-phase liquid/gas reactant streams, non-constant diffusion parameters, changing material properties from layer to layer, chemical reactions (chemical kinetics), and the interaction of reactants with the material layers.

It is a truly exciting topic!

Signature
--Egm6322.s09.xyz 19:09, 30 April 2009 (UTC)

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Using Wikipedia to Teach Writing

Why? 1. Because it's fun!

Presentation based on the article "Are We Ready to Use Wikipedia to Teach Writing? by Robert E. Cummings 30 April 2009 Dr. C. Harris ENG 481

In Outline of Rhetoric, James Herrick states that rhetoric is public-minded and wikis are public documents, which makes a wiki perfect writing space.

When doing creative writing it is a way to make students feel creative. (Frances) its fun to be in control of your own site!

Wikipedia allows students to see what their peers think while allowing students to freely express themselves whether they're right or wrong. (Brittany)

Wikipedia is a great tool for students to view the opinions of others and to express their own ideas. Wikis can also be a great way to encourage creative writing--Cassandra

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Talk:Using Wikipedia to Teach Writing

Open discussion that can be formatted into a future article

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User:Adjbjr

It is a Great joy to have people like WikiPeople providing philanthropic opportunities like this to enrich peoples lives.

Hi! I'm Antonio DeJesus Biag Jr. aka. Adjbjr. read as (ad jib jar)

I'm Highly Creative, Analytic, and Kingly(I like giving orders). After many years of working in different fields of work I found out that I can't be in one place doing one thing--I'm not a one-work man-- I like doing many things of which I myself have created in my Mind. all of these things, lingering in my head, are of VALUE to the community. I want to Express these thoughts to all humanity and beyond and somehow DIRECT people, People, PEOPLES to their 'Purposes' by creating MEDIA (Films, Movies and TVshows)

It is a Great joy to have people, like WikiPeople, providing philanthropic opportunities like this to enrich peoples lives.

Hi! I'm Antonio DeJesus Biag Jr. aka. Adjbjr. read as (ad jib jar)

I'm Highly Creative, Analytic, and Kingly(I like giving orders). After many years of working in different fields of work I found out that I can't be in one place doing one thing(given order)--I'm not a one-work man that waits for orders-- I like doing many things of which I myself have created in my Mind. These things, lingering in my head, are of VALUE to the community. I want to Express these thoughts to all humanity and beyond and somehow DIRECT people, People, PEOPLES to their 'Purposes' by creating MEDIA- Films, Movies and TVshows to MAKE our world the best there can be with the HELP of individual like you…

I am who am because i will be Who I Will Be due of You.

Antonio DeJesus Biag Jr.

title: User:Adjbjr