Phase Equilibria Tutorial

I created a tutorial which describes phase behavior of a system consisting of three substances: hydrogen, helium, and boron. The main objective of this tutorial is to aid in discovery of how much, if any, boron condenses out of the gaseous phase. The same methods used in this tutorial may also be used to explore the vaporization of boron. Please note that more work may need to be done on Appendix A, and possibly other parts. I value comments and suggestions.

The link is here:
http://www.mediafire.com/?sharekey=0961cf99bccaeeb0ab1eab3e9fa335ca4d776e759cad9414

Hopefully this will be helpful to the open source Polywell.

Book list

Here are a number of books that I own, and their supposed relation to the Polywell concept. Some of the relations are a stretch.

Tadmor, Z., Gogos, C. Principles of Polymer Processing, 2nd Ed. Wiley, NY. 2006

Relevance: Extruders are discussed in this tome, so this knowledge may be useful for transporting Boron.


Fogler, Scott. Elements of Chemical Reaction Engineering. 4th Ed. Pearson Education, 2006

Relevance: There is an interesting parallel between Chemical Kinetics and Nuclear Reaction Kinetics

Abrahamowitz, M. Stegun, I. Handbook of Mathematical Functions. Dover Publications 1965

Relevance: This book is from another era, but it has solutions to a variety of differential equations.

Israelvachvili, J. Intermolecular and Surface Forces. 2nd Ed. Academic Press. 1992

Relevance: This book discusses coulombic forces, but a lot of other stuff too.

Zubrick, James. The Organic Chem Lab Survival Manual. 6th Ed. Wiley, New York. 2004

Relevance: Basic laboratory skills are a must when generating any sort of feedstock

Wooton. Beckenback, Fleming. Modern Analytic Geometry. Houghton Mifflin, Boston, 1972

Relevance: This is a good place to start to think spatially. Questions related to magnetic field distributions may begin to be answered. Vectors included.

Krall, N., Trivelpiece, A. Principles of Plasma Physics. San Francisco Press. San Francisco, 1986

Relevance: This is an excellent reference for the study of Plasma Physics. It is a little advanced however, so a mathematical methods course may be required.

Chen, Francis. Introduction to Plasma Physics and Controlled Fusion, Vol: 1, 2nd ed. Plenum Press, New York, NY, 1984

Relevance: This is a commonly used textbook in the plasma physics schools

Bittencourt, J.A., Fundamentals of Plasma Physics, 3rd ed., Springer, New York, 2004


Relevance: This a good place to start learning plasma physics. It is not nearly as scary as the text by Krall.

Ida, Nathan. Engineering Electromagnetics, 2nd Ed., Springer, New York, 2004

Relevance: This is prerequisite for learning plasma physics. Many of the Maxwell Equations are discussed here. However, for a complete study, one may also want to consult electrodynamics. See the on-line book, Electromagnetic Field Theory by Bo Thide. (http://www.plasma.uu.se/CED/Book)

Callister, W. Materials Science and Engineering: An Introduction, 6th Ed. Wiley, New York, NY., 2003

Relevance: Every system atrophies.

Welty et. al, Fundamentals of Momentum, Heat, and Mass Transfer, 4th Ed. Wiley, New York, NY, 2001

Relevance: This is an elementary and practical understanding of transport phenomena.

Seader, J., Henley, E. Separation Process Principles, Wiley, New York, NY, 1998

Relevance: Have you ever wanted to have a chemistry lab on a large scale? This may be useful for purifying boron?

Riedel, S., Nilsson, J. Electric Circuits, 6th Ed. Prentice Hall, Upper Saddle River, NJ, 1999


Relevance: This is obvious. We will be dealing with electricity.

Tipler, P. Physics: For Scientists and Engineers (Vol 3: Modern Physics: Quantum Mechanics, Relativity, and The Structure of Matter), 4th Ed. W. H. Freeman., New York, NY, 1990

Relevance: Strange things occur at small scales.

Tipler, P. Physics: For Scientists and Engineers (Vol 2: Electricity and Magnatism), 4th Ed. W. H. Freeman., New York, NY, 1990

Relevance: This is a good step towards understanding the behaviour in the reactor.

Tipler, P. Physics: For Scientists and Engineers (Vol 1: Mechanics, Oscillations and Waves, Thermodynamics), 4th Ed. W. H. Freeman., New York, NY, 1990
Silberberg, M. Chemistry: The Molecular Nature of Matter and Change. 3rd Ed. Mc-Graw Hill., New York, NY, 2003

Relevance: A good source for looking up odd things that are forgotten.

Hornback, J. Organic Chemistry. Thomson Learning. Belmont, CA. 1998

Relevance: Boron Chemistry is similar to Carbon Chemistry.

Davis, Stephen R. C++ for Dummies. 3rd Ed. IDG Books, Foster City, CA 1998

Relevance: Virtual Polywell, Process Controls

Jones, B. Aitken, P. Sams Teach Yourself C in 21 Days, 6th Ed. Sams Publishing, Indianapolis, IN, 2003

Relevance: Virtual Polywell, Process Controls

Tester, J. Modell, M. Thermodynamics and Its Applications, 3rd Ed. Prentice Hall, Upper Saddle River, NJ, 1997

Relevance: Boron, Hydrogen, and Helium Phase Equilibria, Electrochemistry – D2O splitting

Prausnitz, J., Lichtenthaler, R., Azevedo, E. Molecular Thermodynamics and Fluid Phase Equilibria, 3rd Ed. Prentice Hall, Upper Saddle River, NJ, 1999

Relevance: Boron, Hydrogen, and Helium Phase Equilibria

Sandler, S. Chemical, Biochemical, and Engineering Thermodynamics, 4th Ed., Wiley, New York, NY, 2006

Relevance: Boron, Hydrogen, and Helium Phase Equilibria


Cengel, Y. Boles, M. Thermodynamics: An Engineering Approach, 4th Ed. Mc-Graw Hill, New York, NY, 2002

Relevance: Prometheus

Potter, M. Somerton, C. Thermodynamics for Engineers (Schaum’s Outlines), Mc-Graw Hill, New York, NY, 1993

Relevance: Prometheus

Winnick, J. Chemical Engineering Thermodynamics, Wiley, New York, NY, 1997

Relevance: Boron, Hydrogen, and Helium Phase Equilibria

Laidler, K., Meiser, J., Sanctuary, B. Physical Chemistry, 4th Ed., Hougton Mifflin, Boston, MA, 2003

Relevance: Boron, Hydrogen, and Helium Phase Equilibria

Chapra, S., Canale, R. Numerical Methods for Engineers, 4th Ed. Mc-Graw Hill, New York, NY, 2002

Relevance: Virtual Polywell

Gerald, Wheatley. Applied Numerical Analysis, 7th Ed. Pearson Education, New York, NY, 2004

Relevance: Virtual Polywell

Bird, R., Stewart, W., Lightfoot, E. Transport Phenomena, 2nd Ed, Wiley, New York, NY, 2002

Relevance: This is a slightly more advanced treatment of heat, momentum, and mass transfer.

Seborg, D., Edgar, T., Mellichamp, D. Process Dynamics and Control, 2nd Ed, Wiley, New York, NY, 2004

Relevance: for keeping the Polywell reactor running smoothly.

Moore, J., Weatherford, L. Decision Modeling: With Microsoft Excel, 6th Ed. Prentice Hall, Upper Saddle River, NJ, 2001

Relevance: what it the best way to get organized when we go big?

Peters, M., Timmerhaus, K., West, R. Plant Design and Economics for Chemical Engineers, 5th Ed. Mc-Graw Hill., New York, NY, 2003


Relevance: how much does it cost to pursue this project?

Perry, R., Green, D. Perry’s Chemical Engineer’s Handbook, 7th Ed. Mc-Graw Hill, New York, NY, 1997


Relevance: the Polywell reactor setup would be very much like a chemical plant.

Axelrod, R., Cooper, C. The St. Martin’s Guide to Writing, 6th Ed. Bedford/St. Martins, Boston, MA, 2001

Relevance: do you want people to believe you?

Markel, M. Technical Communication, 7th Ed., Bedford/St. Martins, Boston, MA, 2004

Relevance: do you want people to believe you?

Alred, G., Brushaw, C., Oliu, W. Handbook of Technical Writing, 7th Ed. Bedford/St. Martins, Boston, MA, 2003

Relevance: do you want people to believe you?

Mankiw, N. Principles of Macroeconomics, 3rd Ed. Thompson-South-Western, Mason, OH, 2003

Relevance: how is the market behaving? How does my project float around it? Peters and Timmerhaus could help compliment this book.

Stewart, J. Calculus, 4th Ed. Brooks/Cole, Pacific Grove, CA, 1999

Relevance: So I wanted to do vector Calculus but…

Edwards, C., Penney, D. Differential Equations and Boundary Value Problems: Computing and Modeling, 3rd Ed., Pearson Education, Upper Saddle River, NJ, 2004

Relevance: Them Equations look funny, are there any analytical solutions?

Thompson, S. Gardner, M. Calculus Made Easy, St. Martins Press, New York, NY 1998

Relevance: So I wanted to do vector Calculus but…

Saxon, J. Geometry, Trigonometry, Algebra III: An Incremental Development, Saxon Publishers, Norman, OK, 1985

Relevance: This is useful to have in the toolbox, as are all books on this subject

Jaeger, L. Cartesian Tensors in Engineering Science, Pergamon Press, London, England, 1966

Relevance: to help with the transport book, and electromagnetics book

Hibbeler, R, Engineering Mechanics: Statics, 10th Ed., Prentice Hall, Upper Saddle River, NJ,

Relevance: to protect people from dynamics.

Oxlade, C., Stockley, C., Wertheim, J. The Usborne Illustrated Dictionary of Physics: The facts you need to know at a glance, Usborne Publishing, London, England, 1986

Relevance: this is handy as a physics reference

Chandler, D. Introduction to Modern Statistical Mechanics, Oxford University Press, New York, NY 1987

Relevance: helpful with plasma physics, but mainly the equilibrium case

Huang, K. Statistical Mechanics, Wiley, New York, NY, 1965

Relevance: helpful with plasma physics

Parker, S. Eyewitness Science: Electricity, Dorling Kindersley, New York, NY, 1992

Relevance: a layman’s introduction to electricity. This is a kids’ book.

Rockwell, T. Reactor Shielding Design Manual, Von Nostrand, Princeton, NJ, 1956

Relevance: be safe

Epstein, L., Relativity Visualized, Insight Press, San Francisco, CA, 1997


Relevance: A great introduction to relativity

Hawking, S. A Brief History of Time, Bantam Books, New York, NY, 1996

Relevance: A curious science book.

Farlow, Stanley J., Partial Differential Equations for Scientists and Engineers, Mineola, NY, 1973

Relevance: A differential equation book that is arguably easier to understand than Edward’s and Penny, but it is lacking in computer applications

Frenkel, D., Smit, B., Understanding Molecular Simulation: From Algorithms to Applications, 2nd Ed., Academic Press, San Diego, CA, 2002

Relevance: It is curious that modeling molecular interactions or plasma interactions is a similar feat

Shilov, Georgi E., Elementary Real and Complex Analysis, Dover, Mineola, NY, 1973

Relevance: So the Calculus

Shilov, Georgi E., Linear Algebra, Dover, Mineola, NY, 1977

Relevance: for Plasma Physics

Strogatz, Steven H., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, and Engineering, Perseus Books, Cambridge, MA, 1994

Relevance: Questionable

Roman, Steven., Advanced Linear Algebra, 3rd Ed. Springer, NY, 2008


Relevance: for Plasma Physics

Roman, Steven.,Writing Excel Macros with VBA, 2nd Ed., O’Reilly, 2002

Relevance: for making Excel do things it would not normally do

Tenenbaum, M., Pollard, H., Ordinary Differential Equations, Dover, Mineola, NY, 1963

Relevance: Everything changes, hopefully on due to one factor

Kubo, R., Toda, M., Hashitsume, N., Statistical Physics II: Nonequilibrium Statistical Mechanics, 2nd Ed., Springer, Berlin, 1998

Relevance: so the plasma we are dealing with is not at equilibrium. Moui comprende?

Non-Maxwellian Distribution

Apparently I have fallen into the trap of applying the Maxwell-Boltzmann distribution to a system that doesn't follow it, due to various constraints that I don't fully understand. That said, the term "ergodic" does not apply to the Polywell concept since it is not in an equilibrium state. Chandler speaks of a quasi-ergodic case where a certain energy barrier prevents every energy state from being reached (p. 162). If every energy state was reached, we would be at equilibrium. According to Chandler, it is possible to apply Non-Boltzmann sampling to get around the problem of lack of ergodicity (p. 169).

- A novice just trying to get started

Source:
Chandler, David; Introduction to Modern Statistical Mechanics,
Oxford University Press, 1987

My Understanding of the Polywell Concept

Bussard’s Polywell concept is a theoretically new way of producing power, but that is the only thing alien about it. In a physical sense, it is merely a construct of both mass and energy streams. While the conservation of mass and the conservation of energy is not respected in the Polywell itself, it is respected everywhere else. However, this lack of respect should not be a hindrance since nuclear reactions are understood just as chemical reactions are. In an analogous sense, the Bussard Polywell, and its auxiliary equipment, is just a chemical plant. Raw materials come in, are purified, proceed to a reactor, and through a separation process produce both waste and product streams. The figure below shows a simplified model of the Polywell system.

The mass stream containing raw materials can consist either a mixture containing the reactive fuel, or a purified form of the reactive fuel. The choice would depend on economies of scale, and in-house technological advantages. In short, the choice would depend on economics. The reactive fuel itself would be boron, in the ideal case, since it would produce no lasting radioactive byproducts, as Dr. Bussard suggested.


Depending on the purity of the raw materials, the priming stage may range anywhere separation equipment to a unit to ensure that the reactor feed is at the correct temperature and pressure. In the latter case, no purification would be needed. In the former case, the separation equipment would need to be combined with equipment to ensure the correct temperature and pressure of the reactor feed.


Around the reactor itself, a vacuum pumping system would be necessary. However, it is not shown on the diagram. It is important though that the net mass efflux of air into and out of the reactor be zero. In the ideal case of course, this steady state condition would be maintained at very low concentrations of air to avoid interactions of the reaction within the reactor with the air surrounding the reactor.


In the post processing stage, products from the reactor would need to be dealt with. Doing this, would require a second set of separation equipment to extract unreacted portions of the reactor feed from reacted portions.


In addition, the waste would need to be conditioned such that it is safe to release to the environment or sell as an end product. In the case of boron as a reactor feed, the waste would be helium. This of course has a wide range of uses ranging from balloon animals to cryogenics. In other cases, when deuterium and tritium are used as a reactor feed neutrons will need to be dealt with.


It should be noted that understanding the behavior inside the Polywell is of critical importance to understand the flow rates exhibited in all other stages of the system. In the chemical plant sense, we would need to know the kinetics of the reaction. However, since we are dealing with a nuclear reaction, kinetics is instead called neutronics. That is we need to know how much of the reactor feed is converted during one pass through the reactor by understanding the nuetronics. To do this, we need to predict the number of collisions of the particles if we assume that fusion only occurs during collisions. Therefore, we need to know the dynamics of the system. Rather than running endless experiments, it would be useful to understand mathematically this behavior so to limit the number of experiments needed. Allen and Tildesley suggest the following algorithm for understanding the relation of computer simulations with experimental results. The figure below presents a modification of a flow chart given in chapter one of their book.

Here is the reason for the relations in the diagram:


When we consider plasma within the Polywell, we can analyze it in two ways. Either we can view it experimentally or we can make some sort of hypothesis about how it will behave and make use of the hypothesis. The hypothesis is normally used in one of two ways: either by simulating the position and movement of each particle within the plasma over time, or by predicting the number of positions that all particles can exist in along with the probability of existence of each position in a time where all positions are realized. Assuming these two methods are equivalent is saying that plasma is ergodic. In the first case, the simulation is a dynamics simulation. It is similar to molecular dynamics. One program that does molecular dynamics is LAMMPS. In the second case, theories are made to describe the behavior. The theory just described for this case is statistical mechanics. One important daughter of statistical mechanics is the Monte Carlo simulation. As the name implies, Monte Carlo is a random simulation that generates possibilities for positions of the particles. One should note that a Monte Carlo simulation is computationally easier than tracking the position of countless particles as in the dynamics simulation. However, due to its removal from the reality of the situation that dynamics simulations more easily entail, the predictions that are produced may not always be needed. What is needed for statistical mechanics to work, and therefore Monte Carlo to work, is the correct partition function describing the number of positions and the probabilities of these positions. Monte Carlo, and its parent statistical mechanics, must therefore be compared to the dynamics simulation. However, the dynamics simulation may not reflect reality either, so it must therefore be compared to experimental data. One must note that the data from the experiment may not reveal everything about the behavior of the system itself. In the case of movement of electrons within the plasma, one cannot measure both the position and the velocity accurately. In measuring one, the other is perturbed by the act of performing the measurement. This lack of ability of measuring both quantities accurately is known as the Heisenberg Uncertainty Principle. However, if a dynamics simulation can model the velocities and positions of electrons, and these dynamics can be related to a reading from experimental data that is dependent on these dynamics, then some idea of the positions and velocities can be deduced. One should note though that this idea may not be exact since many measurements such as temperature and pressure are based on averages over all particles. From statistics, one realizes that the average can be the same while the distribution of the velocities and positions changes. Statistical mechanics can help with this problem since it can be related to dynamics simulations. However, a statistical mechanics model may be related to a dynamics model and even match the experimental results but still not match the exact positions and velocities of the actual system. I cannot give the solution to the answer this problem. However, it is likely that it does not need to be known. I believe only the probability of collisions of particles needs to be known.

Sources:

M.P Allen and T.J Tildesley; Computer Simulation of Liquids,
Oxford University Press, 1989

Inspired by:

Chandler, David; Introduction to Modern Statistical Mechanics,
Oxford University Press, 1987