Assessment and Applications of the Conversion of Chemical Energy to Mechanical Energy Using Model Rocket Engines

To provide the first-year engineering students with a hands-on experience in an engineering application using both chemistry and physics, this team project uses a set of chemical and physical energy concepts and MS Excel based analysis. The main objective of the project is to calculate how much of the potential maximum possible chemical energy is converted into propulsion when using model rocket engines with solid fuel. The secondary objective is to determine the effects of increasing conversion rates on the performance of a model rocket. The solid fuel or propellant used in common model rocket engines is black powder. Compared to composite and hybrid engines, engines with black powder are cheaper and easier to ignite. Affordability of this propellant has made it possible to test fire many engines of different sizes. In addition, solid model rocket engines provide a good analogy to solid rocket boosters used in some of today’s launch vehicles. Rockets are momentum engines, thus, it is unusual to consider them in terms of energy, but energy is felt by observers even in model rocket launches. Total impulse is the measure of momentum imparted to the vehicle and depends on several processes including the chemical energy of the propellant and the useful kinetic energy of the exhaust. The project centers around calculation of the total energy released by the combustion of the reactants in model rocket engines of various types (A through F). The propulsion energy is a small fraction of the total energy released during combustion where a significant part of the total is lost heat. Many students enjoyed this activity as they learned how to code several sets of chemical balance and physical energy equations using MS Excel. Each team wrote a detailed technical report that explains the overall project. They used field pictures and the graphs to illustrate various parts of the project. They also included an essay on alternative propulsion means to explore the outer Solar system and beyond. An anonymous learning survey was developed, implemented, and analyzed to assess the educational effect of this project. The survey results and anecdotal evidence show this was a good and a challenging learning experience that was also too demanding for some of the students

Statistical Mechanics of maximal independent sets

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation

What is the meaning of the graph energy after all?

For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$, the energy of the graph is defined by $E(G)=\sum_{j=1}^{n}|\lambda_{j}|$. Myriads of papers have been published in the mathematical and chemistry literature about properties of this graph invariant due to its connection with the energy of (bipartite) conjugated molecules. However, a structural interpretation of this concept in terms of the contributions of even and odd walks, and consequently on the contribution of subgraphs, is not yet known. Here, we find such interpretation and prove that the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the traces of even powers of the adjacency matrix. We then use such result to find bounds for the energy in terms of subgraphs contributing to it. The new bounds are studied for some specific simple graphs, such as cycles and fullerenes. We observe that including contributions from subgraphs of sizes not bigger than 6 improves some of the best known bounds for the energy, and more importantly gives insights about the contributions of specific subgraphs to the energy of these graphs

Self-consistent equation for an interacting Bose gas

We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential $V(r)$ such that 0<\int d\br V(r) = a<\infty. Expressing the partition function by the Feynman-Kac functional integral yields a classical-like polymer representation of the quantum gas. With Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation $\rho (\mu)=F(\mu-a\rho(\mu))$ between the density $\rho$ and the chemical potential $\mu$, valid in the range of convergence of Mayer series. The function $F$ is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling V_{\gamma}(\br)=\gamma^{3}V(\gamma r) we prove that in the mean-field limit $\gamma\to 0$ only tree diagrams contribute and function $F$ reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of Mayer series (vicinity of Bose-Einstein condensation) and study dominant corrections to mean field. At lowest order, the form of function $F$ is shown to depend on single polymer partition function for which we derive lower and upper bounds and on the resummation of ring diagrams which can be analytically performed.Comment: 33 pages, 6 figures, submitted to Phys.Rev.

Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs

The effectiveness of stochastic algorithms based on Monte Carlo dynamics in solving hard optimization problems is mostly unknown. Beyond the basic statement that at a dynamical phase transition the ergodicity breaks and a Monte Carlo dynamics cannot sample correctly the probability distribution in times linear in the system size, there are almost no predictions nor intuitions on the behavior of this class of stochastic dynamics. The situation is particularly intricate because, when using a Monte Carlo based algorithm as an optimization algorithm, one is usually interested in the out of equilibrium behavior which is very hard to analyse. Here we focus on the use of Parallel Tempering in the search for the largest independent set in a sparse random graph, showing that it can find solutions well beyond the dynamical threshold. Comparison with state-of-the-art message passing algorithms reveals that parallel tempering is definitely the algorithm performing best, although a theory explaining its behavior is still lacking.Comment: 14 pages, 12 figure