scholarly journals Assessment of a three-dimensional line-of-response probability density function system matrix for PET

2012 ◽  
Vol 57 (21) ◽  
pp. 6827-6848 ◽  
Author(s):  
Rutao Yao ◽  
Ranjith M Ramachandra ◽  
Neeraj Mahajan ◽  
Vinay Rathod ◽  
Noel Gunasekar ◽  
...  
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Three Dimensional ◽  
Response Probability ◽  
Function System ◽  
System Matrix

scholarly journals Derivation of Equations for a Size Distribution of Spherical Particles in Non-Transparent Materials

2021 ◽  
Vol 5 (4) ◽  
pp. 53-60
Author(s):  
Daniel Gurgul ◽  
Andriy Burbelko ◽  
Tomasz Wiktor
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Size Distribution ◽  
Density Function ◽  
Cross Section ◽  
Three Dimensional ◽  
Spherical Particles ◽  
Two Dimensional ◽  
Mathematical Formulas ◽  
Transparent Materials

This paper presents a new proposition on how to derive mathematical formulas that describe an unknown Probability Density Function (PDF3) of the spherical radii (r3) of particles randomly placed in non-transparent materials. We have presented two attempts here, both of which are based on data collected from a random planar cross-section passed through space containing three-dimensional nodules. The first attempt uses a Probability Density Function (PDF2) the form of which is experimentally obtained on the basis of a set containing two-dimensional radii (r2). These radii are produced by an intersection of the space by a random plane. In turn, the second solution also uses an experimentally obtained Probability Density Function (PDF1). But the form of PDF1 has been created on the basis of a set containing chord lengths collected from a cross-section.The most important finding presented in this paper is the conclusion that if the PDF1 has proportional scopes, the PDF3 must have a constant value in these scopes. This fact allows stating that there are no nodules in the sample space that have particular radii belonging to the proportional ranges the PDF1.

A probability density function method for turbulent mixing and combustion on three-dimensional unstructured deforming meshes

2000 ◽  
Vol 1 (2) ◽  
pp. 171-190 ◽  
Author(s):  
S Subramaniam ◽  
D. C. Haworth
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Turbulent Mixing ◽  
Three Dimensional ◽  
Quantitative Agreement ◽  
Particle Number Density ◽  
Time Dependent ◽  
Ic Engine ◽  
Density Control

A hybrid Lagrangian-Eulerian methodology is developed for numerical simulation of turbulent mixing and combustion in arbitrary three-dimensional time-dependent geometric configurations. The context is a probability density function (PDF) based approach intended for modelling in cylinder processes in reciprocating piston internal combustion (IC) engines. Issues addressed include mean estimation, particle tracking and particle number-density control on three-dimensional unstructured deforming meshes. The suitability of the methodology for statistically time-dependent three-dimensional turbulent flow with large density variations is demonstrated via simulations of turbulent freon vapour/air mixing on an unstructured deforming mesh representing an idealized IC engine [13]. Computed profiles of mean and r.m.s. freon mole fractions show good quantitative agreement with measurements. Moreover, inherent advantages of the Lagrangian-Eulerian PDF approach are demonstrated, compared to Eulerian finite volume solutions of an (approximately) equivalent set of moment equations. The new approach is, by design, compatible with existing computational fluid dynamics codes that are used for multidimensional modelling of in-cylinder thermal fluids processes. This work broadens the accessibility of PDF methods for practical turbulent combustion systems.

Path Integral Method for Nonlinear Systems Under Levy White Noise

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Alberto Di Matteo ◽  
Antonina Pirrotta
Keyword(s):  
Probability Density Function ◽  
Nonlinear Systems ◽  
Probability Density ◽  
Density Function ◽  
Path Integral ◽  
Response Probability ◽  
Stability Index ◽  
Integral Solution ◽  
Stable Random Variables ◽  
White Noises

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

Development and Application of a Transported Probability Density Function Method on Unstructured Three-Dimensional Grids for the Prediction of Nitric Oxides

2013 ◽  
Vol 136 (3) ◽  
Author(s):  
Andreas Fiolitakis ◽  
Peter Ess ◽  
Peter Gerlinger ◽  
Manfred Aigner
Keyword(s):  
Nitric Oxide ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Flame Structure ◽  
Hybrid Approach ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Nitric Oxides ◽  
German Aerospace

The present work explores the capability of the transported probability density function (PDF) method to predict nitric oxide (NO) formation in turbulent combustion. To this end a hybrid finite-volume/Lagrangian Monte Carlo method is implemented into the THETA code of the German Aerospace Center (DLR). In this hybrid approach the transported PDF method governs the evolution of the thermochemical variables, whereas the flow field evolution is computed with a Reynolds-averaged Navier–Stokes (RANS) method. The method is used to compute a turbulent hydrogen-air flame and a methane-air flame and computational results are compared to experimental data. In order to assess the advantages of the transported PDF method, the flame computations are repeated with the “laminar chemistry” approach as well as with an “assumed PDF” method, which are both computationally less expensive. The present study reveals that the transported PDF method provides the highest accuracy in predicting the overall flame structure and nitric oxide formation.

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