Method of False Rates for Radioactive Decay Chain Calculation
Issue:
Volume 5, Issue 3, September 2020
Pages:
27-31
Received:
18 December 2019
Accepted:
31 December 2019
Published:
14 September 2020
Abstract: Many engineering applications involving radioactive materials requires the time history of the radioactivity of nuclides within a decay chain. The system of differential equations with initial conditions or initial value problem describing radioactive decay of a parent and daughter nuclides was posited by Ernest Rutherford who was awarded a Noble Prize in 1910 for this work. Harry Bateman (1910) provided an analytic solution to the radioactive decay chain problem with the constraint that initial inventory of all daughter elements is zero. Required data for the decay chain calculation consists of the parent and all daughters’ radioactive half-life. The half-life for essentially all radionuclides has been established and is available from multiple sources. Solutions other than Bateman’s (non-zero initial conditions) can be computed analytically but become unwieldy for longer decay chains. For this reason, many applications use a numerical solution. However, a numerical solution can require constraints on the time step size. The proposed method of false rates provides a unique algorithm for the decay chain activities. The method treats the decay chain with arbitrary initial conditions and the calculation is analytic or exact. The method is unexpectedly simplistic. An example decay chain calculation compares the solutions by the method of false rates with a numerical method. The comparison is a verification of the method of false rates calculation. The method of false rates is easily coded as a stand-alone application or as a sub-module of a more general code such as a contaminant transport model.
Abstract: Many engineering applications involving radioactive materials requires the time history of the radioactivity of nuclides within a decay chain. The system of differential equations with initial conditions or initial value problem describing radioactive decay of a parent and daughter nuclides was posited by Ernest Rutherford who was awarded a Noble P...
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Axisymmetric Flow Through a Cylinder with Porous Medium
Muhammad Umar Farooq,
Abdul Rehman,
Naveed Sheikh,
Manzoor Iqbal
Issue:
Volume 5, Issue 3, September 2020
Pages:
32-35
Received:
2 October 2020
Accepted:
20 October 2020
Published:
27 October 2020
Abstract: The present work is a struggle to establish a mathematical appearance of the conduct of axisymmetric fluid flow in a moving cylinder confined in a porous medium. The fluid is assumed to be flowing through the annular region formed between two concentric smooth cylinders for the case when the outer cylinder is kept fixed while the inner cylinder is assumed to be moving with a constant velocity along the axial direction and is also assumed to be rotating with a constant angular velocity with reference to the centre line along the axial axis. Firstly, the conducting equations of motion are obtained in the form of a system of coupled non-linear partial differential equations with corresponding boundary conditions. The system is then transformed into a new set of coupled non-linear ordinary differential equations using a set of suitable similarity transformation. The problem is then solved using the fourth order numerical technique, the Runge-Kutta-Shooting method. The concluding results are derived for non- dimensional coupled differential equations. In the end the results are graphically presented and the behaviour of porosity parameter over the fluid flow is examined. The observed results indicated that with increasing values of the Reynolds’s numbers the non-dimensional linear and axial velocities also increases.
Abstract: The present work is a struggle to establish a mathematical appearance of the conduct of axisymmetric fluid flow in a moving cylinder confined in a porous medium. The fluid is assumed to be flowing through the annular region formed between two concentric smooth cylinders for the case when the outer cylinder is kept fixed while the inner cylinder is ...
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