scholarly journals Modeling the Swelling of Hydrogels with Application to Storage of Stormwater

Water ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 34
Author(s):  
Hans O. Åkerstedt ◽  
T. Staffan Lundström ◽  
I. A. Sofia Larsson ◽  
Jiri Marsalek ◽  
Maria Viklander
Keyword(s):  
Diffusion Coefficient ◽  
Moving Boundary ◽  
Volume Growth ◽  
Water Volume ◽  
Storage Device ◽  
Maximum Volume ◽  
Stefan Problems ◽  
Porous Bodies ◽  
Swelling Rate ◽  
Geometrical Dimensions

The swelling effect in hydrogel bodies or sponge-like porous bodies (SPB) used in a specific stormwater storage concept of the down-flow type is considered. A macroscopic swelling model is proposed, in which water is assumed to penetrate into the hydrogel by diffusion described by diffusion equations together with a free-moving boundary separating the interface between the water and hydrogel. Such a type of problem belongs to the certain class of problems called Stefan-problems. The main objective of this contribution is to compare how the theoretical total amount of absorbed water is modified by the inclusion of swelling, when compared to the previously studied SPB devices analyzed only for the effect of diffusion. The results can be summarized in terms of the geometrical dimensions of the storage device and the magnitude of the diffusion coefficient D. The geometrical variables influence both the maximum possible absorbed volume and the time to reach that volume. The diffusion coefficient D only influences the rate of volume growth and the time to reach the maximum volume of stored water. The initial swelling of the hydrogel SPB grows with time (Dt) until the steady state is reached and the swelling rate approaches zero. In all the cases considered, the swelling in general increases the maximum possible absorbed water volume by an amount of 14%.

scholarly journals The Modeling of Nitrogen Mass Transport in CoCr Alloys

2016 ◽  
Vol 36 (331) ◽  
pp. 19-26
Author(s):  
Akvilė Petraitienė
Keyword(s):  
Diffusion Coefficient ◽  
Plasma Nitriding ◽  
Nitrogen Concentration ◽  
Nitrogen Adsorption ◽  
Nitrogen Diffusion ◽  
Depth Profiles ◽  
Swelling Rate ◽  
Cocr Alloy ◽  
Kinetics Of ◽  
Good Agreement

Abstract The kinetics of plasma nitriding of CoCr alloy below temperatures of nitrides formation and mechanisms of nitrogen penetration are analyzed by proposed kinetic modeling in this article. Proposed nitrogen diffusion model is based on the trapping – detrapping (TD) model and developed taking into account the effect of the concentration dependent diffusivity of nitrogen, nitrogen adsorption on the surface of alloy and surface swelling process. The model indicates the influence of chromium atoms to nitrogen atoms diffusivity. The model consists of time and depth dependent diffusion, which is described by a partial differential equation, and it is solved by using Crank – Nicolson finite difference method. By fitting of experimental nitrogen depth profiles, it is shown that nitrogen diffusion coefficient varies with nitrogen concentration according to Einstein-Smoluchowski relation. Nitrogen depth profiles in plasma nitrided medical grade CoCr alloy (ISO 5831 – 12) at T = 400 °C for 1, 4 and 20 hours calculated on the basis of this model are in good agreement with experimental nitrogen profiles. Furthermore, the swelling process is showed and analyzed, derived the dependency of swelling rate on nitriding duration – the swelling rate is inversely proportional to the square root of nitriding duration. The obtained diffusion coefficient value and the swelling process rates satisfy the experimental data form Ref. The derived model explains physical processes during plasma nitriding and allows obtaining nitrogen depth profiles for any requisite nitriding duration.

scholarly journals Ertragskundliche Aspekte und Strukturdynamik der Buchen-Lichtwuchsdurchforstung

2018 ◽  
Vol 169 (6) ◽  
pp. 332-339 ◽  
Author(s):  
Joachim Klädtke
Keyword(s):  
Volume Growth ◽  
Basal Area ◽  
Structural Diversity ◽  
Maximum Volume ◽  
Negative Effects ◽  
Single Tree ◽  
Stand Productivity ◽  
Single Tree Selection ◽  
Volume Production ◽  
Equilibrium Growth

Effects of heavy selective thinnings in beech stands with regard to stand productivity and structural diversity Based on thinning trials in beech stands initiated in the early 1970s, the effects of selective thinnings in favour of 100 to 120 future crop trees were analysed with regard to volume productivity and the stands' structural diversity. The results show that these kinds of thinnings reduce volume productivity by approximately 6% compared to the maximum volume growth. The reduction in volume growth is mostly caused by the fact that the strongly released crop trees have reduced their height growth for the benefit of a lateral crown expansion, while basal area growth was not affected. Calculations indicate that silvicultural systems working with only 50 future crop trees per hectare may decrease the maximum volume production by about 12%. Furthermore, the results show that the intense crown releases caused by selection thinnings increase the horizontal and vertical diversity of beech stands, since the trees in the understorey profit from better light conditions. The indices calculated for diameter and height diversity of the selection thinning plots approach the values being typical for single tree selection forests (“Plenterwälder”). Already after three to four consecutive selection thinnings, the beech stands' diameter distributions displayed an exponentionally decreasing shape typical for single tree selection forests close to equilibrium. Growth simulations revealed that it might even be possible to directly transform beech stands by selection thinnings into single tree selection structures. However, presumably negative effects on timber quality to be expected under single tree selection structures should be considered as a critical aspect with regard to theoretically possible transformation.

scholarly journals Some recent results for heat-diffusion moving boundary problems

1985 ◽  
Vol 32 (3) ◽  
pp. 437-460
Author(s):  
James M. Hall ◽  
Jeffrey N. Dewynne
Keyword(s):  
Stefan Problem ◽  
Moving Boundary ◽  
Formal Series ◽  
Heat Diffusion ◽  
Upper And Lower Bounds ◽  
Two Phase ◽  
Boundary Problems ◽  
Stefan Problems ◽  
Analytical Approximations ◽  
Cylinders And Spheres

Integral formulations for the three classical single phase Stefan problems involving the infinite slab and inward solidifying cylinders and spheres are utilized to generate standard analytical approximations. These approximations include the pseudo steady state estimate, large Stefan number expansions, upper and lower bounds, approximations based on integral iteration and related results such as formal series solutions. In order to demonstrate the applicability and limitations of the integral formulations three generalizations of the classical stefan problem are considered briefly. These problems are diffusion with two simultaneous chemical reactions, a Stefan problem with two moving boundaries and the genuine two phase Stefan problem.

Nodal Integral and Finite Difference Solution of One-Dimensional Stefan Problem

2003 ◽  
Vol 125 (3) ◽  
pp. 523-527 ◽  
Author(s):  
James Caldwell ◽  
Svetislav Savovic´ ◽  
Yuen-Yick Kwan
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Moving Boundary ◽  
Finite Difference Methods ◽  
Melting Process ◽  
Time Dependent ◽  
One Dimensional ◽  
Stefan Problems ◽  
Finite Difference Solution ◽  
Very High

The nodal integral and finite difference methods are useful in the solution of one-dimensional Stefan problems describing the melting process. However, very few explicit analytical solutions are available in the literature for such problems, particularly with time-dependent boundary conditions. Benchmark cases are presented involving two test examples with the aim of producing very high accuracy when validated against the exact solutions. Test example 1 (time-independent boundary conditions) is followed by the more difficult test example 2 (time-dependent boundary conditions). As a result, the temperature distribution, position of the moving boundary and the velocity are evaluated and the results are validated.

scholarly journals Strontium Leachability of Hydrofracture Grouts for Sludge-Slurries

1982 ◽  
Vol 15 ◽  
Author(s):  
M. T. Morgan ◽  
E. W. Mcdaniel ◽  
J. G. Moore ◽  
H. E. Devaney ◽  
L. R. Dole
Keyword(s):  
Diffusion Coefficient ◽  
Apparent Diffusion Coefficient ◽  
Moving Boundary ◽  
Distilled Water ◽  
Curing Time ◽  
Bedding Planes ◽  
Apparent Diffusion ◽  
Shale Formation ◽  
Series Of Experiments ◽  
Weak Trend

ABSTRACTThis paper summarizes a series of experiments to determine the strontium leachability of cement-based sludge-slurry hydrofracture grouts.The hydrofracture process has been used at ORNL since 1966 for the routine disposal of intermediate-level waste (ILW) solutions. In this process, cement and other additives are mixed with a waste stream to form grout, which is then injected into a shale bed at a pressure sufficient to cause fracturing along the horizontal bedding planes. This injected grout soon hardens, fixing the radionuclides between layers of the massive Conasauga shale formation.Modified IAEA dynamic leach tests on hydrofracture grout specimens showed improved leach resistance (by a factor of 3 to 5) as the curing time increased from 28 d to 91 d. A weak trend toward lower leachability with increased amounts of dry solids was observed.The moving boundary-plus-diffusion leach model fits the dynamic leach data successfully in most cases. An apparent diffusion coefficient of 5 × 10−12 cm2/s and a moving boundary coefficent of 1 × 10−7 s−1 were obtained when one of the best grout compositions was leached in distilled water.

Development of singularities in some moving boundary problems

1995 ◽  
Vol 6 (5) ◽  
pp. 491-507 ◽  
Author(s):  
J. R. King
Keyword(s):  
Power Law ◽  
Moving Boundary ◽  
Moving Boundary Problems ◽  
Boundary Singularity ◽  
Local Behaviour ◽  
Boundary Problems ◽  
Stefan Problems ◽  
Ill Posed ◽  
Impermeable Boundary ◽  
Well Posed

We investigate the local behaviour at the boundary singularity for the following moving boundary problems: (i) Hele-Shaw flows in which the interface is initially non-analytic; (ii) power-law Hele-Shaw flows in which the interface contains a corner; (iii) Stefan problems in which the interface contains a corner. Both well-posed (‘injection’) and ill-posed (‘suction’) problems are considered. Related results for corner development in the presence of an impermeable boundary are also noted.

A Four-Step Fixed-Grid Method for 1D Stefan Problems

2010 ◽  
Vol 132 (11) ◽  
Author(s):  
M. Tadi
Keyword(s):  
Free Boundary ◽  
Moving Boundary ◽  
Free Boundary Problems ◽  
Grid Method ◽  
Moving Boundary Problems ◽  
Boundary Problems ◽  
One Dimensional ◽  
Nonlinear Part ◽  
Stefan Problems ◽  
Fixed Grid

This note is concerned with a fixed-grid finite difference method for the solution of one-dimensional free boundary problems. The method solves for the field variables and the location of the boundary in separate steps. As a result of this decoupling, the nonlinear part of the algorithm involves only a scalar unknown, which is the location of the moving boundary. A number of examples are used to study the applicability of the method. The method is particularly useful for moving boundary problems with various conditions at the front.

scholarly journals COMPUTATION OF ONE DIMENSIONAL ONE PHASE STEFAN PROBLEMS

2020 ◽  
Author(s):  
V.G. Naidu ◽  
P. Kanakadurga Devi
Keyword(s):  
Single Phase ◽  
Moving Boundary ◽  
Moving Boundary Problem ◽  
Initial Time ◽  
Neumann Condition ◽  
Type Problem ◽  
Governing Equations ◽  
Accurate Analysis ◽  
One Dimensional ◽  
Stefan Problems

To design an efficient device or to calculate the performance of existing device requires an accurate analysis of parameters involved in the system. In this work, an efficient front tracking finite difference method is developed to solve one dimensional single phase moving boundary problem with Neumann condition. The basic difficulty apart from the need to find the moving boundary presented, that there is no domain for the first phase at initial time. This difficulty is handled by the age old principle of basic mathematics. Naturally, giving symbolic names to the unknowns by modelling the problem, governing equations are developed with the conditions of the Stefan type problem, solved it and compared the obtained solutions with existing results wherever possible.

Sign in / Sign up

Export Citation Format

Share Document