## THERMAL SCIENCE

International Scientific Journal

### EXPLICIT FINITE DIFFERENCE SOLUTION FOR CONTAMINANT TRANSPORT PROBLEMS WITH CONSTANT AND OSCILLATING BOUNDARY CONDITIONS

**ABSTRACT**

For constant and oscillating boundary conditions, the 1-D advection-diffusion equation with constant coefficients, which describes a contaminant flow, is solved by the explicit finite difference method in a semi-infinite medium. It is shown how far the periodicity of the oscillating boundary carries on until diminishing to below appreciable levels a specified distance away, which depends on the oscillation characteristics of the source. Results are tested against an analytical solution reported for a special case. The explicit finite difference method is shown to be effective for solving the advection-diffusion equation with constant coefficients in semi-infinite media with constant and oscillating boundary conditions. [Project of the Serbian Ministry of Education, Science and Technological Development, Grant no. 171011]

**KEYWORDS**

PAPER SUBMITTED: 2019-07-22

PAPER REVISED: 2019-10-04

PAPER ACCEPTED: 2019-10-07

PUBLISHED ONLINE: 2019-11-17

**THERMAL SCIENCE** YEAR

**2020**, VOLUME

**24**, ISSUE

**3**, PAGES [2225 - 2231]

- Rao, C. S., Environmental Pollution Control Engineering, 3rd reprint, Wiley Eastern Ltd, New Delhi, 1995
- Ebach, E. H., White, R., Mixing of fluids flowing through beds of packed solids, Journal of American Institute of Chemical Engineers 4 (1958) pp.161-164
- Fry, V.A., Istok, J.D., Guenther, R.B., Analytical solutions to the solute transport equation with rate-limited desorption and decay, Water Resources Research 29 (1993) pp. 3201-3208
- Lin, C., Ball, W.P., Analytical modeling of diffusion-limited contamination and decontamination in a two-layer porous medium, Advances in Water Resources 21 (1998) pp. 297-313
- Chen, J.S., Liu, C.W., Liao, C.M., Two-dimensional Laplace-transformed power series solution for solute transport in a radially convergent flow field. Advances in Water Resources 26 (2003) pp. 1113-1124
- Derya, A., Billur, I.E.B., Necati, Ö., The Dirichlet problem of a conformable advection-diffusion equation, Thermal Science 21 (2007) pp. 9-18.
- Karahan, H., Implicit finite difference techniques for the advection-diffusion equation using spreadsheets, Advances in Engineering Software. 37 (2006) pp. 601-608
- Huang, Q., Huang, G., Zhan, H., A finite element solution for the fractional advection-dispersion equation, Advances in Water Resources 31 (2008) pp. 1578-1589
- Zhao, C., Valliappan, S., Transient infinite element for contaminant transport problems, International Journal for Numerical Methods in Engineering 37 (1994) pp. 113-1158
- Zhao, C., Valliappan, S., Numerical modelling of transient contaminant migration problems in infinite porous fractured media using finite/infinite element technique: theory, International Journal for Numerical and Analitical Methods in Geomechanics 18 (1994) pp. 523-541
- Wang, W., Dai, Z., Li, J., Zhou, L., A hybrid Laplace transform finite analytic method for solving transport problems with large Peclet and Courant numbers, Computers & Geosciences 49 (2012) pp. 182-189
- Yeh, G.T., Comparisons of successive iteration and direct method to solve finite element equations of aquifer contaminant transport, Water Resources Research 21 (1985) pp. 272-280
- Sudicky, E.A., Mclaren, R.G., The Laplace transform Galerkin technique for large-scale simulation of mass transport in discretely fractured porous formations, Water Resources Research 28 (1992) pp. 499-514
- Russell, T.F., Celia, M.A., An overview of research on Eulerian-Lagrangian localized adjoint methods (ELLAM), Advances in Water Resources 25 (2002) pp. 1215-1231
- Dai, Z., Samper, J., Wolfsberg, A., Levitt, D., Identification of relative conductivity models for water flow and solute transport in unsaturated compacted bentonite, Physics and Chemistry of the Earth 33 (2008) pp. S177-S185
- Yadav, R. R., Jaiswail, D. K., Gulrana, Two-dimensional solute transport for periodic flow in isotropic porous media: an analytical solution, Hydrological Processes 26 (2012) pp. 3425−3433
- Jaiswal, D.K, Kumar, A., Kumar, N., Yadav, R.R., Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media, Journal of Hydro-environment Research 2 (2009) pp. 254-263
- Savović, S., Caldwell, J., Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions, International Journal of Heat and Mass Transfer 46 (2003) pp. 2911−2916
- Savović, S., Caldwell, J., Numerical solution of Stefan problem with time-dependent boundary conditions by variable space grid method, Thermal Science 13 (2009) pp. 165−174
- Savović, S., Djordjevich, A., Finite difference solution of the one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media, International Journal of Heat and Mass Transfer 55 (2012) pp. 4291−4294
- J. D. Anderson, Computational Fluid Dynamics, McGraw-Hill, New York, 1995.
- Djordjevich, A., Savović, S., Finite difference solution of two-dimensional solute transport with periodic flow in homogenous porous media, Journal of Hydrology and Hydromechanics 65 (2017) pp. 426−432.