Abstract:
The paper presents a formal exact solution of the one dimensional advection-diffusion equation (ADE) with constant coefficients in a finite domain. The numerical solution of the one-dimensional advection diffusion equation is obtained by using explicit centered difference schemes with FTBSCS and FTCS techniques for prescribed initial and boundary data. Numerical results for both the schemes are compared in terms of accuracy by error estimation with respect to exact solution of the and also, the numerical features of the rate of convergence are presented graphically.
Keywords: Advection Diffusion Equation, Exact solution, Explicit Finite Difference Schemes, Rate of Convergence.
1. Introduction
Advection–diffusion equation (ADE)
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Due to the growing surface and subsurface hydro-environment degradation and the air pollution, the advection–diffusion equation has drawn significant attention of hydrologists, civil engineers and mathematical modelers. The analytical/numerical solutions along with an initial condition and two boundary conditions help to understand the contaminant or pollutant concentration distribution behavior through an open medium like air, rivers, lakes and porous medium like aquifer. It has wide applications in other disciplines too, like soil physics, petroleum engineering, chemical engineering and …show more content…
We describe Mathematical Models for comparison of the error based on ([4], [5], [6], [7], [8], [9], [14], [15]). Based on the study of the general finite difference method for the second order linear partial differential equation ( [1], [2], [3], [11], [12], [13],[16], [17]), we develop an explicit finite difference scheme for our model treated as ADE as an IBVP with two sided boundary conditions in section 3. In section 3, we also establish the stability condition of the numerical schemes. In section 4, we present an algorithm for the numerical solution and we develop a computer programming code for the implementation of the numerical schemes and perform numerical simulations in order to verify the behavior for various parameters. In section 5, a comparison of errors for both the techniques with respect to an exact solution is projected herein in terms of accuracy. Numerical features of the rate of convergence are also presented graphically. Finally the conclusions of the paper are given in the last section.
2. Advection Diffusion Equation
The one-dimensional advection-diffusion equation [1] is given as
∂C/∂t+u ∂C/∂x=D (∂^2 C)/(∂x^2 )………(1) where C represents the solute concentration [ML-3] at x along longitudinal direction at time t, D is the solute dispersion, if it is independent of position and time, is called dispersion coefficient [L2T-1], t = time [T]; x = distance [L] and,
For Herbert Run the conductivity level was 687µS/cm. The Turbidity level was 0 FAU and the Nitrate level was 0.02ppm. I accept my hypothesis and reject parts of my hypothesis. I reject that both streams have a high turbidity level. Both streams’ turbidity level is zero.
Summary Determining the concentration of a liquid can be a tricky process involving complex procedures if it were not for science’s ability to test a substance’s absorbency through spectrophotometry. The experiment was carried out to discover the concentration of Red Dye #40 in several common soft drinks. The samples of the dye were diluted, and tested using a spectrophotometer. The absorbencies of these samples were then recorded, and a standard line curve with the concentration equation and R2 value was created with these results. Using the absorbencies of the dye samples, the concentrations of the soda samples were determined using the slope equation provided by the graphing software.
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\section{Facility Static and Dynamic Control}\label{Calibr} The facility calibration is the transfer function between the oscillating gauge pressure $P_C(t)$ in the chamber (described in ~\autoref{Sub31}) and the liquid flow rate $q(t)$ in the distributing channel, i.e. the test section. Due to practical difficulties in measuring $q(t)$ within the thin channel, and being the flow laminar, this transfer function was derived analytically and validated numerically as reported in ~\autoref{Sub32} and ~\autoref{Sub33}. \subsection{Pressure Chamber Response}\label{Sub31} Fig.\ref{fig:2a} shows three example of pressure signals $P_C(t)$, measured in the pneumatic chamber.
In this lab we used two processes called Diffusion and Osmosis. Diffusion is the movement of molecules from areas of high concentration to areas of low concentration. Diffusion is a process that requires no energy and involves smaller non-polar molecules. In Figure 1 you can see the molecules spreading throughout the glass from the area of high concentration, so that the areas with low concentration are filled evenly as well. The other process was osmosis.
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1. 150 ml of boiled water was poured into each of the three beakers labeled A, B, C. 2. Five tea bags were soaked for the time given by the manufacturer (two minutes) , in beaker A (Control). The teabags were immediately removed after the time elapsed. 3.
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Rivers that passed through urban areas became a holder for human waste products. Factories began releasing pollutants directly into rivers and streams. By the 1850’s inhabitants began experiencing epidemics of cholera and typhoid. In 1969, chemical waste releasaed into Ohio’s Cuyahoga River caused it to burn into flames and the waterway became a symbol of how insutrial pollution was destroying America’s natural resources. Aditionally, mining activties also affected water pollution by increasing the amount of toxic elements released into the environment.
The gradient gave the value of K, the rate constant for the reaction. Figure 2 shows the plotted graph Figure 2. From the
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