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# Chem Hydrogeology GLY 5828

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This 6 page Class Notes was uploaded by Adriana Boyle on Monday October 12, 2015. The Class Notes belongs to GLY 5828 at Florida International University taught by Michael Sukop in Fall. Since its upload, it has received 30 views. For similar materials see /class/221776/gly-5828-florida-international-university in Geology at Florida International University.

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Date Created: 10/12/15

Heat Diffusion Equation Derivation of Diffusion Equation The fundamental mass balance equation is where ZIZP ZO ZL 2A I inputs P production 0 outputs L losses A accumulation 1 Assume that no chemical is produced or lost within the control volume and hence 2P 2L 0 So 1 simpli es to ZI ZOZA 2 Considering a control volume cell where there is uX in the X direction only we have where Jxlx indicates the mass uX density ML39ZT39I in the X direction at the point X Jxlx gtJxxAx Ay X X AX In order to satisfy 2 we must have J J WAyAz EMAyAZ 3 at That is the ux into the left wall times the area over which it occurs minus the ux out of the right wall times its area in any interval of time At must equal the change in chemical mass in the control volume Diffusion occurs in response to concentration gradients e g 6C6X 0001 in accordance with Fick s law J DE 4 where J is the areal ux ML39ZT39I and D is the diffusion coefficient LZT39I Incorporating Fick s law into equation 3 we have D D AyAZ AxAyAZ 6x 6x A it Dividing both sides by AXAyAz gives D Dag 59 axxw 6C 6 Ax at Obviously we are just taking the gradients of the gradients and if we shrink AX to differential size and assume that D is constant in space we have a j D6x 7 x it or equivalently 62C 6C 8 6x2 6t This is the 1D version of the widely applicable Heat Equation Finite Difference Expression of Heat Equation Most partial differential equations can not be solved analytically Numerical solutions that reduce the problem to algebraic equations are often necessary These equations can then be solved by direct or iterative methods The essence of numerical methods for PDEs lies in converting the differentials to finite differences Consider the following concentration values on a linear domain from X to X AX 6C6XIXAxz C IXAX The gradient of C in the X direction at X AXZ ie BhBXIXAxz would be linearly appr0Ximated by CX 6C C C E xAx x xAx x 6x Ax2 xAx x Ax This should be a reasonable appr0Ximation when AX is small enough What about second order derivatives like 62C6X2 Remember from Calculus that zC X2 can be written as 66C6X6X So basically we need a gradient of gradients for the second order derivative We have already estimated the gradient BCaxIxAxz We could look the other direction and compute a gradient at X AX 2 6C C xrAx x A 10 6x xiAxZ xxAx Ax CIMX 6C6xIxAxz CIX aCaxImz Clx x l v I v I X AX I Estimate here I X I Estimate here I X AX Estimate here Then taking the gradient between 6C xIxmxz and BCBXIXAxz gives a 62C N 6x 6x2 E 6x xAx2 xrAxZ C W x x W 11 Axl We need a nite difference estimate for aCat too This is a little more complicated because now we have to include time tAt C X tAt X AX The simplest approach is to use the known concentration from the preVious time step At the start of any numerical solution the Initial Condition will be the time 0 concentration So we can approximate the derivative aCat by ACAt or ac N AC C C 12 61 At At Putting it all together The heat equation is 62C 6C 6x2 5 13 Let s replace the derivative terms with their nite difference approximations C 2C D maxim C mm M At C C xrAprz m rpm 14 Written this way nearly all of the concentrations all of those with the t 7 At subscript are known from the previous time step or from the initial condition vat is the only exception and it is what we are trying to determine Let s rearrange to solve for vat DAI C C Cmim T39E 2C C Maxim 15 n Anzac arm This is the llly explicit nite difference approximation for the Heat equation It is explicit because it works with the known concentrations from the previous time step While this is relatively simple it is only stable when the factor DAtAx2 is less or equal to 12 This can be a serious limitation Say we want to simulate a diffusion coefficient of 1039 2 mzs391 gas in soil That means AtAx2 has to be less than 002 s cm39z If we wanted to simulate the process with one second time steps we d need space steps of 002 cm Simulating a 2 m soil profile would require calculations at 10000 space nodes There are other approaches for the time derivative that avoid this stability restriction Initial and Boundary Conditions Initial and boundary conditions must be specified when differential equations like the heat equation are to be solved Initial conditions simply specify the concentrations throughout the problem domain at time zero Boundary conditions specify either 1 The concentrations at the boundaries or 2 The chemical ux at the boundaries usually zero The fixed concentration boundary concept is simple The chemical ux boundary is slightly more difficult We go back to Fick s law 1 D 16 6x and notice that if aCax 0 then there is no ux The finite difference expression we developed for aCax is ac CxAx Cx 2 17 ax xAx2 AX Setting this to 0 is equivalent to C C xAx x So if x is the last point inside the problem domain we could put some ghost points outside the domain at x Ax Then if we make the concentration at the ghost points equal to the concentration inside the domain there will be no ux Often the boundary conditions are constant in time but they need not be Analytical Solution A number of analytical solutions for the heat equation are available Here we present one for the following conditions 0 Initial condition Cx0 0 0 Left boundary condition C0t l 0 Right boundary condition Coot 0 Note that the right boundary is at infinite distance The solution is c erfc Dt 19 Analytical solutions can be used to check the results of finite difference computer programs when the boundary conditions are equivalent Summary Hopefully you have the idea that we ve come full circle in our derivation of the Heat equation and its finite difference approximation We began with a macroscopic consideration of uxes and chemical mass changes in time and incorporated concentration gradients via Fick s law Then we showed how this led to a second order partial differential equation the Heat equation Our development of the finite difference expression for the Heat equation was essentially the same process in reverse we considered how to express the derivatives in terms of the concentration and eventually derived a simple algebraic equation that can be used to solve the Heat equation numerically

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