CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

Analytical solution of the 1D advection-dispersion equation

Register Blogs Community New Posts Updated Threads Search

Like Tree1Likes
  • 1 Post By LuckyTran

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   June 29, 2021, 11:28
Default Analytical solution of the 1D advection-dispersion equation
  #1
New Member
 
Anelechi Ibekwe
Join Date: Jun 2021
Posts: 1
Rep Power: 0
burges is on a distinguished road
The Ogata and Banks analytical solution of the ADE for a continuous source of infinite duration and a 1D domain with initial condition: C(x,0) = C_{i}, and BCs: C(0,t) = C_{o}, and C(x_{end},t) = 0.

is given as:

C(x,t) = \frac{Co}{2}[erfc(\frac{x-vt}{2\sqrt{Dt/R}}) + exp(\frac{vx}{D/R})erfc(\frac{x-vt}{2\sqrt{Dt/R}})]

where C [mol/L] is the concentration, x [m] is the distance, R is the retardation factor, D [m2/day] is the effective dispersion/diffusion, v [m/day] is the flow velocity, Ci [mol/L] is the initial concentration in the column, and Co [mol/L] is the influent (or injected) concentration.

The time-evolution of production and degradation of species obtained from the numerical solution of the ADE shows that specie production increases with time, whereas degradation (inversion of the production) reduces with time. However, the analytical equation above only matches the production (Ci = 0, Co > 0) but not the degradation (Ci > 0, Co = 0). So, I have been looking for the equivalent analytical expression that describes the degradation process (meaning that Ci would be contained therein). I am thus hoping that there is someone who may know this equation or could point me in the right direction. Thank you in advance.
burges is offline   Reply With Quote

Old   July 1, 2021, 00:00
Default
  #2
Senior Member
 
Lucky
Join Date: Apr 2011
Location: Orlando, FL USA
Posts: 5,747
Rep Power: 66
LuckyTran has a spectacular aura aboutLuckyTran has a spectacular aura aboutLuckyTran has a spectacular aura about
Abuse linearity. Define a new dynamic variable equal to K(x,t)=C(x,t)-Ci, re-cast the BC's and IC in terms of this new variable and you can make it work backwards. Basically, just do a substitution. As you've noticed, the solution only works for Ci=0 and Co>0 but not Ci>0 and Co=0. But it still works for Ki=0 and Ko>0 (which also satisfies your desire for Ci>0 and Co=0). =)
aero_head likes this.
LuckyTran is offline   Reply With Quote

Reply

Tags
analytical solution, cfd, convection-diffusion, numerical computation


Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Solver Validation chemFoam using analytical Solution Hanzo OpenFOAM Running, Solving & CFD 5 April 4, 2019 09:08
Calculation of the Governing Equations Mihail CFX 7 September 7, 2014 07:27
Exact solution of Burgers equation mcaro Main CFD Forum 3 January 25, 2011 07:46
Analytical solution of poisson equation kostaspan Main CFD Forum 1 April 22, 2010 13:24
IcoFoam parallel woes msrinath80 OpenFOAM Running, Solving & CFD 9 July 22, 2007 03:58


All times are GMT -4. The time now is 16:27.