Message Boards Message Boards

Solve a PDE with boundary conditions (chemical adsorption in fixed beds)?

GROUPS:

Dear Wolfram team:

I have been trying for week to solve a system of 2 partial differential equations describing the adsorption of a chemical substance on a fixed bed (for example, a column of activated carbon). The 2 equations are the following, taken from McCabe (1993):

Description of eq 1

Description of eq 2

Unfortunately I cannot get past the general solution (with arbitrary constants) because when I try to put boundary conditions the Mathematica program fails. Maybe I am using the wrong command or syntax, or maybe there are too much or too few boundary conditions.

I have left attached the program, where I tryed to simplify the problem combining both equations in a third. Thank you in advance for your help.

Best regards,

Alberto Silva

Answer
9 days ago

Dear Wolfram team:

Any idea about how to solve the boundary value problem? The result should be an adsorption wave approximately like this:

Figure from McCabe (1993)

With an exponential profile that moves way from the origin as time passes.

However, all the solutions I can get are static (i.e. independent of time) something that is absurd with waves. How can I get a solution that depends both of space (x coordinate) and time?

Thank you in advance,

Best regards,

Alberto Silva

Answer
6 days ago

I will suggest posting the equations in the actual not rather than an attachment, and using only simple Latin alphabet ascii characters as variables (no subscripts, no Greek letters, etc.) This will make it much easier for readers who may wish to play around with the system in question.

POSTED BY: Daniel Lichtblau
Answer
6 days ago

The meaning of terms in the code (I have put a "->" in the legend where I have subtitututed a greek or script letter with a latin one) is this:

  1. epsilon -> "e" (void fraction, unitless)
  2. Kappa -> "Kd" (mass transfer coefficient, assumed constant, units: 1 / time)
  3. "a" : surface area of adsorbing particles
  4. "c": aqueous concentration of target substance (units:mass/volume)
  5. "w": adsorbed concentration of target substance in the solid particles (units:mass/volume)
  6. "u": bulk liquid velocity (units: lenght/time)

Now I will post the code. I will greatly appreciate your help.

ClearAll["Global`*"]
{x, t, t1, eqT, bcon, c, c0} =.

eq1 := u D[c[t, x], x] + e D[c[t, x], t] + (1 - e) D[w[t, x], x]
eq2 := -a Kd c[t, x] + (1 - e) D[w[t, x], x]
eqT := FullSimplify[eq1 - eq2]

bcon1 := {DirichletCondition[c[t, x] == c0  , {x == 0, t >= t1}], 
  DirichletCondition[c[t, x] == c0 E^(-((x a Kd)/u)) , {t == t1}], 
  DirichletCondition[c[t, x] == c0*cb , {x == L, t == tb}]}

GeneralSolution -> FullSimplify [DSolve[{eqT == 0}, {c[t, x]}, {x, t}]]
RelevantSolution -> 
 FullSimplify [DSolve[{eqT == 0, bcon1}, {c[t, x]}, {x, t}]]
Answer
5 days ago

Group Abstract Group Abstract