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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}]]

Dear Wolfram team:

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

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