# 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):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 Attachments:
 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: epsilon -> "e" (void fraction, unitless) Kappa -> "Kd" (mass transfer coefficient, assumed constant, units: 1 / time) "a" : surface area of adsorbing particles "c": aqueous concentration of target substance (units:mass/volume) "w": adsorbed concentration of target substance in the solid particles (units:mass/volume) "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}]] `