Does it make sense my results of a pde equation from a metalurgy problem?

Hello , I have a metalurgy problem in which I want to model a CSTR reactor , but whose equation is a diffusion-convection equation that is being addresed next, by the way I tried to solve it with the pdepe solver. I adjointed the pdf file so can be better understandable.

As assumption I am working only with the gold in ore equation because cyanide and oxygen equation are supposedly constant in time (not real but ideally) ,so the following equation is left :

kinetic rate is:

which "t" means residence time.

Initial conditions are : t=0 ; Cau(z,t)=0

Boundary conditions are:

z=0 ; Cau(0,t)=Cau_0

z=L ; dC/dt =0

The code is the following.

x = 0:1:100;
t = 0:1:100;
m = 0;
sol = pdepe(m,@pdefun,@pdeic,@pdebc,x,t);
u1 = sol(:,:,1);
surf(x,t,u1)

zlabel('Mineral conc (x,t)')
xlabel('Reactor length')
ylabel('Residence Time')  

function [c,f,s] = pdefun(x,t,u,dudx) % Equation to solve
U=10;
E=0.1e-8;
c = 1;
k=0.0017;
a=1.3703;
b=1.071;
cc=0.1195;
Ox=8;
cn=159;

f = E.* dudx;% - U.*u;
y =k*((u-0.17)^a)*((cn)^b)*((Ox)^cc) ;
s = -y -U.*dudx ;
end
% ---------------------------------------------
function u0 = pdeic(x) % Initial Conditions
u0 = 1;
end
% ---------------------------------------------
function [pl,ql,pr,qr] = pdebc(xl,ul,xr,ur,t) % Boundary Conditions
Cai=1;
pl = ul-Cai;
ql = 0;
pr = 0;
qr = 1;
end
% ---------------------------------------------

My question arise from the fact that when using a low mineral concentration feed, my reactor length spans a large length , on the contrary when feeding with a high mineral concentration the residence time and reactor length reduces. Should´t be the other way around? Don´t know if the code is broke or its not properly written or there is a reason thereby. Thanks in advance for the help.

[Torsten]

I don't know exactly what you mean by "my reactor length spans a large length" or "reactor length reduces". You define the reactor length as 100, don't you ?

[Raúl Rivera]

Yes , you right. I mean, the curve shows an asynthote like this , so its not necesary to build a 100 meters long reactor just pick the length when the conversion suffices the desired conversion , I better explain it with "residence time" .Using 100 mg/kg gold-ore.

Compared with a 1mg/kg ore

I am doubtful if these are trustworthy results because residence time is practically the same, despite there is a big gap in concentrations. Supossedly an upper concentration value should get more residence time .Thanks for the help

[Torsten]

C_Au^inf does not depend on the concentration C_Au ? I cannot believe that it's 0.17 independent of C_Au.

[Raúl Rivera]

Yes , C_Au^inf does not depend on the concentration C_Au, because that is the concentration after a infinite leaching time , it is took from lab experiments , as much the other exponential values, just the gold concentration is a variable.

[Torsten]

Ok, then you must accept from the form of your equation for r_1 that the reaction rate in case of high concentrations of C_Au is much higher than for low concentrations. Thus there is no reason not to believe that the time to reach equilibrium of 0.17 is almost independent of the concentration of C_Au at the inlet.