| Author | Message | ||
     Spyridon Gerontas (sgag2) New member Username: sgag2 Post Number: 1 Registered: 08-2004 |
Hello, I am trying to do the mathematical modelling which describes the oxygen concentration profile in a tubular reactor by using Flexpde 4.0 student version. The reactor is composed of two concentric cylindrical zones corresponding to two regions: the lumen and the gel-cell region. The oxygen is fed through the lumen, diffuses through the gel and is consumed by the cells. The governing equation for the lumen (steady state) is v*dz(c)=d1*(1/r)*dr(c)+d1*drr(c), where v:axial velocity, d1:oxygen diffusivity in the lumen, c:oxygen concentration. The governing equation for the gel-cell region is d2*(1/r)*dr(c)+d2*drr(c)=q, where q oxygen uptake rate and d2 diffusivity in the gel. The boundary conditions are at r=0 dr(c)=0 (radial symmetry) at the interface between lumen and gel d1*(dr(c)-)=d2*(dr(c)+) , c-=c+ At the outer radius of the reactor dr(c)=0 no flux out of the reactor my problem is how to define the boundary condition in the interface as there is a jump in the value of dr(c). ******************script************************** TITLE 'Tubular reactor' SELECT COORDINATES XCYLINDER VARIABLES c ! oxygen concentration DEFINITIONS d ! oxygen diffusivity a ! maximum velocity q ! oxygen uptake rate i=0.17^2 ! radius of the lumen EQUATIONS c: div(d*grad(c))=q*i*(1/0.15)+a*i*(1/33)*(1-(r^2))*dz(c)+d*(1-((0.17/33)^2))*dzz(c ) ! it has been used dimensionless analysis ! the term d*(1-((0.17/33)^2))*dzz(c) has been added to adjust the term div(d*grad(c)) in dimensionless form BOUNDARIES REGION 1 ! lumen a=3.6 ! maximum velocity in the lumen- flow assumed laminar q=0 ! no reaction in the lumen d=3*10^(-5) ! oxygen diffusivity in the lumen START(0,1) value(c)=1 ! inlet value of concentration LINE TO (0,0) natural(c)=0 ! radial symmetry LINE TO (1,0) LINE TO (1,1) natural(c)=0 ! I add it in purpose to show that it is assumed that the flux in the interface of the regions in 0. ! By removing this term the result is the same LINE TO FINISH REGION 2 ! gel with immobilized cells a=0 ! no convection in the gel q=1.055*10^(-3) ! oxygen uptake rate (assuming zero-order kinetics) d=2.22*10^(-5) ! oxygen effective diffusivity in the gel START(0,1) LINE TO (1,1) LINE TO (1,0.2/0.17) natural(c)=0 ! no flux out of the tubular reactor LINE TO (0,0.2/0.17) LINE TO FINISH PLOTS CONTOUR(c) ELEVATION(c) FROM (0.5,0) TO (0.5,0.2/0.17) ELEVATION(c) FROM (1,0) TO (1,0.2/0.17) ELEVATION(c) FROM (0,0) TO (0,0.2/0.17) END ****************************************************** Also i would like to ask if the problem can be solved for pulsatile flow in the lumen instead of laminar(v=vmax(1-(r/r1)^2)*(1+sin(ωt) instead of v=vmax(1-(r/r1)^2). Thanks in advance!!! Spyridon | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 211 Registered: 06-2003 |
You have a term div(d*grad(c)). FlexPDE will integrate this by parts and assume that the surface term normal(d*grad(c)) is continuous across interfaces. This means that you will automatically have a jump in dr(c) equalt to the diffusivity ratio d1/d2 at the material interface. This enforces conservation of c. You don't have to do anything special. See the Help Index under Natural Boundary conditions. I don't understand the second question. |