1 tpbv3.mcd
Solve substrate/product concentration profiles in a spherical gel (WITH PRODUCT INHIBITION)
One two-point boundary value (TPBV) problem solved with "sbval"; will take 3 minutes.
Instructor: Nam Sun Wang
.
d2s
dr2.
2
r
ds
dr .
φ2v( )s B.C.: s( )1 1 ds( )0
dr 0
.
d2p
dr2.
2
r
dp
dr .
φ2v( ),s p B.C.: p( )1 p1 dp( )0
dr 0dp( )1
dr
ds( )1
dr
p1 1 (automatically true)
Dimensionless model parameters and rate expression: i ..1 25 si.
i 0.05
β1Γ10 γ1 j ..1 25 pj.
j 0.1
v( ),s p s
1s
β
.
Γs2.
γp
V,i j v ,sipj
0.131
0.014
V
φ7
Transform the above equation into two 1st-order
ODEs with the relationship: p=1+p1-s
dsdr( ),,r s z z
dzdr( ),,r s z .
φ2v( ),s 1 p1 s .
2 if ,,r 0 .
φ2
3v( ),s 1 p1 s z
r
ODE( ),r y dsdr ,,r y1y2
dzdr ,,r y1y2
Use sbval function to evaluate the initial conditions:
guess10 y initial( ),r guess guess1
0yfinal( ),r y y11← target is 1
y0 sbval ,,,,,guess 0 1 ODE y initial yfinal
=y0 0.295 ← MathCAD's guess of s(0)
Set the initial condition:
yinitial
y01
0
Integrate ODE N 100 i ..1 N
yout rkfixed ,,,,y initial 0 1 N ODE
Use our own variable names
riyout ,i 1 siyout ,i 2 dsdriyout ,i 3
Values at r=1
=sN0.986 =dsdrN1.388
