Chromatographic Reactions of Three Components
​Housam Binous

​Three species
A
1
,
A
2
, and
A
3
move in an electrostatic, centrifugal, or velocity field and mix according to the cyclic reaction system:
A
1
k
e
⇆
k
e
A
2
k
e
⇆
k
e
A
3
k
e
⇆
k
e
A
1
. All reaction rate constants are assumed to be equal to
k
e
=
-2
10
-1
s
.​The three governing equations for the concentrations
c
i
of the species
A
i
are​​
∂
c
i
∂t
=
D
i
2
∂
c
i
∂
2
x
-
v
i
∂
c
i
∂x
-2
k
e
c
i
+
k
e
c
j
+
k
e
c
m
,​where
{i,j,m
} are three permutations of
{1,2,3}
.​The boundary and initial conditions are
lim
x±∞
c
j
(x,t)=0
and
c
j
(x,0)=
1
3
δ(x)
.​All components are assumed to be sufficiently similar to have identical diffusion coefficients

D
1
=
D
2
=
D
3
=
-5
10
2
cm
s.
The velocities of species
A
1
,
A
2
, and
A
3
are taken to be
v
1
=0.02cm/s
,
v
2
=-0.03cm/s
, and
v
3
=0.04cm/s
.​An expression for the reduced first central moment,

μ
1j
, the second and third central moments (
μ
2j
and
μ
3j
) were derived in the paper by H. Binous and B. J. McCoy in 1992 (see Ref. 1).​These moment expressions are used to construct the concentration profile using the following truncated series of the Gram-Charlier expansion:​​
c
j
(x,t)=
1
3
-
2
X
j

1+
μ
3j
8
3
X
j
-12
X
j
6
1.5
2
μ
2j


2π
μ
2j
,​where
X
j
=
x-

μ
1j

2
μ
2j
.​This notebook shows the concentration profiles of the species
A
1
,
A
2
, and
A
3
at different times. In particular, Fig. 4 of Ref. 1 is reporduced. In accordance with the initial conditions, peaks are very high at small dimensionless time
τ=
k
e
t.
For large values of
τ
, all peaks merge into one. Thus, (1) separation of peaks occurs for small
τ
; (2) the species
A
1
,
A
2
, and
A
3
behave as one component for large
τ
. For intermediate values of
τ
, higher-order terms may be needed to adequately represent the pulse. The concentrations of the species
A
1
,
A
2
, and
A
3
are shown as red, blue, and green peaks, respectively.​The gray peak corresponds to
3
∑
i=1
c
i
(i.e., the concentration of the combined species). This combined concentration exhibits multiple peaks or a broad asymmetric peak for small
τ
. This behavior is actually observed in reverse-phase liquid chromatographic separations where one protein can undergo conformational changes and be present in multiple states.​​
​ ​Reference:​[1] H. Binous and B. J. McCoy, “Chromatographic Reactions of Three Components: Application to Separations,” Chemical Engineering Science, 47(17/18), 1992 pp. 4333–4343. https://doi.org/10.1016/0009-2509(92)85112-O​
In[]:=
v
1
=210^-2;
v
2
=-310^-2;
v
3
=410^-2;(*velocitiesofspecies
A
1
,
A
2
,and
A
3
*)​​
In[]:=
ke=10^-2;(*allreactionrateconstantsareassumedtobeequalto
k
e
=
-2
10
-1
s
*)
In[]:=
Dif=10^-5;(*Assumeidenticaldiffusioncoefficients
D
1
=
D
2
=
D
3
=
-5
10
2
cm
s*)​​
In[]:=
(*reducedfirstcentralmoment*)​​μ[p1,1,t_]:=(2
v
1
-
v
2
-
v
3
)(1-Exp[-3ket])/(9ke)+(
v
1
+
v
2
+
v
3
)t/3;​​μ[p1,2,t_]:=(2
v
2
-
v
1
-
v
3
)(1-Exp[-3ket])/(9ke)+(
v
1
+
v
2
+
v
3
)t/3;​​μ[p1,3,t_]:=(2
v
3
-
v
2
-
v
1
)(1-Exp[-3ket])/(9ke)+(
v
1
+
v
2
+
v
3
)t/3;
In[]:=
(*secondandthirdcentralmoments*)
In[]:=
μ[2,1,t_]:=-(2
v
1
-
v
2
-
v
3
)^2Exp[-6.ket]/(81ke^2)-2(2
v
1
^2-
v
2
^2-
v
3
^2-2(
v
1
v
2
+
v
1
v
3
-2
v
2
v
3
))tExp[-3ket]/(27ke)+8(
v
1
^2+
v
2
^2+
v
3
^2-(
v
1
v
2
+
v
1
v
3
+
v
2
v
3
))Exp[-3.ket]/(81ke^2)+(-4
v
1
^2-7
v
2
^2-7
v
3
^2+4
v
1
v
2
+4
v
1
v
3
+10
v
2
v
3
)/(81.ke^2)+​​(54Difke+4(
v
1
^2+
v
2
^2+
v
3
^2-
v
1
v
2
-
v
1
v
3
-
v
2
v
3
))t/(27.ke);​​μ[2,2,t_]:=-(2
v
2
-
v
1
-
v
3
)^2Exp[-6.ket]/(81ke^2)-2(2
v
2
^2-
v
1
^2-
v
3
^2-2(
v
1
v
2
+
v
2
v
3
-2
v
1
v
3
))tExp[-3ket]/(27ke)+8(
v
1
^2+
v
2
^2+
v
3
^2-(
v
1
v
2
+
v
1
v
3
+
v
2
v
3
))Exp[-3.ket]/(81ke^2)+(-4
v
2
^2-7
v
1
^2-7
v
3
^2+4
v
1
v
2
+4
v
2
v
3
+10
v
1
v
3
)/(81.ke^2)+​​(54Difke+4(
v
1
^2+
v
2
^2+
v
3
^2-
v
1
v
2
-
v
1
v
3
-
v
2
v
3
))t/(27.ke);​​μ[2,3,t_]:=-(2
v
3
-
v
2
-
v
1
)^2Exp[-6.ket]/(81ke^2)-2(2
v
3
^2-
v
2
^2-
v
1
^2-2(
v
3
v
2
+
v
1
v
3
-2
v
2
v
1
))tExp[-3ket]/(27ke)+8(
v
1
^2+
v
2
^2+
v
3
^2-(
v
1
v
2
+
v
1
v
3
+
v
2
v
3
))Exp[-3.ket]/(81ke^2)+(-4
v
3
^2-7
v
2
^2-7
v
1
^2+4
v
3
v
2
+4
v
1
v
3
+10
v
2
v
1
)/(81.ke^2)+​​(54Difke+4(
v
1
^2+
v
2
^2+
v
3
^2-
v
1
v
2
-
v
1
v
3
-
v
2
v
3
))t/(27.ke);
In[]:=
μ[3,j_,l_,m_,t_]:=-2(2
v
j
-
v
l
-
v
m
)^3Exp[-9ket]/(729ke^3)-2(2
v
j
-
v
l
-
v
m
)(-4
v
j
^2+4
v
j
v
l
-4
v
l
^2+4
v
j
v
m
-4
v
m
^2+4
v
m
v
l
)+3ket(2
v
j
^2-2
v
j
v
l
-
v
l
^2-2
v
j
v
m
-
v
m
^2+4
v
m
v
l
)Exp[-6ket]/(243ke^3)-(2
v
j
-
v
l
-
v
m
)(-2
v
j
^2+2
v
j
v
l
+16
v
l
^2+2
v
j
v
m
+16
v
m
^2-34
v
m
v
l
)+36ket(-
v
j
^2+
v
j
v
l
-
v
l
v
m
+
v
j
v
m
)+9ke^2t^2(
v
j
^2-
v
j
v
l
+
v
l
^2-
v
j
v
m
+
v
m
^2-
v
m
v
l
)Exp[-3ket]/(243ke^3)+2(2
v
j
-
v
l
-
v
m
)(-11
v
j
^2+11
v
j
v
l
+13
v
l
^2+11
v
j
v
m
+13
v
m
^2-37
v
m
v
l
)+9ket(
v
j
^2-
v
j
v
l
-2
v
l
^2-
v
j
v
m
-2
v
m
^2+5
v
m
v
l
)(729ke^3);
In[]:=
X
j_
[x_,t_]:=(x-μ[p1,j,t])
2μ[2,j,t]
;
In[]:=
(*concentrationprofileusingthetruncatedseriesexpansion*)
In[]:=
c
j_
[x_,t_]:=1/3Exp[-
X
j
[x,t]^2]1+μ[3,j,Delete[{1,2,3},j][[1]],Delete[{1,2,3},j][[2]],t](8
X
j
[x,t]^3-12
X
j
[x,t])(6(2μ[2,j,t])^1.5)
2πμ[2,j,t]
;
In[]:=
τ=0.001;(*dimensionlesstime*)
In[]:=
plt=PlotEvaluate
c
1
[x,100τ],
c
2
[x,100τ],
c
3
[x,100τ],
3
∑
i=1
c
i
[x,100τ],{x,-τ10,τ10},PlotRangeAll,FrameTrue,PlotStyle{{Thick,Darker@Red},{Thick,Darker@Blue},{Thick,Darker@Green},{Thick,Darker@Gray}},AxesOrigin{-τ10,0},FrameLabel{Style["position",14],Style["concentration of all species",14]},AspectRatio->1,GridLines->Automatic,ImageSize{400,400}
Out[]=