Lightfoot investigates the effect of chemical reaction on the rate of gas absorption in an agitated tank. Dissolved gas A reacts with liquid B. Reaction is irreversible first-order. Example of such situation would be absorption of SO2 or H2S in aqueous NaOH solution.A simple model involves the assumptions:The governing equation and boundary conditions are:-=0(z=0)=(z=δ)=Our results are compared to those given in Fig. 2 of Ref. 1 and in Fig. 18.4-4 of Ref. 2.
AB
2
c
A
2
z
k
1
c
A
c
A
c
A0
c
A
c
Aδ
DSolve yields an expression for (z)
c
A
In[]:=
DSolve[{DABcA''[z]==k1cA[z],cA[0]==cA0,cA[δ]==cAδ},cA,z]
Out[]=
cAFunction{z},-cA0-cAδ+cA0+cAδ+
-
k1
zDAB
2
k1
zDAB
k1
δDAB
2
k1
δDAB
2
k1
zDAB
k1
δDAB
-1+
2
k1
δDAB
Taking into account the dimensionless position, ζ=z/δ, the Thiele modulus and parameter B=cAδ/cA0,
the concentration of A becomes equal to: Csch[ϕ] (B Sinh[ζ ϕ]+Sinh[ϕ-ζ ϕ])
the concentration of A becomes equal to: Csch[ϕ] (B Sinh[ζ ϕ]+Sinh[ϕ-ζ ϕ])
In[]:=
cA[ζ_]:=-cA0-cAδ+cA0+cAδ+1/cA0/.z->ζδ/.cAδ->BcA0/.k1->ϕ^2DAB/δ^2//FullSimplify
-
k1
zDAB
2
k1
zDAB
k1
δDAB
2
k1
δDAB
2
k1
zDAB
k1
δDAB
-1+
2
k1
δDAB
In[]:=
cA[ζ]
Out[]=
Csch[ϕ](BSinh[ζϕ]+Sinh[ϕ-ζϕ])
Checking our expressions for (ζ) and B versus their counterparts given in Refs. 1-2
c
A
In[]:=
Csch[ϕ](BSinh[ζϕ]+Sinh[ϕ-ζϕ])-1/Sinh[ϕ](Sinh[ϕ]Cosh[ϕζ]+(B-Cosh[ϕ])Sinh[ϕζ])//FullSimplify
Out[]=
0
In[]:=
Solve[-DABS/δcA'[1]==Vk1B,B]/.k1->ϕ^2DAB/δ^2//FullSimplify
Out[]=
B
Sδ
SδCosh[ϕ]+VϕSinh[ϕ]
In[]:=
B=
Sδ
SδCosh[ϕ]+VϕSinh[ϕ]
Out[]=
Sδ
SδCosh[ϕ]+VϕSinh[ϕ]
Plotting, , the ratio of rate of absorption with chemical reaction to rate of physical absorption to tank containing no dissolved gas
N
-
AB
c
A
z
z=0
AB
c
A0
δ
z=0
In[]:=
N
Out[]=
ϕ(VϕCosh[ϕ]+SδSinh[ϕ])
SδCosh[ϕ]+VϕSinh[ϕ]
In[]:=
VoverSD={10^-6,10^2,10^4,10^6,10^8,10^20};
In[]:=
p=ShowTableLogLinearPlot/.V->SδVoverSD[[i]],{ϕ,10^-5,10},PlotRange->{{10^-5,10},{0,2}},AspectRatio->1,GridLines->Automatic,Frame->True,FrameLabel->Style["Thiele Modulus",16],Style"",16,PlotStyle->{Dashed,Hue[i/6],Thickness[0.0075]},{i,1,6}
N
N
Out[]=
In[]:=
img=
;
In[]:=
Show[p,ImageSize->400{1,1},Prolog->{Raster[ImageData[img,DataReversed->True],{Scaled[{0.01,-0.005}],Scaled[{1,1.}]}]}]
Out[]=
Nice discussion of Figure above -- discussion reproduced from Ref. 2
References
References
◼
[1] E. N. Lightfoot, Steady State Absorption of a Sparingly Soluble Gas in an Agitated Tank with Simultaneous Irreversible First-order Reaction, AIChE J ., 4, 499–500 (1958). https://doi.org/10.1002/aic.690040425
◼
[2] R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot, and Daniel Klingenberg, Introductory Transport Phenomena, NY: Wiley, 2015.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Gas absorption with chemical reaction in an agitated tank
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, September 28, 2025
https://community.wolfram.com/groups/-/m/t/3552263
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, September 28, 2025
https://community.wolfram.com/groups/-/m/t/3552263