Collision integrals versus dimensionless temperature​
Housam Binous and Ahmed Bellagi

The collision integrals,
Ω
D
(for mass diffusivity) and
Ω
k
=
Ω
μ
(for thermal conductivity and viscosity), are computed based on the Lennard–Jones potential.​Collision integrals are dimensionless functions of the temperature and of the intermolecular potential field for one molecule of A and one molecule of B.​​Appendix K of Ref. 1 tabulates values of
Ω
D
and
Ω
k
=
Ω
μ
, as a function of κ×T /
ϵ
AB
where κ is the Boltzmann constant, equal to 1.38
-16
10
ergs/K, and
ϵ
AB
is the energy of molecular interaction for the binary system A and B, a Lennard–Jones parameter, in ergs.​​Neufeld and coworkers (Ref. 2) developed empirical equation in order to provide estimates of these collision integrals. Such equation makes computerized calculations more convenient.​​
Ω
D
=A×
-B
TS
+C×
-D×TS

+E×
-F×TS

+G×
-H×TS

​​and​​
Ω
k
=
Ω
μ
=A×
-B
TS
+C×
-D×TS

+E×
-F×TS

+G×
-H×TS

-R×
B
TS
×Sin(P-S×
W
TS
)
​​where
TS=κ×T/
ϵ
AB
​​​Here, we use NonlinearModelFit in order to fetch new sets of the empirical constants that yield slightly smaller sum of squared residuals.​​References :​[1] J. R. Welty, G. L. Rorrer and D. G. Foster, Fundamentals of Momentum, Heat, and Mass Transfer, 7th Edition, NY: Wiley, 2019.[2] P. D. Neufeld, A. R. Janzen, and R. A. Aziz, J. Chem. Phys., 57, 1100 (1972).
Ω
D
=A
-B
TS
+C
-D×TS

+E
-F×TS

+G
-H×TS

In[]:=
Clear[A1,B1,C1,D1,E1,F1,G1,H1,R1,S1,W1,P1]
In[]:=
OMEGA11=A1/TS^B1+C1/Exp[D1TS]+E1/Exp[F1TS]+G1/Exp[H1TS]
Out[]=
C1
-D1TS

+
-F1TS

E1+
-H1TS

G1+A1
-B1
TS
In[]:=
data={{0.3,2.662},{0.35,2.476},{0.4,2.318},{0.45,2.184},{0.5,2.066},{0.55,1.966},{0.6,1.877},{0.65,1.798},{0.7,1.729},{0.75,1.667},{0.8,1.612},{0.85,1.562},{0.9,1.517},{0.95,1.476},{1.,1.439},{1.05,1.406},{1.1,1.375},{1.15,1.346},{1.2,1.32},{1.25,1.296},{1.3,1.273},{1.35,1.253},{1.4,1.233},{1.45,1.215},{1.5,1.198},{1.55,1.182},{1.6,1.167},{1.65,1.153},{1.8,1.116},{1.85,1.105},{1.9,1.094},{1.95,1.084},{2.,1.075},{2.1,1.057},{2.2,1.041},{2.3,1.026},{2.4,1.012},{2.5,0.9996},{2.6,0.9878},{2.7,0.977},{2.8,0.9672},{2.9,0.9576},{3.,0.949},{3.1,0.9406},{3.2,0.9328},{3.3,0.9256},{3.4,0.9186},{3.5,0.912},{3.6,0.9058},{3.7,0.8998},{3.8,0.8942},{3.9,0.8888},{4.,0.8836},{4.1,0.8788},{4.2,0.874},{4.3,0.8694},{4.4,0.8652},{4.5,0.861},{4.6,0.8568},{4.7,0.853},{4.8,0.8492},{4.9,0.8456},{5.,0.8422},{6.,0.8124},{7.,0.7896},{8.,0.7712},{10.,0.7424},{20.,0.664},{30.,0.6232},{40.,0.596},{50.,0.5756},{60.,0.5596},{70.,0.5464},{80.,0.5352},{90.,0.5256}};(*datafromappendixKofRef.1*)
In[]:=
p1=ListPlot[data,Frame->True,GridLines->Automatic,AspectRatio->0.75,ImageSize->Medium,FrameLabel->{Style["dimensionless temperature",16],Style["collision integral",16]},PlotStyle->Darker@Gray,PlotRange->{{0,90},{0,3}}]
Out[]=
In[]:=
mdl=NonlinearModelFit[data,OMEGA11,{A1,B1,C1,D1,E1,F1,G1,H1},TS]
Out[]=
FittedModel
1.64
-4.45TS

+1.21
-19TS

+0.285
1

+
1.07
0.157
TS

In[]:=
sol=Normal@mdl
Out[]=
1.63583
-4.45147TS

+1.21352
-1.8126TS

+0.284661
-0.599496TS

+
1.06612
0.157468
TS
In[]:=
mdl["ParameterTable"]
Out[]=
Estimate
Standard Error
t-Statistic
P-Value
A1
1.06612
0.00114345
932.372
1.57477×
-139
10
B1
0.157468
0.000301141
522.906
1.05414×
-122
10
C1
1.63583
0.03284
49.8121
1.14103×
-54
10
D1
4.45147
0.075705
58.8001
2.17399×
-59
10
E1
1.21352
0.0296889
40.8744
4.26357×
-49
10
F1
1.8126
0.0420876
43.0673
1.45906×
-50
10
G1
0.284661
0.0137204
20.7472
7.08289×
-31
10
H1
0.599496
0.015775
38.0029
4.60085×
-47
10
In[]:=
A1N=1.06036;B1N=0.15610;C1N=0.19300;D1N=0.47635;E1N=1.03587;F1N=1.52996;​​G1N=1.76474;H1N=3.89411;(*constantsfromRef.2*)
In[]:=
{A1New,B1New,C1New,D1New,E1New,F1New,G1New,H1New}=​​{1.06612,0.157468,1.63583,4.45147,1.21352,1.8126,0.284661,0.599496}(*constantsobtainedusingNonlinearModelFit*)
Out[]=
{1.06612,0.157468,1.63583,4.45147,1.21352,1.8126,0.284661,0.599496}
In[]:=
Grid[Transpose@{{" ","A","B","C","D","E","F","G","H"},{"Our Fit ",A1New,B1New,C1New,D1New,E1New,F1New,G1New,H1New},{"Ref. 1 ",A1N,B1N,C1N,D1N,E1N,F1N,G1N,H1N}},Background->{None,{Lighter[Yellow,.5],{White,Lighter[Blend[{Blue,Green}],.6]}}},Frame->All,ItemStyle->12,Spacings->{1.5,1.5}]
Out[]=
Our Fit
Ref. 1
A
1.06612
1.06036
B
0.157468
0.1561
C
1.63583
0.193
D
4.45147
0.47635
E
1.21352
1.03587
F
1.8126
1.52996
G
0.284661
1.76474
H
0.599496
3.89411
Our fit has a lower sum of squared residuals
In[]:=
ourfit=Sum[(data[[i]][[2]]-sol)^2/.TS->data[[i]][[1]],{i,1,Length[data]}]
Out[]=
0.0000128369
In[]:=
fitNeufeld=Sum[(data[[i]][[2]]-(A1N/TS^B1N+C1N/Exp[D1NTS]+E1N/Exp[F1NTS]+G1N/Exp[H1NTS]))^2/.TS->data[[i]][[1]],{i,1,Length[data]}]
Out[]=
0.000250318
In[]:=
res=mdl["FitResiduals"];
In[]:=
Sum[res[[i]]^2,{i,1,Length[data]}]
Out[]=
0.0000128369
In[]:=
mdl["ANOVATableSumsOfSquares"][[2]]
Out[]=
0.0000128369
Ω
k
=
Ω
μ
=A×
-B
TS
+C×
-D×TS

+E×
-F×TS

+G×
-H×TS

+R×
B
TS
×Sin(S×
W
TS
-P)
In[]:=
Clear[A1,B1,C1,D1,E1,F1,G1,H1,R1,S1,W1,P1]
In[]:=
OMEGA22=A1/TS^B1+C1/Exp[D1TS]+E1/Exp[F1TS]+10^-4R1TS^B1Sin[S1TS^W1-P1]
Out[]=
C1
-D1TS

+
-F1TS

E1+A1
-B1
TS
-
R1
B1
TS
Sin[P1-S1
W1
TS
]
10000
In[]:=
data={{0.3,2.785},{0.35,2.628},{0.4,2.492},{0.45,2.368},{0.5,2.257},{0.55,2.156},{0.6,2.065},{0.65,1.982},{0.7,1.908},{0.75,1.841},{0.8,1.78},{0.85,1.725},{0.9,1.675},{0.95,1.629},{1.,1.587},{1.05,1.549},{1.1,1.514},{1.15,1.482},{1.2,1.452},{1.25,1.424},{1.3,1.399},{1.35,1.375},{1.4,1.353},{1.45,1.333},{1.5,1.314},{1.55,1.296},{1.6,1.279},{1.65,1.264},{1.7,1.248},{1.8,1.221},{1.85,1.209},{1.9,1.197},{1.95,1.186},{2.,1.175},{2.1,1.156},{2.2,1.138},{2.3,1.122},{2.4,1.107},{2.5,1.093},{2.6,1.081},{2.7,1.069},{2.8,1.058},{2.9,1.048},{3.,1.039},{3.1,1.03},{3.2,1.022},{3.3,1.014},{3.4,1.007},{3.5,0.9999},{3.6,0.9932},{3.7,0.987},{3.8,0.9811},{3.9,0.9755},{4.,0.97},{4.1,0.9649},{4.2,0.96},{4.3,0.9553},{4.4,0.9507},{4.5,0.9464},{4.6,0.9422},{4.7,0.9382},{4.8,0.9343},{4.9,0.9305},{5.,0.9269},{6.,0.8963},{7.,0.8727},{8.,0.8538},{10.,0.8242},{20.,0.7432},{30.,0.7005},{40.,0.6718},{50.,0.6504},{60.,0.6335},{70.,0.6194},{80.,0.6076},{90.,0.5973}};(*datafromappendixKofRef.1*)
Our fit has a lower sum of squared residuals