Laminar Friction Factors for a Concentric Annulus
​Authors: Housam Binous and Ahmed Bellagi

The friction factors for concentric annulus at low Reynolds numbers are given by
f=
A
Re
Dh
=
64
Re
eff
. Hence,
A=64*ζ
where
ζ
is a shape factor, which depends on b/a. ​​
​​Some numerical values of
f*
Re
Dh
and
D
eff
/
D
h
=1/ζ
are given in Table 6.3. Here, we reproduce this Table.​​
​​A plot of A versus D/d has been reported previously. Here, we also plot A versus D/d.​​
​​​References:​[1] O. C. Jones, Jr., J. C. M. Leung, An Improvement in the Calculation of Turbulent Friction in Smooth Concentric Annuli, J. Fluids Eng., Dec 1981, 103(4): 615-623. DOI: 10.1115/1.3241781[2] Frank M. White, Fluid Mechanics, 4th edition, McGraw-Hill, 2000.
In[]:=
(*RadiusratioandcorrespondingshapefactorfromRef.1*)​​ratio={0.0,10^-8,10^-5,10^-3,10^-2,0.05,0.10,0.15,0.20,0.30,0.50,0.70,1.00};​​​​shapeFactor={1.0,0.9457,0.9132,0.8569,0.7989,0.7419,0.7161,0.7021,0.6930,0.6820,0.6719,0.6681,0.6667};
In[]:=
(*interpolationfunction*)​​factor=Interpolation[Transpose[{ratio,shapeFactor}],InterpolationOrder->1];
In[]:=
ratioTable={0.0,0.00001,0.001,0.01,0.05,0.1,0.2,0.4,0.6,0.8,1.0};
In[]:=
(*DataintheformofaTablesimilartoTable6.3ofRef.2*)​​Grid[Prepend[Transpose[{ratioTable,64/factor[ratioTable],factor[ratioTable]}],{"Radius ratio b/a","f*Re_Dh","Shape factor 1/ζ"}],Frame->All,Background->{None,{Lighter[Blend[{Blue,Red}],.3],{Lighter[Blend[{Blue,Darker@Red}],.8],Lighter[Red,.9]}}},ItemSize->{8.4,2.0},Frame->Darker[Gray,.6],ItemStyle->14,Spacings->{Automatic,.8}]
Out[]=
Radius ratio b/a
f*Re_Dh
Shape factor 1/ζ
0.
64.
1.
0.00001
70.0832
0.9132
0.001
74.6878
0.8569
0.01
80.1102
0.7989
0.05
86.265
0.7419
0.1
89.373
0.7161
0.2
92.3521
0.693
0.4
94.5417
0.67695
0.6
95.5224
0.67
0.8
95.861
0.667633
1.
95.9952
0.6667
In[]:=
(*DataintheformofaPlots*)
In[]:=
ratioplot=Table[i,{i,0.001,1,0.001}];
In[]:=
ListPlot[Quiet[Transpose[{1/ratioplot,64/factor[ratioplot]}]],PlotRange->{{1.0,20},{78,98}},Frame->True,GridLines->Automatic,PlotStyle->Darker@Red,AspectRatio->1,Joined->True,FrameLabel->{Style["D/d",16],Style["A",16]}]
Out[]=
In[]:=
(*comparisonwithpreviouslypublishedplot*)
In[]:=
img=
;
In[]:=
ratioplot=Table[i,{i,1,10,0.5}];
In[]:=
p=ListPlot[Quiet[Transpose[{ratioplot,64/factor[1/ratioplot]}]],PlotRange->{{1.0,10},{88,98}},Frame->True,GridLines->Automatic,PlotStyle->Darker@Red,PlotMarkers->Automatic,AspectRatio->1,FrameLabel->{Style["D/d",16],Style["A",16]}];
In[]:=
Show[p,Prolog->{Raster[ImageData[img,DataReversed->True],{Scaled[{0.0,-0.02}],Scaled[{1,1}]}]}]
Out[]=