Horizontal Slug Flow

Data (Reid et al 1957)

In[]:=
g=9.81;

Superficial Velocities

In[]:=
ULS=0.65;UGS=1.5;

Mixture velocity

In[]:=
vS=ULS+UGS
Out[]=
2.15

Internal diameter of the pipe

In[]:=
d=0.154;A=
π
2
d
4
Out[]=
0.0186265

Liquid properties at standard condition (20 C, 1 atm)

In[]:=
ρL=998.7;μL=0.0010;σ=0.0731;

Gas properties at standard condition (20 C,1 atm)

In[]:=
ρG=1.21;μG=0.000018;

Empirical coefficients

In[]:=
k=0.0682;m=2/3;ϵ=0.00007
Out[]=
0.00007
lsl=lS/l
lfl=lF/l

Analysis

Translational velocity

In[]:=
vT=1.2vS+0.54Sqrt[gd]
Out[]=
3.24373

Governing equations

In[]:=
eq1=ULS==vSHLSlsl+vFHLF(1-lsl)
Out[]=
0.652.15HLSlsl+HLF(1-lsl)vF
In[]:=
eq2=UGS==vS(1-HLS)lsl+vG(1-HLF)(1-lsl)
Out[]=
1.52.15(1-HLS)lsl+(1-HLF)(1-lsl)vG
In[]:=
eq3=(vT-vF)HLF==(vT-vS)HLS
Out[]=
HLF(3.24373-vF)1.09373HLS
In[]:=
eq4=
τFSF-τISI
AF
==
τGSG+τISI
AG
Out[]=
SFτF-SIτI
AF

SGτG+SIτI
AG

Shear stresses

In[]:=
τF:=0.5fFρL
2
vF
In[]:=
τG:=0.5fGρG
2
vG
In[]:=
τI:=0.5fIρG
2
(vG-vF)

Friction factors

In[]:=
fI=0.0142;
In[]:=
eq5=fF==0.0013751+
1
3
20000
ϵ
Dhf
+
1000000
ReF
Out[]=
fF0.0013751+
1/3
1.4
Dhf
+
1000000
ReF
In[]:=
Dhf:=
4AF
SF
In[]:=
ReF:=
ρLvFDhf
μL
In[]:=
eq6=fG==0.0013751+
1
3
20000
ϵ
Dhg
+
1000000
ReG
Out[]=
fG0.0013751+
1/3
1.4
Dhg
+
1000000
ReG
In[]:=
Dhg:=
4AG
(SG+SI)
In[]:=
ReG:=
ρGvGDhg
μG

Geometrical variables

In[]:=
AF:=
2
d
(θ-0.5Sin[2θ])
4
In[]:=
AG:=
2
d
(π-θ+0.5Sin[2θ])
4
In[]:=
SF:=dθ
In[]:=
SG:=d(π-θ)
In[]:=
SI:=dSin[θ]
In[]:=
HLF:=
θ-0.5Sin[2θ]
π
In[]:=
HLS:=1-
vS
Sqrt[gd]
-Fo
vS
Sqrt[gd]
+2400
-3
4
Bo
In[]:=
Bo:=
(ρL-ρG)g
2
d
σ
In[]:=
Ifd<Sqrt[2]0.025,Fo=0,Fo=2.61-2
2
0.025
d

Out[]=
2.46296
In[]:=
HLS
Out[]=
1.09614

Solving

In[]:=
SOL=Solve[{eq1,eq2,eq3,eq4,eq5,eq6},{θ,vF,vG,fF,fG,lsl}]