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Chapter Three
Two-Phase Flow Contents 1- Introduction. 2- Notations and relations. 3- Flow patterns. 4- Homogeneous flow model. 5- Separated flow model. 6- Examples. 7- Problems; sheet No. 3 1- Introduction: Background Two-phase flow is widely seen in the nature: Rain, snow, smog, dust-flow, etc. Two-phase Flow: Is a flow of two-phase mixture of a substance with the same chemical composition. (e.g., Water-steam flow, water-ice flow, etc.) Two-Component Flow: Is a flow of two-component mixture with different chemical composition. (e.g., air-water flow, oil-water flow, air-dust flow, etc.) Two- phase flows are related to the phase-change Phenomena. (e.g., boiling, condensation, freezing, melting, solidification, crystallization). 2- Notation and relations: Subscripts
f = fluid g= gas l= liquid h= enthalpy
Void
Thu
And
Mas
Tota
Volu
Tota
Qua
And
And
d fraction
us
d
ss flow ra
al mass fl
umetric fl
al volume
ality (or m
d
d in a therm
n α :
ate Wg, W
low rate W
flow rate Q
etric flow
mass quali
modynami
Wf:
W:
Qg, Qf:
rate Q:
ity) x:
ic equilibrrium,
Mas
And
Ther
Volu
Thu
Volu
Rela
ss flux (or
d
refore
umetric fl
us
umetric q
ative velo
r superfic
flux (or su
quality β:
city
cial veloci
uperficial
ty) j:
velocity) j:
3- Flow patterns: Vertical flow patterns
(a), (d), (f), (h) and (i): Flow patterns frequently observed in large diameter tubes. (b), (c), (e) and (g): Flow patterns specially appear in capillary tubes.
Horizontal flow patterns
Flow pattern maps
(1) Vertical flow [Hewitt & Roberts, 1969]
1
(2) Horizontal flow [Baker, 1954]
(3) Generalized map [Taitel & Dukler, 1976]
a+b : F vs. X c : K vs. X d : T vs. X
.
.
.
.
4- H Con
Mom
dzdp
−
dd
−
Ener
dzdi
For
dd
−
dF
wτ
=wτ
Homogen
ntinuity eq
mentum E
dzdp
zp
⎠⎞
⎜⎝⎛−=
Adzdp
=1
rgy Equat
(dzd
+21
dw=0, the
(1vdz
dp
h
=
pdZwτ=
w : wall sh
⎜⎝⎛=
21
hhf ρ
neous flo
quation:
Equation:
ZF dzdp⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
gdzdF
+
tion:
( ) gu h + s2
e energy e
21(
dZdE
+
hear stress
⎟⎠⎞2
hhu
ow mode
aZ dzdp⎟⎠⎞
⎜⎝⎛−
vg
h
+θsin
dd
=θsin
equation be
( )2udzd
h
s ,p:tube p
el:
dzdu
AW h+
dzdw
dzdq
−
ecomes:
sin θg+
perimeter
h
)θ
hf : f For
−
⎜⎝⎛−
d
Get
McA
Cicc
Duk
hμ =
For
For
friction fa
a tube of d
dzdp
F)( =−
=⎟⎠⎞ 1
F Adzdp
=2
t
Adams
μ1
chitti
hμ =
kler
ff
jjμ +=
laminar fl
hf =
turbulent
actor
diameter,
dzdF
A1
( ) =1w p
Aτ
DvGf hh
22
gh
xμ
+=1
(gxμ +=
gg ufj
=+
low:
hRe16
=
flow (Bla
D
⎜⎝⎛
21
hfAp ρ
f
xμ−
+1
) fx μ−1
[ ggh x μυρ
asius corre
⎟⎠⎞2
hhuρ
f
( )g x−+ 1
elation):
) ]ffv μ
When using the McAdams Correlations,
1
11−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛−=⎟
⎠⎞
⎜⎝⎛−
g
fg
f
fg
foF
xvv
xdzdp
dzdp
μμ
Laminar flow
41
11−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛−=⎟
⎠⎞
⎜⎝⎛−
g
fg
f
fg
foF
xvv
xdzdp
dzdp
μμ
Turbulent Flow
Thus
(Laminar Flow)
41
2 11−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+=
g
fg
f
fgfo x
vv
xμμ
φ (Turbulent flow)
Acceleration Pressure Drop
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛− h
h
a
vA
WdzdG
dzdu
Gdzdp
Where
( )[ ]fgh vxxv
dzd
dzdv
−+= 1
= ( )dzdxv
dpdv
xdpdv
xdzdp
gffg +⎥⎦
⎤⎢⎣
⎡−+ 1
1
2 11−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+=
g
fg
f
fgfo x
vv
xμμ
φ
⎟⎠⎞
⎜⎝⎛+=
AdzdWGv
dzdvG h
h 12
And
( ) }1)1(2
dzdA
Axvv
dpdv
xdpdv
xdzdp
dzdxvG
dzdp
gfffg
gfa
+−⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡−++=⎟
⎠⎞
⎜⎝⎛−
Gravitational Pressure Drop
fgfZ xvvg
dzdp
+=⎟
⎠⎞
⎜⎝⎛−
θsin
Finally
( ) ( )
( ) ⎥⎦
⎤⎢⎣
⎡−++
+++−++
=⎟⎠⎞
⎜⎝⎛−
dpdv
xdpdv
xG
xvvg
dzdA
AxvvG
dzdxvGxvv
DGf
dzdp
fg
fgffgfgffgf
h
a 11
sin12
2
222 θ
5- Separated flow model: Pressure Drop Correlations:-
aZF dzdp
dzdp
dzdp
dzdp
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−=−
Where:
dzdF
Adzdp
F1)( =−
( )[ ] θαρρα sin1 gdzdp
gfZ
+−=⎟⎠⎞
⎜⎝⎛−
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
+=⎟⎠⎞
⎜⎝⎛−
αα 111 222
fg
a
vxvxAdz
dA
Wdzdp
For a circular tube of diameter, D
22
2 2)( fo
ffofo
foF
F DvGf
dzdp
dzdp φφ =⎟
⎠⎞
⎜⎝⎛−=⎟
⎠⎞
⎜⎝⎛−
22
2 2)( go
ggogo
goF
F DvGf
dzdp
dzdp φφ =⎟
⎠⎞
⎜⎝⎛−=⎟
⎠⎞
⎜⎝⎛−
2
222 2
)( ggg
gg
FF D
vxGfdzdp
dzdp φφ =⎟
⎠⎞
⎜⎝⎛−=⎟
⎠⎞
⎜⎝⎛−
Where gfgofo φφφφ ,,, are the two-phase frictional multipliers related to each other as follows:
( )fof
goggo
fof
ggg
fo
fffo fv
fvfvfv
xff
x 222222 1 φφφφ ==−=
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−
−+=⎟
⎠⎞
⎜⎝⎛−
αα 111 222
fg
a
vxvxAdz
dA
Wdzdp
( ) ( )
( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
−⎟⎠⎞
⎜⎝⎛∂∂
+⎭⎬⎫
⎩⎨⎧
−
−−= 2
2
2
22
1
1.
1122
ααα
ααgf
p
fg vxvxx
vxxvdzdxG
( ) ( )( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
++ 2
2
2
2222
11
.11
ααα
ααgf
x
fg vxvxpdp
dvxdpdvx
dzdpG
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−
−+−
αα 11 222
fg vxvxdzdA
AG
( ) 222
2 12)( f
fff
fF
F DvxGf
dzdp
dzdp φφ
−=⎟
⎠⎞
⎜⎝⎛−=⎟
⎠⎞
⎜⎝⎛−
Finally,
2
1
HH
dzdp
=−
Where
( ) ( )( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
−⎟⎠⎞
⎜⎝⎛+
⎭⎬⎫
⎩⎨⎧
−−
−+= 2
2
2
222
2
1 11
.11222
ααα
ααφ gffg
foffo vxvx
dxdvxxv
dzdxG
DvGf
H
( ) ( )( ) θρααρ
ααsin1
11 222
gvxvx
dzdA
AG
fgfg −++
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−
−+−
and
( ) ( )( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
−⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
++= 2
2
2
2222
2 11
.111
ααα
ααgffg vxvx
dpd
dpdvx
dpdvx
GH
6- Examples: 1- Using the homogeneous flow pressure drop method, calculate the two phase pressure drop for up flow in vertical tube of 10mm internal diameter that is 2m long .The flow is adiabatic ,the mass flow rate is 0.02 kg/s and the vapor quality is 0.05 .The fluid is R-123 at a saturation temperature of
3οC and saturation pressure of 0.37 bar ,whose physical properties are : ρl= 1518 kg/m3 , ρg= 2.6 kg/m3 , µg= 12.6×10-6 Pa.s , µl= 58.56×10-5 Pa.s Solution:
1
11 0.9685
1 50.3
∆ 50.3 2 9.81 90 ∆ 987
1 10.000557 .
4571
0.079
. 0.00961
∆2
4953
Total pressure drop =∆ ∆ 5.94 .
University of Technology Sheet No. 3 Mechanical Engineering Dep. Two-Phase Flow Fluid Mechanics II (3 rd year) 2011/2012 1- Find the flow pattern when 4 kg/s of steam –water mixture of quality 20% at 20 bar flows in a circular tube of internal diameter 0.1 m: a) When the flow is vertically upward. b) When the flow is horizontal. The physical properties required are: ρl= 850 kg/m3, ρg = 10 kg/m3 , μl= 128*10-6 Pa.s , μg= 16*10-6 Pa.s 2- A steam –water mixture of quality 0.1 at 5 bar flow through a smooth vertical round tube of diameter 0.05m.The total flow rate is 0.6 kg/s .Calculate: a) The homogeneous void fraction. b) Homogeneous gravitational pressure gradient. c) Homogeneous frictional pressure gradient. The physical properties required are: ρl= 915 kg/m3, ρg = 2.67 kg/m3 , μl=180*10-6 Pa.s , μg= 14*10-6 Pa.s
[0.97 ; 255 N/m3 ; 547 N/m3]
3- Consider mixture of water (Wl=0.42 kg/s) and air (Wg=0.01 kg/s) flowing upward in a vertical pipe (D=25mm ,L=45cm). Given friction factor of (f=0.079/Re0.25).Find the total pressure drop ,the volumetric flow rates ,void fraction ,and mean water and air velocity. Using homogeneous model. the physical properties required are: ρl= 915 kg/m3, ρg = 1.2 kg/m3 ,μl= 1*10-3 Pa.s , μg= 1.8*10-5 Pa.s
[3169 N/m2 ; 0.00833m3/s ; 0.0042m3/s ; 0.952 ; 17.65m/s]
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