GAS HOLDUP AND PRESSURE DROP CHARACTERISTICS
3.2. Theory of frictional pressure drop
Gas Holdup and Pressure Drop Characteristics flow in helical coil. From the literature review, it is seen that correlations based on experiments exist for predicting the pressure drop, but none of these approaches have taken into account the contacting mechanism between the phases, effect of bubble formation, phase interaction due to movement of bubbles and the wettability effect on pressure drop in helical coil system and also with non-Newtonian system. A model is formulated based on mechanical energy balance within the framework of dynamic interaction of the phases. The model includes the effect of bubble formation, phase interaction or slip due to movement of gas as slug, friction at the wall of the conduit and the wettability effect of liquid on the pressure drop.
Chapter-3
58
2
1 2
D D b
Dc
(3.3)
Incorporating Dc instead of D in the mathematical definition of Dean number (the Dean number is defined as Re (d/D)0.5, a significant number for identifying the dominating force between inertia and viscous force of flow in a curved tube) results a new number, referred to as the helical coil number, He, which can be expressed as
5 . 2 0 5
. 0
1 2 Re
c e
c t n
n D
D b D
He d
(3.4)
As the pitch increases for the same coil curvature, the centrifugal force decreases at the same flow rate thus weakening the secondary flow field, and ultimately straight tube behavior is appeared (Lin and Tarbell 1980; Yamamoto et al., 1995; Vashisth et al.,2008). Applying a force balance to a differential element of a pipe, the wall shear stress can be expressed as,
f
w dL
dp
d
4 (3.5)
Viscous shear occurs primarily at the pipe wall. The friction factor is a dimensionless group representing the pressure loss due to friction and is defined as the ratio of wall shear stress (w) to kinetic energy per unit volume which represents the relative importance of wall shear stress to the total energy losses. The Fanning friction factor is expressed as
2
2
f Uw
(3.6)
Eqs. (3.5) and (3.6) yields
d U f dL
dp
f
2 2
(3.7)
TH-1484_10610718
Gas Holdup and Pressure Drop Characteristics It is well known that friction factors have been successfully correlated for single-phase flow with two dimensionless groups called Reynolds number and relative roughness. For gas-liquid two- phase flow through a pipe, Eq. (3.1) becomes
D d u d
u g f
dL
dp m m m m
m
2 2
2 2
2
(3.8)
where the subscript m refers to the mixture of gas and liquid. Normally mixture properties such as density and viscosity are calculated as weighted averages of the individual phase properties with input fraction or holdup of the phases as weighting factors. For mixture, density,
l l g g
m
. The mixture velocity is defined as the sum of the superficial velocities ofindividual phases as
u
m u
sg u
sl.3.3. Experimental
The schematic of the experimental setup used in the present investigation are given in Figure. 2.2 in Chapter 2. The equipment and apparatus consisted of a cylindrical storage tank (S), centrifugal pump (Pm) for circulating the fluid, two rotameters (make: Five Stars Ltd., Mumbai) of different ranges RL1 and RL2 for measuring the flow rate of the fluid, and U-tube manometer (U) to measure the pressure difference across the helically coiled tube. Valves (V1 to V8) were used to control the flow rate of the liquid. The storage tank was of 100 liters capacity, the bottom side of the tank was connected to the suction side of the pump (Pm) through a globe valve (V1). Another globe valve (V8) was provided at the bottom of the storage tank to facilitate the periodic cleaning. Through the compressor, air was passed to the inlet of the tube. The discharge end of the pump was divided into two lines, one directly connected to rotameters (RL1& RL2) and the other to a by-pass line.
Chapter-3
60
Table 3.1: Summary of work on two-phase hydrodynamics in helical coil reported in literature
Author Parameters Working Fluid Correlation Remarks
Mujawar and Rao (1981)
dt = 1.21 cm; Dc = 17.4, 25.4, 61, 121 cm; p = 1.9, 2.6, 5.7, 13.6 cm; n
= 2, 4, 6, 8; Lc = 262, 437, 480, 770 cm
Air-water, Helium, Sodium alginate (0.3-0.5%), SCMC (0.5-1.0%).
2
2 1
1 X X
C
l
, G
1 . 85 /( E
g)
4.60c
,d g C V n L C
p L
p
c sg b g L sg
tp
2
) )(
(
X is the Lockhart –Martinelli parameter ϕ is the pressure drop multiplier
Given a
correlations for holdup and frictional pressure drop
Rangacharyulu and
Davies (1984)
dt = 11, 13 mm; QG = 1 - 10 m3/s; QL = 0.04 - 0.75 m3/s; 18.48< Dc/dt
< 23.4
Air - water, Air- glycerol, Air-Isobutyl alcohol
4 3
0.18
3.6668 . 0 2
/ 2 1
/ )
/(
) (
/ /
Re 05 . 0 1
c i l
l l
o t c i l
R R g
C u D
g d
) /(
)
( 2 2
0 l
l l g g g l l g
g R C
R C R R
C
C0 is the stratified speed of sound, (m/hr)
Done pressure drop and holdup by using
Lockhart and Martinelli
TH-1484_10610718
Gas Holdup and Pressure Drop Characteristics Hart et al.
(1988)
dt = 14.66 ± 0.04 mm;
do = 17.66 ± 0.04 mm; n
= 7, 8, 10, 11, 12, 13; Lc
= 17 m
Air - Water
(70 )
090 . 1 0
5 . 1
De f De
fc s ,
0 Re Re
critRecrit = Critical Reynolds number
Friction factor for a pressure drop
Awwad et al.
(1995)
dt = 12.7, 19.1 25.4, 38.1 mm; DC = 330, 340, 350, 360, 640, 650, 660 mm; α : 0.5 – 200;
n = 4 - 14
Air -water 1/2
2 2
1 1 12
1 ] 1 [
/
X X X
F C
X
n d
L
1 . 2 0
1 . 0
c t t l c
t
d D
d gd U D
Fr d F
X is the Lockhart –Martinelli parameter
Given correlation for pressure drop and holdup
Xin et al.
(1996)
dt = 12.7, 19.1, 25.4, 38.1 mm; Dc = 305, 609 mm
Air - water
6 . 0 2
1
2 1 65.45
1 1 20
/
d
L F
X X
X
, for Fd 0.1
7 . 1 2
1
2 1 434.8
1 1 20
/
d
L F
X X
X
, for Fd0.1
2 . 0 2
/ 1
) tan 1
(
D
F d Fd r
Produced correlation for pressure drop and holdup
Chapter-3
62 Chen and Guo
(1999)
dt = 39 mm; Dc = 265, 522.5 mm; Lc : 4320, 6760 mm
Oil- air - water
2
2 1
1 )
( X X
f C
L
,
where
l tp
L dL
dP ) (
)
2 (
,
lg 2
) (
) (
dL X dP ll
X is the Lockhart –Martinelli parameter
Given a correlation for holdup
Ali (2001) dt = 0.603, 0.464 cm Dc = 1.6225, 22.448 cm; p = 1,5 cm LC =223.8, 112.3, 111.5, 226.3, 113.2, 426, 213 cm; n = 3, 6
Air - water EuGrhc21.88Re0.9, Re500
3 /
Re 2
25 .
5
rhc
EuG , 500Re6300
5 /
Re 2
56 .
0
rhc
EuG , 6300Re10,000
5 /
Re 1
09 .
0
rhc
EuG , Re10,000 where Grhc 0.85e0.15/Lc
Given correlation for pressure drop in single phase and multi-phase flow
Mandal and Das (2003)
di = 0.01, 0.013 m;
Dc = 0.131 - 0.222 m;
Θ = 0, 4, 8, 12;
n = 12, 13, 15, 16, 19, 20
Water, 1% Amyl alcohol, 30% Glycerin
Q hg Q
M P M
P
g l
g l tp
ftp
Produced
correlation for Pressure drop
TH-1484_10610718
Gas Holdup and Pressure Drop Characteristics Biswas and Das
(2006)
dt = 9.33, 9.7, 12 mm;
Dc = 176, 178, 216, 266 mm;
Air - CMC Pfl,
(eff)wtp/(eff)wl
Pfl Given correlation for pressure dropGupta et al.
(2011)
dt = 4.5, 6.5, 7.94, 9.53, 12.01 mm;
Lc = 3.98,5 m;
Dc = 114.3, 266.85, 323.1, 462, 655 mm;
p = 40, 60, 100, 200, 300 mm
Air - water fc fs(10.903NGn0.227), for
N
Gn 70
)525 . 0 1
( Gn0.516
s
c f N
f , for
N
Gn 70
where fc fs(1aNGnb ),
f
s 16 / N
Re
2 2 2
2 Re
Re ( / ( / )
) / (
t t
c
t c
Gn D d p d
d N D
N
N
NGn = Germano number
Developed correlation for coil and the straight tube
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64
The by-pass was provided with a globe valve V2 to control the flow through rotameters. Two more valves V3 and V4 were provided at the inlet of rotameters to operate the rotameters either separately or in combination. Two solenoid valves were fixed in the inlet (entrance) and another one in outlet (Exit). Solenoid valves (SV1 and SV2) (Techno company) are of range of maximum 10 kg/cm2). The helical coils were made using transparent thick-walled polyethylene tube. Two pressure taps (PT1 and PT2) at the beginning and end of the coil were provided to measure pressure drop across the coil. These taps were connected to the limbs of a U-tube manometer (M1). One more U-tube manometer (M2) is used for measuring the pressure drop in horizontal side which is connected to the pressure taps (PT3 and PT4) in horizontal tube at its outlet. One more tap is created for collecting fluid for holdup. Compressor (C) is connected to the gas rotameter (RG), and this gas rotameter connected the inlet tube for two phase flow. Water and the air are sent to the inlet of the coil tube and with the second coil onwards it gets constant flows of air and water. The background of the test section consists of graph sheet to know the size of the bubble/drop flowing through the test section. The equivalent bubble diameter which is observed in the coil is measured by the image analysis software (Digimizer) from air gap length observed in test section. Water and air are pumped into the test section of helical coil by using centrifugal pump (for water) and compressor (for Air). The flow rate of water is measured through pre-calibrated rotameters RL1 and RL2. The flow rate of air is measured through the gas pre-calibrated rotameter. Holdup at different flow rates is measured by quick closing valve technique. The quick closing valves (Solenoid valves) SV1and SV2 in the setup are placed at the two ends of the test section. These quick closing valves (Solenoid valves) are arranged in the entry of the coil and exit of the coil. There is electrically adjustment for simultaneous closing of the two valves to trap the flowing mixture instantaneously. Once the system reaches steady state
TH-1484_10610718
Gas Holdup and Pressure Drop at particular flow rate of two fluids, the flowing mixture is arrested by closing these two valves in the test section instantaneously. By knowing the volume of liquids, gas holdup is calculated from the following equation
T L T
g v
v v
(3.9)
where, g= gas holdup, vL= volume of water, vT= total volume of gas liquid mixture. Physical properties of the systems are given in Table 2.1 in chapter 2. The uncertainty of the experimental data is shown in Table 3.2.
Table 3.2: Typical uncertainties of the gas holdup at constant tube diameter (dt = 0.015 m), coil diameter (Dc = 0.117 m)
Usl
(m/s)
Usg
(m/s)
Range of mean* N
Range of STDEV*
(×10-3)
Range of uncertainty*
(×10-3)
Range of relative uncertainty* % 0.189-1.132 0.094 0.438 - 0.601 6.0 4.415-5.432 1.802- 2.217 0.299 - 0.507 0.189-1.132 0.189 0.452 - 0.667 6.0 2.486-4.236 1.014-1.735 0.225 - 0.261 0.189-1.132 0.283 0.467 - 0.713 6.0 2.817-4.035 1.652-7.411 0.355 - 1.089 0.189-1.132 0.377 0.503 - 0.748 6.0 4.023- 4.681 1.645-1.917 0.256 - 0.328
* The theory to calculate the values is given in chapter 2 in equations (2.10 – 2.12)
Each value of gas holdup in 3rd column corresponds to the mean of 6 data points (i.e., N=6).
Relative uncertainty is calculated from standard uncertainty and mean value of the corresponding data set.
Chapter-3
66