Gas holdup analysis by different correlations and models

4 6 8 10 12 14 0.05

0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.45 5% Paraffin liquid 15% Paraffin liquid 25% Paraffin liquid 35% Paraffin liquid at u_sg= 2.55×10^-2 m/s

εg (-)

uj (m/s)

(a)

4 6 8 10 12 14

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

5% Kerosene 15% Kerosene 25% Kerosene 35% Kerosene at u_sg= 2.55×10^-2 m/s

εg (-)

uj (m/s)

(b)

Figure 4.4: Variations of gas holdup with liquid jet velocity at different (a) paraffin liquid (b) kerosene volume concentrations

081 . 162 1 . 2 42 . 1 586 . 0 238 .

0 Re

Re 48 .

3 × ^{− −}

= _{j g r}

g We Fr H

ε (4.14)

The correlation was developed based on 160 experimental data at different operating conditions.

The predicted values

ε

_g from the correlation were compared with the experimental values, which are shown in Figure 4.5. The overall percentage error between experimental and predicted values was found to be +12% and -12%. The ranges of the variables of groups for which the correlation is valid are: 27.82 < Re_g < 111.28, 81514.4 < Re_j < 731406.7, 1.30 < We < 17.36, 0.06

< Fr < 0.18, 19 < Hr < 41. The theory based on which it was followed for the analysis of variance is given in Appendix I.

0.0 0.1 0.2 0.3 0.4 0.5

0.0 0.1 0.2 0.3 0.4 0.5

5% Paraffin liquid 15% Paraffin liquid 25% Paraffin liquid 35% Paraffin liquid 5% Kerosene 15% Kerosene 25% Kerosene 35% Kerosene

ε g - predicted (-)

εg - experimental (-) + 13%

- 12 %

Figure 4.5: Parity plot of calculated values and experimental values of gas holdup for different concentrations of paraffin liquid and kerosene

4.5.2 Analysis by Lockhart-Martinelli correlation

In the present analysis, it is found that the model Equation (4.1), which is developed for horizontal flow, does not fit well with the experimental values of vertical downflow, so an attempt has been made to modify the correlation in terms of quality (x). The quality is given by the ratio of the mass flow rate of gas and the mass flow rate of a gas-liquid-liquid mixture, which is expressed as

dl cl dl cl g g

g g m

g

Q Q

Q m

x m

−

+ −

=

= ρ ρ

ρ

&

&

(4.15)

The volumetric flow rate of gas Qg is calculated from the volumetric flow rate of liquid Qcl-dl and the gas holdup as

(

g

)

dl cl g g

Q Q

ε ε

= − ⁻

1 (4.16)

By substituting Equation (4.16) in the Equation (4.15) the gas holdup can be expressed as,

( )







 −

+

=

− x

x

dl cl

g

g 1

1 1

ρ

ε ρ (4.17)

Based on the present study, the quality (x) is correlated with the physical and operating conditions of the present experiment. The correlation can be expressed by different significant dimensionless groups based on the dimensional analysis. The functionality of the correlation can be obtained by the multiple regression analysis (by Microsoft Excel data analysis tool) which is expressed as

53 . 0 90 . 1 66 . 0 65 . 0

10Re Re

10 15 .

1 1 − − − −

×

− =

X x H

x

r g

j (4.18)

By substituting Equation (4.18) in the Equation (4.17) the gas holdup can be expressed as,

( )

_^









  ×





 +

=

−

−

−

−

−

53 . 0 90 . 1 66 . 0 65 . 0

10Re Re

10 15 . 1 1

1

X H_r

g j dl

cl g g

ρ

ε ρ _(4.19)

Comparisons between the predicted values of gas holdup (

ε

_g) by modified Lockhart– Martinelli correlation and the experimental data are shown in Figure 4.6. It is seen that the correlation Equation (4.19) fits well within the range of experimental conditions. The correlation coefficient (R²) and the standard error (SE) are 0.96 and 0.21, respectively. The overall percentage error between experimental and predicted values is found to be +16% and -18%. The correlation valid within the ranges: 27.82 < Reg < 111.28, 81514.4 < Rej < 731406.7, 5.81 < X’< 60.71.

0.0 0.1 0.2 0.3 0.4 0.5

0.0 0.1 0.2 0.3 0.4 0.5

5 % Kerosene 15 % Kerosene 25 % Kerosene 35 % Kerosene 5 % Paraffin liquid

15 % Paraffin liquid 25 % Paraffin liquid 35 % Paraffin liquid

ε g- predicted, (-)

ε_g - experimental, (-) + 16%

- 18%

Figure 4.6: Parity plot of calculated values and experimental values of gas holdup (ɛg) by modified Lockhart- Martinelli correlation (Equation (4.19))

4.5.3 Analysis by drift-flux model

The drift flux model is applied in this study for gas-liquid-liquid flow in the downflow column.

For a homogeneous flow, the values of Co and ud correspond to one and zero, respectively. The drift velocity for upward gas-liquid two-phase flow is given by

( )

^{1 4}

2 2 









 −

=

−

−

− dl cl

g dl cl dl cl d

u g

ρ

ρ ρ

σ (4.20)

By using Equation (4.20) and Equation (4.2), the distribution coefficient (C_o) can be expressed for downflow gas-liquid-liquid system as follows:

( )

sl sg

dl cl

g dl cl dll cl g

sg

o u u

g u

C +



























 −

−

−













= ⁻

−

−

4 1

2 2

ρ

ρ ρ

σ ε

(4.21)

The entrainment occurs beyond a minimum entrainment velocity for which the dispersion of gas occurs and results in the fluid circulation inside the column. The circulation of fluid may occur by counteracting the buoyancy effect of gas bubbles and insoluble liquid in the column. At the center of the column the jet kinetic energy is higher than the radial positions. Due to this energy distribution, along the column wall, the gas bubble and the other insoluble liquid move upward and get circulated in the column.

Based on present experimental data, a correlation has been made by dimensional analysis of Buckingham Pi theorem to obtain the values of Co within the range of liquid velocity of 0.04 to 0.14 m/s and the gas velocity of 0.85 × 10^-2 to 2.55 × 10^-2 m/s. The range of drift flux velocity for paraffin-water system is -6.57 to -5.71 and for kerosene-water system is -6.60 to -5.84 as per present experimental conditions. The correlation can be expressed as

42 . 0 029 . 0 23 . 0 08 .

0 Re

Re 5 .

24 ^{− − −}

= Mo Fr

C_{o j g} (4.22)

By regression analysis of Equation (4.22), the correlation coefficient and standard error are found to be 0.977 and 0.050, respectively. The parity plot of experimental versus calculated values of Co by Equation (4.22) is shown in Figure 4.7. The correlation shows good prediction of the experimental values. The overall percentage error between experimental and predicted values is found to be +10% and -7%. The correlation is valid within the range of 27.82 < Reg < 111.28, 81514.4 < Rej < 731406.7, 1.44 × 10^-11 < Mo < 1.01 ×10^-9, 0.06 < Fr < 0.18.

20 40 60 80 100 120 140

20 40 60 80 100 120 140

5% paraffin liquid 15% paraffin liquid 25% paraffin liquid 35% paraffin liquid 5% kerosene 15% kerosene 25% kerosene 35% kerosene

C o- predicted,(-)

C_o- experimental,(-) + 7%

- 10%

Figure 4.7: Parity plot of calculated values and experimental values of distribution parameter (C_o)

4.5.4 Interpretation of gas holdup by slip velocity model

In the present downflow column, the slip velocity is found to be positive or negative. The

positive values of slip velocity specify the higher rise velocity of bubbles relative to downward velocity of liquid. The negative value indicates the downward movement of bubbles with the liquid. Based on the present experimental study, the following relation is developed,

β αε +

= _g

b s

u

u (4.23)

where

74 . 2 91 . 2 78 . 0 5Re 10 64 .

4 × ⁻ _j We⁻ Fr

−

α

= (4.24)

and β = 1.245. The slip velocity is positive for u_{sl <} 0.10 m/s and d_b < 0.66 mm. From the Fig.4.8, it is seen that the slip velocity is a function of concentrations of dispersed liquid.

Typical profiles of slip velocity as a function of bubble size for paraffin liquid-water and kerosene-water are shown in Figure 4.9a and Figure 4.9 b, respectively. The slip velocity increases with the bubble size for the respective dispersed liquids. The average bubble size for paraffin liquid-water and kerosene-water system are 1.21 and 0.76 mm, respectively, within the range of present experimental conditions.

0.05 0.10 0.15 0.20 0.25 0.30 0.35

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

us(m/s)

ε_g (-)

5% Paraffin liquid 15% Paraffin liquid 25% Paraffin liquid 35% Paraffin liquid

(a)

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 -0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

5% Kerosene 15% Kerosene 25% Kerosene 35% Kerosene

us(m/s)

ε_g (-)

(b)

Figure 4.8: Variations of slip velocity with gas holdup at usg = 0.84×10^-2 m/s, for different concentrations of (a) paraffin liquid (b) kerosene

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

us (m/s)

d_b (m) 5% Paraffin liquid 15% Paraffin liquid 25% Paraffin liquid 35% Paraffin liquid

(a)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

-0.3 0.0 0.3 0.6 0.9 1.2 1.5

us (m/s)

d_b (mm) 5% Kerosene

15% Kerosene 25% Kerosene 35% Kerosene

(b)

Figure 4.9: Variations of slip velocity with bubble size at u_sg = 0.84×10^-2 m/s, for 5% of dispersed liquid (a) paraffin liquid (b) kerosene