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Comments on “Two-phase flow and boiling heat transfer in two vertical narrow annuli”
Nuclear Engineering and Design 245 (2012) 241–242
Contents lists available at SciVerse ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Comments on “Two-phase flow and boiling heat transfer in two vertical narrow annuli” In the paper of Changhong et al. (2005), the authors expressed the two-phase frictional multiplier (˚l ) as a function of the Lockhart–Martinelli parameter in turbulent–turbulent flow (Xtt ) in order to predict the frictional pressure drop in the narrow annuli. Their correlation was ˚l =
3.956
(1)
0.536 Xtt
Also, the Chisholm correlation (1967) for turbulent–turbulent flow (C = 20) in terms of ˚2l was suggested as follows: ˚2l = 1 +
20 1 + 2 Xtt Xtt
(2)
Moreover, Mishima and Hibiki (1996) proposed a correlation for the parameter C as a function of the hydraulic diameter (dE ) to take into account the effect of the gap size. Their correlation was ˚2l = 1 +
21(1 − e−0.319dE ) 1 + 2 Xtt Xtt
(3)
Recently, Muzychka and Awad (2010) developed an alternative approach for predicting two-phase frictional pressure drop using superposition of three pressure gradients: singlephase liquid, single-phase vapor and interfacial pressure drop as follows:
dp dz
f,tp
=
dp dz
f,l
+
dp dz
f,i
+
dp dz
(4) f,g
using Eqs. (5)–(7) as follows: ˚2l,i =
(dp/dz)f,i (dp/dz)f,l
= ˚21 − 1 −
=
(dp/dz)f,tp − (dp/dz)f,l − (dp/dz)f,g (dp/dz)f,l
1
(8)
2 Xtt
From Eq. (8), it is clear that ˚2l,i cannot be negative because its minimum value is zero, which corresponds to the case of no contribution to the pressure gradient through phase interaction (i.e. (dp/dz)f,i = 0). Fig. 1 shows ˚l,i versus Lockhart–Martinelli’s parameter Xtt using Eqs. (1)–(3). Fig. 1 graphically illustrates ranges of the Lockhart–Martinelli parameter in turbulent–turbulent flow (Xtt ) for which the two-phase frictional multiplier correlation proposed by Changhong et al. (2005) are inappropriate. It is clear that at Xtt < 0.06 or at Xtt > 12, ˚l,i does not exist for the proposed correlation of Changhong et al. (2005) because ˚2l,i has negative values while ˚l,i exists for the proposed correlations of Chisholm (1967) and Mishima and Hibiki (1996) at all values of Xtt . Moreover, ˚l,i increases first with increasing the Lockhart–Martinelli’s parameter Xtt using the proposed correlation of Changhong et al. (2005) in the range 0.06 ≤ Xtt ≤ 0.1, then it decreases with increasing the Lockhart–Martinelli’s parameter Xtt while ˚l,i decreases always with increasing the Lockhart–Martinelli’s parameter Xtt using the proposed correlations of Chisholm (1967) and Mishima and Hibiki (1996). It can be seen that the present note presents an assessment of formulations for frictional pressure drop correlation for flow in narrow annuli. The Changhong et al. (2005) correlation results in negative values. Two other formulations (the proposed correlations
Rearranging Eq. (4), we obtain
dz
f,i
=
dp dz
f,tp
−
dp dz
f,l
−
dp dz
2 Xtt =
(dp/dz)f,tp (dp/dz)f,l ˚2g ˚21
=
0.1
f,g
Using the definitions of the two-phase multiplier for the liquid phase (˚2l ), and the Lockhart–Martinelli parameter for turbulent–turbulent flow (Xtt ). These two definitions were proposed by Lockhart and Martinelli (1949) as follows: ˚21 =
0.01 100
(5)
(dp/dz)f,l (dp/dz)f,g
(6)
10
100 100
Changhong et al. Chisholm, C = 20 Mishima and Hibiki (d E = 2.2 mm, C = 10.6) Mishima and Hibiki (d E = 3 mm, C = 12.9)
10
10
1
1
(7)
Dividing both sides of Eq. (5) by the single-phase liquid frictional pressure gradient (dp/dz)f,l , we can obtain an expression for the two-phase frictional interfacial multiplier for the liquid phase (˚2l,i ) 0029-5493/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2011.12.029
1
Φl,i
dp
0.1 0.01
0.1
1
10
Xtt Fig. 1. ˚l,i versus Lockhart–Martinelli’s parameter Xtt .
0.1 100
242
Comments on “Two-phase flow and boiling heat transfer in two vertical narrow annuli” / Nuclear Engineering and Design 245 (2012) 241–242
of Chisholm (1967) and Mishima and Hibiki (1996)) are compared against the Changhong et al. (2005) correlation and shown to result in positive values of the two-phase frictional interfacial multiplier for the liquid phase (˚2l,i ). Based on the present note, it is recommended to use the proposed correlations of Chisholm (1967) and Mishima and Hibiki (1996) at all values of Xtt while the proposed correlation of Changhong et al. (2005) can be used in the range 0.06 ≤ Xtt ≤ 12. Also, from Fig. 1, the proposed correlation of Changhong et al. (2005) seems to follow the trends and values of the correlation of Mishima and Hibiki (1996) (dE = 3 mm, C = 12.9) well for the range of 0.2 < Xtt < 2. Thus, the proposed correlation of Changhong et al. (2005) can be used if appropriate limits are applied. References Changhong, P., Yun, G., Suizheng, Q., Dounan, J., Changhua, N., 2005. Two-phase flow and boiling heat transfer in two vertical narrow annuli. Nuclear Engineering and Design 235 (16), 1737–1747. Chisholm, D., 1967. A theoretical basis for the Lockhart–Martinelli correlation for two-phase flow. International Journal of Heat and Mass Transfer 10 (12), 1767–1778.
Lockhart, R.W., Martinelli, R.C., 1949. Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chemical Engineering Progress Symposium Series 45 (1), 39–48. Mishima, K., Hibiki, T., 1996. Some characteristics of air-water two-phase flow in small diameter vertical tubes. International Journal of Multiphase Flow 22 (4), 703–712. Muzychka, Y.S., Awad, M.M., 2010. Asymptotic generalizations for the Lockhart–Martinelli method for two phase flows. ASME Journal of Fluids Engineering 132 (3) (Article (031302)).
M.M. Awad ∗ Mechanical Power Engineering Department, Faculty of Engineering, Mansoura University, Mansoura, 35516, Egypt ∗ Tel.:
+2 050 2244862; fax: +20 50 2244690. E-mail address: m m [email protected] 1 July 2011 23 December 2011 24 December 2011
Contents lists available at SciVerse ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Comments on “Two-phase flow and boiling heat transfer in two vertical narrow annuli” In the paper of Changhong et al. (2005), the authors expressed the two-phase frictional multiplier (˚l ) as a function of the Lockhart–Martinelli parameter in turbulent–turbulent flow (Xtt ) in order to predict the frictional pressure drop in the narrow annuli. Their correlation was ˚l =
3.956
(1)
0.536 Xtt
Also, the Chisholm correlation (1967) for turbulent–turbulent flow (C = 20) in terms of ˚2l was suggested as follows: ˚2l = 1 +
20 1 + 2 Xtt Xtt
(2)
Moreover, Mishima and Hibiki (1996) proposed a correlation for the parameter C as a function of the hydraulic diameter (dE ) to take into account the effect of the gap size. Their correlation was ˚2l = 1 +
21(1 − e−0.319dE ) 1 + 2 Xtt Xtt
(3)
Recently, Muzychka and Awad (2010) developed an alternative approach for predicting two-phase frictional pressure drop using superposition of three pressure gradients: singlephase liquid, single-phase vapor and interfacial pressure drop as follows:
dp dz
f,tp
=
dp dz
f,l
+
dp dz
f,i
+
dp dz
(4) f,g
using Eqs. (5)–(7) as follows: ˚2l,i =
(dp/dz)f,i (dp/dz)f,l
= ˚21 − 1 −
=
(dp/dz)f,tp − (dp/dz)f,l − (dp/dz)f,g (dp/dz)f,l
1
(8)
2 Xtt
From Eq. (8), it is clear that ˚2l,i cannot be negative because its minimum value is zero, which corresponds to the case of no contribution to the pressure gradient through phase interaction (i.e. (dp/dz)f,i = 0). Fig. 1 shows ˚l,i versus Lockhart–Martinelli’s parameter Xtt using Eqs. (1)–(3). Fig. 1 graphically illustrates ranges of the Lockhart–Martinelli parameter in turbulent–turbulent flow (Xtt ) for which the two-phase frictional multiplier correlation proposed by Changhong et al. (2005) are inappropriate. It is clear that at Xtt < 0.06 or at Xtt > 12, ˚l,i does not exist for the proposed correlation of Changhong et al. (2005) because ˚2l,i has negative values while ˚l,i exists for the proposed correlations of Chisholm (1967) and Mishima and Hibiki (1996) at all values of Xtt . Moreover, ˚l,i increases first with increasing the Lockhart–Martinelli’s parameter Xtt using the proposed correlation of Changhong et al. (2005) in the range 0.06 ≤ Xtt ≤ 0.1, then it decreases with increasing the Lockhart–Martinelli’s parameter Xtt while ˚l,i decreases always with increasing the Lockhart–Martinelli’s parameter Xtt using the proposed correlations of Chisholm (1967) and Mishima and Hibiki (1996). It can be seen that the present note presents an assessment of formulations for frictional pressure drop correlation for flow in narrow annuli. The Changhong et al. (2005) correlation results in negative values. Two other formulations (the proposed correlations
Rearranging Eq. (4), we obtain
dz
f,i
=
dp dz
f,tp
−
dp dz
f,l
−
dp dz
2 Xtt =
(dp/dz)f,tp (dp/dz)f,l ˚2g ˚21
=
0.1
f,g
Using the definitions of the two-phase multiplier for the liquid phase (˚2l ), and the Lockhart–Martinelli parameter for turbulent–turbulent flow (Xtt ). These two definitions were proposed by Lockhart and Martinelli (1949) as follows: ˚21 =
0.01 100
(5)
(dp/dz)f,l (dp/dz)f,g
(6)
10
100 100
Changhong et al. Chisholm, C = 20 Mishima and Hibiki (d E = 2.2 mm, C = 10.6) Mishima and Hibiki (d E = 3 mm, C = 12.9)
10
10
1
1
(7)
Dividing both sides of Eq. (5) by the single-phase liquid frictional pressure gradient (dp/dz)f,l , we can obtain an expression for the two-phase frictional interfacial multiplier for the liquid phase (˚2l,i ) 0029-5493/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2011.12.029
1
Φl,i
dp
0.1 0.01
0.1
1
10
Xtt Fig. 1. ˚l,i versus Lockhart–Martinelli’s parameter Xtt .
0.1 100
242
Comments on “Two-phase flow and boiling heat transfer in two vertical narrow annuli” / Nuclear Engineering and Design 245 (2012) 241–242
of Chisholm (1967) and Mishima and Hibiki (1996)) are compared against the Changhong et al. (2005) correlation and shown to result in positive values of the two-phase frictional interfacial multiplier for the liquid phase (˚2l,i ). Based on the present note, it is recommended to use the proposed correlations of Chisholm (1967) and Mishima and Hibiki (1996) at all values of Xtt while the proposed correlation of Changhong et al. (2005) can be used in the range 0.06 ≤ Xtt ≤ 12. Also, from Fig. 1, the proposed correlation of Changhong et al. (2005) seems to follow the trends and values of the correlation of Mishima and Hibiki (1996) (dE = 3 mm, C = 12.9) well for the range of 0.2 < Xtt < 2. Thus, the proposed correlation of Changhong et al. (2005) can be used if appropriate limits are applied. References Changhong, P., Yun, G., Suizheng, Q., Dounan, J., Changhua, N., 2005. Two-phase flow and boiling heat transfer in two vertical narrow annuli. Nuclear Engineering and Design 235 (16), 1737–1747. Chisholm, D., 1967. A theoretical basis for the Lockhart–Martinelli correlation for two-phase flow. International Journal of Heat and Mass Transfer 10 (12), 1767–1778.
Lockhart, R.W., Martinelli, R.C., 1949. Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chemical Engineering Progress Symposium Series 45 (1), 39–48. Mishima, K., Hibiki, T., 1996. Some characteristics of air-water two-phase flow in small diameter vertical tubes. International Journal of Multiphase Flow 22 (4), 703–712. Muzychka, Y.S., Awad, M.M., 2010. Asymptotic generalizations for the Lockhart–Martinelli method for two phase flows. ASME Journal of Fluids Engineering 132 (3) (Article (031302)).
M.M. Awad ∗ Mechanical Power Engineering Department, Faculty of Engineering, Mansoura University, Mansoura, 35516, Egypt ∗ Tel.:
+2 050 2244862; fax: +20 50 2244690. E-mail address: m m [email protected] 1 July 2011 23 December 2011 24 December 2011