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The onset of liquid entrainment from a stratified two-phase region through small branches

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Abstract

A new criterion has been developed to predict the onset of liquid (heavier fluid) entrainment from a stratified two-phase region. The criterion was developed based on the local instability of the interface between two fluids due to the suction effect associated with the discharging of the lighter fluid. To validate the proposed criterion, comparisons were conducted between the measured critical height at the onset and those predicted using a three-dimensional analysis of the flow through two configurations: (1) a single branch mounted on an inclined wall with an inclination angle ranging between −90° and + 60° and (2) dual branches mounted on a vertical wall with the plane passing through the branch centerlines and inclined with an angle α ranging between 0° and 60°. Comparisons demonstrate a very good agreement between the predicted and the measured values for both single and dual branches. This verifies that the onset of liquid entrainment mechanism occurs due to local flow instability of the interface, analogous to Rayleigh–Taylor instability.

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Abbreviations

d :

Branch diameter (m)

Fr :

Froude number of a single branch, as shown in Fig. 1

Fr 1 :

Froude number of the lower branch as shown in Fig. 2

Fr 2 :

Froude number of the upper branch as shown in Fig. 2

g :

Gravitational acceleration (m/s2)

g*:

Dimensionless gravitational acceleration \({\left(\frac{{gr}_{\rm o}}{{v}_{d}^2 }\right)}\)

h OLE :

Critical height corresponding to the onset of liquid entrainment (m)

h :

Distance between the single branch centerline and point (C), as defined in Fig. 1 (m)

h 1 :

Distance between the lower branch centerline and the point (C), as defined in Fig. 2 (m)

h 2 :

Distance between the upper branch centerline and the point (C), as defined in Fig. 2 (m)

\({I_{1}, I_{2}, \ldots, I_{7}}\) :

Integral functions defined in “Appendix”

L :

Distance between the centerlines of the two branches, as shown in Fig. 2 (m)

L*:

Dimensionless distance between the centerlines of the two branches (L/r o )

P A :

Static pressure of the interface at point (A) (N/m 2)

P B :

Static pressure of the interface at point (B) (N/m 2)

P C :

Static pressure of the interface at point (C) (N/m 2)

r o :

Branch radius (m)

S :

Distance between the branch centerline and the point (B) as defined in Fig. 1 (m)

S A :

Distance between the lower branch centerline and the point (A) as defined in Fig. 2 (m)

S B :

Distance between the upper branch centerline and the point (B) as defined in Fig. 2 (m)

\({S^{\ast}, S_{\rm A}^{\ast}, S_{\rm B}^{\ast}}\) :

Dimensionless distances (\({S/r_{o}, S_{\rm A}/r_{o}, S_{\rm B}/r_{o}}\))

V A :

Velocity of fluid at point (A), as defined in Fig. 1

V B :

Velocity of fluid at point (B), as defined in Figs. 1 and 2

V C :

Velocity of fluid at point (C), as defined in Figs. 1 and 2

V d :

Discharge velocity of the single branch (m/s)

\({V_{{d}_1}}\) :

Discharge velocity of the upper branch (m/s)

\({V_{{d}_2}}\) :

Discharge velocity of the lower branch (m/s)

\({V_{{d}_1}^\ast}\) :

Dimensionless discharge velocity of the upper branch \({\left({\frac{V_{d_1 }^\ast }{\sqrt{V_{d_1 }^2 +V_{d_2 }^2}}}\right)}\)

\({V_{{d}_2 }^\ast}\) :

Dimensionless discharge velocity of the lower branch \({\left({\frac{V_{d_2}^\ast }{\sqrt{V_{d_1}^2+V_{d_2 }^2 }}}\right)}\)

V x , V y , V z :

Velocity components in x, y and z directions (m/s)

\({V_{x}^{\ast}, V_{y}^{\ast}, V_{z}^{\ast}}\) :

Dimensionless velocity components in x*, y* and z* directions, \({\left( {\frac{V_x }{\sqrt{V_{d_1 }^2+V_{d_2 }^2 }},\;\frac{V_y }{\sqrt{V_{d_1 }^2 +V_{d_2 }^2}},\;\frac{V_z }{\sqrt{V_{d_1 }^2 +V_{d_2 }^2 }}}\right)}\)

x, y, z :

Cartesian coordinate system, as defined in Fig. 1

x*, y*, z* :

Dimensionless variables (x/r o , y/r o , z/r o )

x 1,y 1,z 1 :

Auxiliary set of Cartesian coordinate system, as defined in Figs. 1 and 2

\({x_1^{\ast} ,y_1^{\ast} ,z_1^{\ast}}\) :

Dimensionless variables

x 2,y 2,z 2 :

Cartesian coordinate system, as defined in Fig. 2

\({x_2^{\ast} ,y_2^{\ast} ,z_2^{\ast}}\) :

Dimensionless variables

\({\rho}\) :

Density of the lighter fluid (kg/m3)

\({\rho+\Delta \rho}\) :

Density of the heavier fluid (kg/m3)

\({\lambda, \beta}\) :

Variables changed from 0 to \({\infty}\).

\({\ominus}\) :

Wall inclination angle, as shown in Fig. 1,

\({\alpha}\) :

The inclination angle of the plane passing through the branch centerlines, as shown in Fig. 2

\({\varPhi ,\varPhi_1 ,\varPhi_2}\) :

Potential functions (m3/s)

\({\varPhi^{\ast},\varPhi _1^\ast ,\varPhi _2^\ast}\) :

Dimensionless potential functions

\({\varPhi_{y^{{\ast}}}^{\ast}}\) :

First derivative of the dimensionless potential function with respect to y*

\({\varPhi_{y^{{\ast}}y{\ast}}^{\ast}}\) :

Second derivative of the dimensionless potential function with respect to y*

\({\varPhi_{z^{{\ast}}}^{\ast}}\) :

First derivative of the dimensionless potential function with respect to z*

\({\varPhi_{z^{{\ast}}z^{{\ast}}}^{\ast}}\) :

Second derivative of the dimensionless potential function with respect to z*

\({\varPhi_{y^{{\ast}}z^{{\ast}}}^{\ast}}\) :

Second derivative of the dimensionless potential function with respect to y* and z*

\({\varPhi_{y_1^\ast }^{\ast}}\) :

First derivative of the dimensionless potential function with respect to \({y_{1}^{\ast}}\)

\({\varPhi_{y_1^\ast y_1^\ast }^{\ast}}\) :

Second derivative of the dimensionless potential function with respect to \({y_{1}^{\ast}}\)

\({\varPhi_{1y_1^\ast }^{\ast}}\) :

First derivative of the dimensionless potential function with respect to \({y_{1}^{\ast}}\)

\({\varPhi_{1y_1^\ast y_1^\ast }^{\ast}}\) :

Second derivative of the dimensionless potential function with respect to \({y_{1}^{\ast}}\)

\({\varPhi_{2y_1^\ast}^{\ast}}\) :

First derivative of the dimensionless potential function with respect to \({y_{1}^{\ast}}\)

\({\varPhi_{2y_1^\ast y_1^\ast }^{\ast}}\) :

Second derivative of the dimensionless potential function with respect to \({y_{1}^{\ast}}\)

\({\varPhi_{1y_2^\ast }^{\ast}}\) :

First derivative of the dimensionless potential function with respect to \({y_{2}^{\ast}}\)

\({\varPhi_{1y_2^\ast y_2^\ast }^{\ast}}\) :

Second derivative of the dimensionless potential function with respect to \({y_{2}^{\ast}}\)

\({\varPhi_{2y_2^\ast }^{\ast}}\) :

First derivative of the dimensionless potential function with respect to \({y_{2}^{\ast}}\)

\({\varPhi_{2y_2^\ast y_2^\ast }^{\ast}}\) :

Second derivative of the dimensionless potential function with respect to \({y_{2}^{\ast}}\)

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Authors and Affiliations

  1. Department of Mechanical Engineering, Assiut University, Assiut, 71516, Egypt

    Mahmoud Ahmed, M. S. Youssef & Ahmed Hamza H. Ali

  2. Department of Energy Resources Engineering, Egypt-Japan University of Science and Technology (E-JUST), New Borg El-Arab City, Alexandria, 21934, Egypt

    Ahmed Hamza H. Ali

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  1. Mahmoud Ahmed
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  2. M. S. Youssef
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  3. Ahmed Hamza H. Ali
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Correspondence to Mahmoud Ahmed.

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Ahmed, M., Youssef, M.S. & Ali, A.H.H. The onset of liquid entrainment from a stratified two-phase region through small branches. Acta Mech 225, 3023–3039 (2014). https://doi.org/10.1007/s00707-014-1098-0

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  • DOI: https://doi.org/10.1007/s00707-014-1098-0

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