Figure 1.
System Schematics.
Figure 1.
System Schematics.
Figure 2.
First four modes axial dimensionless complex frequency as a function of dimensionless single-phase flow velocity.
Figure 2.
First four modes axial dimensionless complex frequency as a function of dimensionless single-phase flow velocity.
Figure 3.
First four modes transverse dimensionless complex frequency as a function of dimensionless single-phase flow velocity.
Figure 3.
First four modes transverse dimensionless complex frequency as a function of dimensionless single-phase flow velocity.
Figure 4.
Nonlinear frequency–amplitude variations for various void fractions.
Figure 4.
Nonlinear frequency–amplitude variations for various void fractions.
Figure 5.
Nonlinear frequency response for various void fractions.
Figure 5.
Nonlinear frequency response for various void fractions.
Figure 6.
Nonlinear frequency–amplitude variations for various temperature differences.
Figure 6.
Nonlinear frequency–amplitude variations for various temperature differences.
Figure 7.
Nonlinear frequency response for various temperature differences.
Figure 7.
Nonlinear frequency response for various temperature differences.
Figure 8.
Nonlinear frequency–amplitude variations for various pressures.
Figure 8.
Nonlinear frequency–amplitude variations for various pressures.
Figure 9.
Nonlinear frequency response for various pressures.
Figure 9.
Nonlinear frequency response for various pressures.
Figure 10.
Nonlinear frequency–amplitude variations for various top tensions.
Figure 10.
Nonlinear frequency–amplitude variations for various top tensions.
Figure 11.
Nonlinear frequency response for various top tensions.
Figure 11.
Nonlinear frequency response for various top tensions.
Figure 12.
(a) Time history and (b) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3.
Figure 12.
(a) Time history and (b) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3.
Figure 13.
(a) Time history and (b) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4.
Figure 13.
(a) Time history and (b) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4.
Figure 14.
(a) Time history and (b) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5.
Figure 14.
(a) Time history and (b) phase plots of uncoupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5.
Figure 15.
(a) Time history and (b) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3.
Figure 15.
(a) Time history and (b) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3.
Figure 16.
(a) Time history and (b) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4.
Figure 16.
(a) Time history and (b) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4.
Figure 17.
(a) Time history and (b) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5.
Figure 17.
(a) Time history and (b) phase plots of uncoupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5.
Figure 18.
(a) Time history and (b) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3, σ of 2.0.
Figure 18.
(a) Time history and (b) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3, σ of 2.0.
Figure 19.
(a) Time history and (b) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3, σ of 2.0.
Figure 19.
(a) Time history and (b) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.3, σ of 2.0.
Figure 20.
(a) Time history and (b) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4, σ of 2.0.
Figure 20.
(a) Time history and (b) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4, σ of 2.0.
Figure 21.
(a) Time history and (b) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4, σ of 2.0.
Figure 21.
(a) Time history and (b) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.4, σ of 2.0.
Figure 22.
(a) Time history and (b) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5, σ of 2.0.
Figure 22.
(a) Time history and (b) phase plots of coupled transverse vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5, σ of 2.0.
Figure 23.
(a) Time history and (b) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5, σ of 2.0.
Figure 23.
(a) Time history and (b) phase plots of coupled axial vibrations of the tip of a cantilever pipe conveying two-phase flow with void fraction of 0.5, σ of 2.0.
Table 1.
Summary of pipe and flow parameters.
Table 1.
Summary of pipe and flow parameters.
Parameter Name | Parameter Unit | Parameter Values |
---|
External Diameter | Do (m) | 0.0113772 |
Internal Diameter | Di (m) | 0.00925 |
Length | L (m) | 0.1467 |
Pipe density | ρpipe (kg/m3) | 7800 |
Gas density | ρGas (kg/m3) | 1.225 |
Water density | ρWater (kg/m3) | 1000 |
Tensile and compressive stiffness | EA (N) | 7.24 × 106 |
Bending stiffness | EI (N) | 1.56 × 103 |
Table 2.
Summary of the linear single-phase solution’s critical flow velocity.
Table 2.
Summary of the linear single-phase solution’s critical flow velocity.
Fluid | Void Fraction | Liquid | Gas | Liquid | Gas | Critical Velocity |
---|
Transverse | Axial |
---|
Single Phase | NA | 0.2 | 0.0 | 1.0 | 0.0 | 5.653 | 14.149 |
Table 3.
Summary of the linear two-phase solution of critical flow velocities.
Table 3.
Summary of the linear two-phase solution of critical flow velocities.
Fluid | Void Fraction | Liquid | Gas | Liquid | Gas | Critical Mixture Velocity |
---|
Transverse * | Axial |
---|
Two-Phase | 0.3 | 0.19998 | 0.00010 | 0.99948 | 0.00052 | 12.505 | 31.634 |
Two-Phase | 0.4 | 0.19997 | 0.00016 | 0.99918 | 0.00082 | 13.349 | 33.750 |
Two-Phase | 0.5 | 0.19995 | 0.00024 | 0.99878 | 0.00122 | 14.613 | 36.966 |
Table 4.
Summary of the linear two-phase solution of critical flow velocities for varying temperature difference.
Table 4.
Summary of the linear two-phase solution of critical flow velocities for varying temperature difference.
Parameter | Void Fraction | Thermal Expansivity | Critical Mixture Velocity |
---|
Transverse * | Axial |
---|
DT = 0 | 0.3 | 0.002 | 12.505 | 31.634 |
DT = 40 | 0.3 | 0.002 | 9.253 | 31.634 |
DT = 50 | 0.3 | 0.002 | 8.237 | 31.634 |
Table 5.
Summary of the linear two-phase solution of critical flow velocities for varying pressurization.
Table 5.
Summary of the linear two-phase solution of critical flow velocities for varying pressurization.
Parameter | Void Fraction | Critical Mixture Velocity |
---|
Transverse * | Axial |
---|
| 0.3 | 12.505 | 31.634 |
| 0.3 | 10.596 | 31.634 |
| 0.3 | 8.237 | 31.634 |
Table 6.
Summary of the linear two-phase solution of critical flow velocities for varying top tensions.
Table 6.
Summary of the linear two-phase solution of critical flow velocities for varying top tensions.
Parameter | Void Fraction | Critical Velocity |
---|
Transverse * | Axial |
---|
| 0.3 | 12.505 | 31.634 |
| 0.3 | 14.155 | 31.634 |
| 0.3 | 10.596 | 31.634 |