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ABSTRACT
This paper presents an algorithm for accurately solving the full two-dimensional governing equations, along with the interface conditions that govern laminar/laminar annular/stratified internal condensing flows. The simulation approach - which can be generalized to adiabatic and evaporating flows, a 3-D level-set technique, and so on - uses a sharp-interface model, separate liquid and vapor domain computational solutions with interface conditions embedded as boundary conditions, and a moving grid technique to locate the dynamic wavy interface (in amplitude and phase) by a method of characteristics solution of the interface tracking equation. The moving grid is spatially fixed for a defined number of instants, but changes when the current marker instant advances in time.
Nomenclature
| Cp | = | Specific heat, J/(kg-K) |
| Frx | = | Froude number in x-direction U/(gxLc)1/2 |
| Fry | = | Froude number in y-direction U/(gyLc)1/2 |
| gx | = | Gravity component in x-direction, m/s2 |
| gy | = | Gravity component in y-direction, m/s2 |
| h | = | Cross-section height of the chanel, m |
| Ja | = | Condensate liquid Jakob number, Cp1 · ΔT/hfg(pin) |
| k | = | Conductivity, W/(m-K) |
| L | = | Length of the channel or test-section, m |
| = | Local interfacial mass flux, kg/m2-s | |
| p0 | = | Steady inlet pressure (also pin), kPa |
| Pr1 | = | Condensate liquid Prandtl number, μ1·Cp1/k1 |
| Rein | = | Inlet vapor Reynolds number, ρ2Uh/μ2 |
| t | = | Nondimensional time |
| = | Mean condensing surface temperature,°C | |
| Tsat(p) | = | Saturation temperature at pressure p,°C |
| U | = | Average inlet vapor velocity in the x-direction, m/s |
| uI | = | Nondimensional velocity in the x-direction |
| vI | = | Nondimensional velocity in the y-direction |
| w | = | Cross-sectional width of the channel, m |
| x, y | = | Nondimensional distances along and perpendicular to the condensing surface |
| xA | = | Nondimensional length of the annular regime |
| Δs | = | Mesh size, m |
| Greek symbols | = | |
| δ | = | Nondimensional value of condensate thickness |
| Δ | = | Physical value of condensate thickness, m |
| µ | = | Viscosity, kg/(m-s) |
| ρ | = | Density, kg/m3 |
| Subscripts | = | |
| 1 or L | = | Represents liquid phase of the flow variable |
| 2 or V | = | Represents vapor phase of the flow variable |
| Superscripts | = | |
| p | = | Represents physical variable, e.g., xp – physical distance along x axis |
| i | = | Value of the flow variable at the interface |
Nomenclature
| Cp | = | Specific heat, J/(kg-K) |
| Frx | = | Froude number in x-direction U/(gxLc)1/2 |
| Fry | = | Froude number in y-direction U/(gyLc)1/2 |
| gx | = | Gravity component in x-direction, m/s2 |
| gy | = | Gravity component in y-direction, m/s2 |
| h | = | Cross-section height of the chanel, m |
| Ja | = | Condensate liquid Jakob number, Cp1 · ΔT/hfg(pin) |
| k | = | Conductivity, W/(m-K) |
| L | = | Length of the channel or test-section, m |
| = | Local interfacial mass flux, kg/m2-s | |
| p0 | = | Steady inlet pressure (also pin), kPa |
| Pr1 | = | Condensate liquid Prandtl number, μ1·Cp1/k1 |
| Rein | = | Inlet vapor Reynolds number, ρ2Uh/μ2 |
| t | = | Nondimensional time |
| = | Mean condensing surface temperature,°C | |
| Tsat(p) | = | Saturation temperature at pressure p,°C |
| U | = | Average inlet vapor velocity in the x-direction, m/s |
| uI | = | Nondimensional velocity in the x-direction |
| vI | = | Nondimensional velocity in the y-direction |
| w | = | Cross-sectional width of the channel, m |
| x, y | = | Nondimensional distances along and perpendicular to the condensing surface |
| xA | = | Nondimensional length of the annular regime |
| Δs | = | Mesh size, m |
| Greek symbols | = | |
| δ | = | Nondimensional value of condensate thickness |
| Δ | = | Physical value of condensate thickness, m |
| µ | = | Viscosity, kg/(m-s) |
| ρ | = | Density, kg/m3 |
| Subscripts | = | |
| 1 or L | = | Represents liquid phase of the flow variable |
| 2 or V | = | Represents vapor phase of the flow variable |
| Superscripts | = | |
| p | = | Represents physical variable, e.g., xp – physical distance along x axis |
| i | = | Value of the flow variable at the interface |
Acknowledgment
This work was supported by NSF Grants CBET-1033591 and CBET-1402702.