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ABSTRACT
This contribution presents the mathematical model to simulate the swirling turbulent gas-droplet flow in a sudden pipe expansion. The set of axisymmetrical steady-state Reynolds averaged Navier–Stokes equations (RANS) for the two-phase flow is utilized. The dispersed phase is modeled by the Eulerian approach. The flow swirl causes an increase in the intensity of heat transfer (more than 1.5 times compared with the nonswirling mist flow at other identical inlet conditions). Evaporation of the droplets leads to a significant increase in the heat transfer intensity in the swirling two-phase flow (more than 2.5 times compared with the single-phase flow).
Nomenclature
| Bi | = | Biot number |
| b1D | = | diffusion parameter of injection, determined with the use of the saturation curve |
| d | = | droplet diameter, m |
| CD | = | drag coefficient of the evaporating droplet, written by taking into account the deviation of the Stokes law |
| CP | = | specific heat capacity, J kg−1 K−1 |
| D | = | diffusion coefficient, m2 s−1 |
| G | = | mass flow rate, kg s−1 |
| J | = | mass flux of steam from the surface of the evaporating droplet, kg m−2 s−1 |
| ER | = | heat transfer enhancement ratio |
| H | = | step height, m |
| = | steam mass concentration at the “vapor-gas mixture–droplet” interface, corresponding to the saturation parameters at droplet temperature TL | |
| KV0 | = | mass concentration of steam far from the droplet |
| k | = | k = ⟨ui′ui′⟩/2 turbulent kinetic energy, m2 s−2 |
| L | = | latent heat of vaporization, J kg−1 |
| ML | = | droplets mass fraction |
| Nu | = | Nusselt number |
| P | = | pressure (Pa) |
| Pr | = | Prandtl number |
| Pij | = | stress production term, m s−2 |
| 2R1 | = | pipe diameter before sudden expansion, m |
| 2R2 | = | pipe diameter after sudden expansion, m |
| Re | = | Re = Um12R1/ν gas-phase Reynolds number |
| ReL | = | ReL = |U − UL|d/ν Reynolds number of the dispersed phase |
| r | = | radial coordinate, m |
| S | = | swirl number |
| Sc | = | Schmidt number |
| Sh | = | Sherwood number |
| StD | = | diffusional Stanton number |
| Stk | = | Stk = τ/τf Stokes number in the mean motion |
| T | = | temperature (K) |
| Ui | = | components of mean velocity of the gas phase, m s−1 |
| ULi | = | components of droplets mean velocity, m s−1 |
| ⟨u′⟩, ⟨v′⟩, ⟨w′⟩ | = | components of velocity pulsation, m s−1 |
| ⟨u′v′⟩ | = | turbulent Reynolds stresses, m2 s−2 |
| ⟨uLi′uLj′⟩ | = | kinetic stresses in the dispersed phase, m2 s−2 |
| x | = | axial coordinate, m |
| xR | = | separation length, m |
| y | = | distance normal to the wall, m |
| We | = | Weber number |
| We* | = | critical Weber number |
| Greek letters | = | |
| Φ | = | Φ = MLρ/ρL volumetric concentration of the dispersed phase |
| ΓE | = | turbulence macroscale, m |
| ΛT | = | the turbulent macroscale time, s |
| ΩtL | = | time of interaction with temperature pulsations of the carrying flow, s |
| ΩLag | = | Lagrangian integral timescales, s |
| ΩE | = | Eulerian integral timescales, s |
| Ωϵ | = | time microscale of turbulence, s |
| αL | = | heat transfer coefficient for the evaporating droplet, W m−2 |
| β | = | blending coefficient |
| ϵ | = | dissipation of the turbulent kinetic energy, m2s−3 |
| ϕij | = | the velocity–pressure–gradient correlation (the pressure term), m s−2 |
| μ | = | dynamic viscosity, kg m−1 s−1 |
| ν | = | kinematic viscosity, m2 s−1 |
| ρ | = | density, kg m−3 |
| σ | = | surface tension, N m−1 |
| τ | = | dynamic relaxation time of droplets, s |
| τf | = | turbulent timescale, s |
| Subscripts | = | |
| 0 | = | parameter under conditions on the pipe axis |
| 1 | = | parameter under inlet conditions |
| A | = | air |
| L | = | droplet |
| P | = | solid particles (without the evaporation process) |
| S | = | parameter in swirling flow |
| T | = | turbulent parameter |
| V | = | steam |
| W | = | parameter on the wall condition |
| m | = | mean-mass parameter |
| Superscripts | = | |
| * | = | parameter on the droplet surface |
| M | = | mist |
| S | = | swirling |
| Acronym | = | |
| LRN | = | low-Reynolds number |
| RANS | = | Reynolds averaged Navier–Stokes equations |
| SMC | = | second moment closure |
| TKE | = | turbulent kinetic energy |
Nomenclature
| Bi | = | Biot number |
| b1D | = | diffusion parameter of injection, determined with the use of the saturation curve |
| d | = | droplet diameter, m |
| CD | = | drag coefficient of the evaporating droplet, written by taking into account the deviation of the Stokes law |
| CP | = | specific heat capacity, J kg−1 K−1 |
| D | = | diffusion coefficient, m2 s−1 |
| G | = | mass flow rate, kg s−1 |
| J | = | mass flux of steam from the surface of the evaporating droplet, kg m−2 s−1 |
| ER | = | heat transfer enhancement ratio |
| H | = | step height, m |
| = | steam mass concentration at the “vapor-gas mixture–droplet” interface, corresponding to the saturation parameters at droplet temperature TL | |
| KV0 | = | mass concentration of steam far from the droplet |
| k | = | k = ⟨ui′ui′⟩/2 turbulent kinetic energy, m2 s−2 |
| L | = | latent heat of vaporization, J kg−1 |
| ML | = | droplets mass fraction |
| Nu | = | Nusselt number |
| P | = | pressure (Pa) |
| Pr | = | Prandtl number |
| Pij | = | stress production term, m s−2 |
| 2R1 | = | pipe diameter before sudden expansion, m |
| 2R2 | = | pipe diameter after sudden expansion, m |
| Re | = | Re = Um12R1/ν gas-phase Reynolds number |
| ReL | = | ReL = |U − UL|d/ν Reynolds number of the dispersed phase |
| r | = | radial coordinate, m |
| S | = | swirl number |
| Sc | = | Schmidt number |
| Sh | = | Sherwood number |
| StD | = | diffusional Stanton number |
| Stk | = | Stk = τ/τf Stokes number in the mean motion |
| T | = | temperature (K) |
| Ui | = | components of mean velocity of the gas phase, m s−1 |
| ULi | = | components of droplets mean velocity, m s−1 |
| ⟨u′⟩, ⟨v′⟩, ⟨w′⟩ | = | components of velocity pulsation, m s−1 |
| ⟨u′v′⟩ | = | turbulent Reynolds stresses, m2 s−2 |
| ⟨uLi′uLj′⟩ | = | kinetic stresses in the dispersed phase, m2 s−2 |
| x | = | axial coordinate, m |
| xR | = | separation length, m |
| y | = | distance normal to the wall, m |
| We | = | Weber number |
| We* | = | critical Weber number |
| Greek letters | = | |
| Φ | = | Φ = MLρ/ρL volumetric concentration of the dispersed phase |
| ΓE | = | turbulence macroscale, m |
| ΛT | = | the turbulent macroscale time, s |
| ΩtL | = | time of interaction with temperature pulsations of the carrying flow, s |
| ΩLag | = | Lagrangian integral timescales, s |
| ΩE | = | Eulerian integral timescales, s |
| Ωϵ | = | time microscale of turbulence, s |
| αL | = | heat transfer coefficient for the evaporating droplet, W m−2 |
| β | = | blending coefficient |
| ϵ | = | dissipation of the turbulent kinetic energy, m2s−3 |
| ϕij | = | the velocity–pressure–gradient correlation (the pressure term), m s−2 |
| μ | = | dynamic viscosity, kg m−1 s−1 |
| ν | = | kinematic viscosity, m2 s−1 |
| ρ | = | density, kg m−3 |
| σ | = | surface tension, N m−1 |
| τ | = | dynamic relaxation time of droplets, s |
| τf | = | turbulent timescale, s |
| Subscripts | = | |
| 0 | = | parameter under conditions on the pipe axis |
| 1 | = | parameter under inlet conditions |
| A | = | air |
| L | = | droplet |
| P | = | solid particles (without the evaporation process) |
| S | = | parameter in swirling flow |
| T | = | turbulent parameter |
| V | = | steam |
| W | = | parameter on the wall condition |
| m | = | mean-mass parameter |
| Superscripts | = | |
| * | = | parameter on the droplet surface |
| M | = | mist |
| S | = | swirling |
| Acronym | = | |
| LRN | = | low-Reynolds number |
| RANS | = | Reynolds averaged Navier–Stokes equations |
| SMC | = | second moment closure |
| TKE | = | turbulent kinetic energy |