Modelling of heat transport in two-phase flow and of mass transfer between phases using the level-set method

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1 Modelling of heat transport in two-phase flow and of mass transfer between phases using the level-set method TGTC-3 Magnus Aashammer Gjennestad and Svend Tollak Munkejord SINTEF Energy Research 1

2 Overview of presentation 2 Motivation The mathematical model Some simulation results Conclusions SINTEF Energy Research 2

3 Overview of presentation 3 Motivation The mathematical model Some simulation results Conclusions SINTEF Energy Research 3

4 Motivation 4 Purpose of work SINTEF Energy Research 4

5 Motivation 4 Purpose of work To gain insight into fundamental phenomena occurring in heat exchangers in liquefaction plants. SINTEF Energy Research 4

6 Motivation 4 Purpose of work To gain insight into fundamental phenomena occurring in heat exchangers in liquefaction plants. Basic hypothesis SINTEF Energy Research 4

7 Motivation 4 Purpose of work To gain insight into fundamental phenomena occurring in heat exchangers in liquefaction plants. Basic hypothesis A thorough understanding of the processes and phenomena occurring at a small-scale level in the heat exchanger is necessary to obtain an improved understanding of the heat exchanger, its design and operation. SINTEF Energy Research 4

8 Motivation 5 Why do we need an increased understanding and why use mathematical models and do simulations? SINTEF Energy Research 5

9 Motivation 5 Why do we need an increased understanding and why use mathematical models and do simulations? Liquefaction of natural gas requires energy. Naturally, we want to make the liquefaction process as energy-efficient as possible. SINTEF Energy Research 5

10 Motivation 5 Why do we need an increased understanding and why use mathematical models and do simulations? Liquefaction of natural gas requires energy. Naturally, we want to make the liquefaction process as energy-efficient as possible. A thorough understanding is necessary to design more efficient heat exchangers. SINTEF Energy Research 5

11 Motivation 5 Why do we need an increased understanding and why use mathematical models and do simulations? Liquefaction of natural gas requires energy. Naturally, we want to make the liquefaction process as energy-efficient as possible. A thorough understanding is necessary to design more efficient heat exchangers. There seems to be a general consensus that numerical simulation is one of the most promising approaches for studying phenomena such as heat transfer characteristics and condensation/boiling. SINTEF Energy Research 5

12 Overview of presentation 6 Motivation The mathematical model Some simulation results Conclusions SINTEF Energy Research 6

13 The flow model 7 SINTEF Energy Research 7

14 The flow model 7 The flow model consists of the incompressible Navier Stokes equations. SINTEF Energy Research 7

15 The flow model 7 The flow model consists of the incompressible Navier Stokes equations. We assume incompressibility, u = 0, SINTEF Energy Research 7

16 The flow model 7 The flow model consists of the incompressible Navier Stokes equations. We assume incompressibility, and momentum balance, u = 0, ρ ( t u + (u ) u) = p + µ u + f b + f s. SINTEF Energy Research 7

17 The flow model 7 The flow model consists of the incompressible Navier Stokes equations. We assume incompressibility, and momentum balance, u = 0, ρ ( t u + (u ) u) = p + µ u + f b + f s. SINTEF Energy Research 7

18 The flow model 7 The flow model consists of the incompressible Navier Stokes equations. We assume incompressibility, and momentum balance, u = 0, ρ ( t u + (u ) u) = p + µ u + f b + f s. SINTEF Energy Research 7

19 The flow model 7 The flow model consists of the incompressible Navier Stokes equations. We assume incompressibility, and momentum balance, u = 0, ρ ( t u + (u ) u) = p + µ u + f b + f s. SINTEF Energy Research 7

20 The flow model 7 The flow model consists of the incompressible Navier Stokes equations. We assume incompressibility, and momentum balance, u = 0, ρ ( t u + (u ) u) = p + µ u + f b + f s. SINTEF Energy Research 7

21 The flow model 7 The flow model consists of the incompressible Navier Stokes equations. We assume incompressibility, and momentum balance, u = 0, ρ ( t u + (u ) u) = p + µ u + f b + f s. SINTEF Energy Research 7

22 The heat transport model 8 SINTEF Energy Research 8

23 The heat transport model 8 Heat transfer is modelled by keeping track of temperature. SINTEF Energy Research 8

24 The heat transport model 8 Heat transfer is modelled by keeping track of temperature. We introduce an advection-diffusion equation for the temperature, ρc p ( t T + u T ) = κ T. SINTEF Energy Research 8

25 The heat transport model 8 Heat transfer is modelled by keeping track of temperature. We introduce an advection-diffusion equation for the temperature, ρc p ( t T + u T ) = κ T. SINTEF Energy Research 8

26 The heat transport model 8 Heat transfer is modelled by keeping track of temperature. We introduce an advection-diffusion equation for the temperature, ρc p ( t T + u T ) = κ T. SINTEF Energy Research 8

27 The heat transport model 8 Heat transfer is modelled by keeping track of temperature. We introduce an advection-diffusion equation for the temperature, ρc p ( t T + u T ) = κ T. SINTEF Energy Research 8

28 The heat transport model 8 Heat transfer is modelled by keeping track of temperature. We introduce an advection-diffusion equation for the temperature, ρc p ( t T + u T ) = κ T. SINTEF Energy Research 8

29 The heat transport model 8 Heat transfer is modelled by keeping track of temperature. We introduce an advection-diffusion equation for the temperature, ρc p ( t T + u T ) = κ T. To model temperature-driven flows, we need to introduce a temperature-dependent buoyancy force, f b = ρg (1 β (T T )), where β is the thermal expansion coefficient and T is a reference temperature. SINTEF Energy Research 8

30 The level-set method 9 SINTEF Energy Research 9

31 The level-set method 9 The interface Γ is implicitly defined as the zero isocontour of the level-set function φ, Γ = { x φ (x) = 0 }. SINTEF Energy Research 9

32 The level-set method 9 The interface Γ is implicitly defined as the zero isocontour of the level-set function φ, Γ = { x φ (x) = 0 }. The level-set function φ is the signed distance to the interface, min y Γ x y if x in Phase 1, φ (x) = min y Γ x y if x in Phase 2. SINTEF Energy Research 9

33 The level-set method 10 SINTEF Energy Research 10

34 The level-set method 10 The time-evolution of the level-set function is determined by t φ + w φ = 0. where w is the interface velocity. SINTEF Energy Research 10

35 The level-set method 10 The time-evolution of the level-set function is determined by t φ + w φ = 0. where w is the interface velocity. The normal vector field n is defined in terms of φ as n = φ. SINTEF Energy Research 10

36 An example level-set function 11 SINTEF Energy Research 11

37 The phase transition model 12 SINTEF Energy Research 12

38 The phase transition model 12 We model mass transfer between phases as a result of vaporisation and/or condensation. SINTEF Energy Research 12

39 The phase transition model 12 We model mass transfer between phases as a result of vaporisation and/or condensation. It is assumed that the interface temperature is equal to the saturation temperature of the liquid. SINTEF Energy Research 12

40 The phase transition model 12 We model mass transfer between phases as a result of vaporisation and/or condensation. It is assumed that the interface temperature is equal to the saturation temperature of the liquid. Any resulting discontinuity in heat flux q = κ T at the interface is used to calculate the mass flux ṁ = [ q n] h, where h is the specific enthalpy difference of the phase transition. SINTEF Energy Research 12

41 The phase transition model 13 The mass flux is used to calculate the jump in velocity [u] at the interface, [ ] 1 [u] = ṁ n. ρ SINTEF Energy Research 13

42 The phase transition model 13 The mass flux is used to calculate the jump in velocity [u] at the interface, [ ] 1 [u] = ṁ n. ρ The interface velocity is found from w = u 1 ṁ ρ 1 n. SINTEF Energy Research 13

43 Overview of presentation 14 Motivation The mathematical model Some simulation results Conclusions SINTEF Energy Research 14

44 15 Lava lamp SINTEF Energy Research 15

45 15 Lava lamp Photo: Nir Schneider, CC BY 2.0. SINTEF Energy Research 15

46 15 Lava lamp Why simulate the lava lamp? Photo: Nir Schneider, CC BY 2.0. SINTEF Energy Research 15

47 15 Lava lamp Why simulate the lava lamp? The lava lamp is a canonical example of a temperature-driven two-phase flow. Photo: Nir Schneider, CC BY 2.0. SINTEF Energy Research 15

48 15 Lava lamp Why simulate the lava lamp? The lava lamp is a canonical example of a temperature-driven two-phase flow. Frequency of blob exchange oscillations can be compared to experiment. Photo: Nir Schneider, CC BY 2.0. SINTEF Energy Research 15

49 Lava lamp 16 SINTEF Energy Research 16

50 Lava lamp t = 500 s y [m] x [m] SINTEF Energy Research 16

51 Lava lamp t = 500 s y [m] y [m] t = 766 s x [m] x [m] SINTEF Energy Research 16

52 Lava lamp t = 500 s y [m] y [m] t = 766 s x [m] x [m] t = 775 s y [m] x [m] SINTEF Energy Research 16

53 Lava lamp t = 500 s y [m] y [m] t = 766 s x [m] x [m] t = 775 s y [m] y [m] t = 1214 s x [m] x [m] SINTEF Energy Research 16

54 Boiling film 17 SINTEF Energy Research 17

55 Boiling film 17 t = 0.3 s y [m] x [m] SINTEF Energy Research 17

56 Boiling film 17 t = 0.3 s y [m] x [m] y [m] x [m] t = 0.4 s SINTEF Energy Research 17

57 Boiling film 17 t = 0.3 s t = 0.5 s y [m] y [m] x [m] x [m] y [m] x [m] t = 0.4 s SINTEF Energy Research 17

58 Boiling film 17 t = 0.3 s t = 0.5 s y [m] y [m] x [m] x [m] y [m] y [m] x [m] x [m] t = 0.4 s t = 0.6 s SINTEF Energy Research 17

59 Boiling film 18 SINTEF Energy Research 18