Numerical and analytical study of film boiling in a planar liquid jet

International Communications in Heat and Mass Transfer 46 (2013) 42–48

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Numerical and analytical study of film boiling in a planar liquid jet☆ Woorim Lee, Gihun Son ⁎ Department of Mechanical Engineering, Sogang University, Seoul 121–742, South Korea

a r t i c l e

i n f o

Available online 7 June 2013 Keywords: Direct numerical simulation Film boiling Level-set method Liquid jet impingement

a b s t r a c t Direct numerical simulation of film boiling in a planar liquid jet is performed by solving the conservation equations of mass, momentum and energy in the liquid, vapor and air phases. The liquid–air and liquid– vapor interfaces are tracked by a sharp-interface level-set method, which is modified to include the effect of phase change at the liquid–vapor interface. An analytical model to predict the vapor film thickness and wall heat flux in the stagnation region is also developed by simplifying the momentum and energy equations in the liquid and vapor phases. The computational results show a stable vapor film formation on the wall. The effects of jet subcooling, jet velocity, and wall temperature on the vapor film thickness and boiling heat transfer are investigated. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Liquid jet impingement is one of the primary cooling techniques of hot metal plates in the metal production industry. When the plate has a very high temperature, as in its initial cooling stage, film boiling occurs and a continuous vapor film is formed between the hot plate and the impinging liquid. An understanding and prediction of the boiling process is essentially important to achieve the desired mechanical and metallurgical properties of metals. Several experimental studies of film boiling in liquid jet impingement were reported in the literature [1–4], but most of them were conducted under transient (or quenching) conditions, where film, transition, and nucleate boiling modes appear sequentially or simultaneously on the wall surface and thus the hydrodynamic and thermal characteristics of film boiling are not clearly separated from the other boiling modes. Also, considering that the temperature and heat flux distributions on the wall surface were obtained from the extrapolation of the measured data below the wall surface using an inverse heat conduction technique, the transient experimental data are expected to include the transient thermal effect of heater solid as well as the characteristics of boiling. Only a few experimental studies were conducted under steadystate conditions. Robidou et al. [5,6] performed steady-state experiments of entire boiling regimes in a planar water jet, having a width of 1 mm. Their data showed that, under the conditions of a jet velocity of 0.8 m/s and a subcooling of 16 °C, the film boiling regime

☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author at: Department of Mechanical Engineering, Sogang University, Shinsu-dong, Mapo-ku, Seoul 121–742, South Korea. E-mail address: [email protected] (G. Son). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.05.021

started at a wall temperature of about 450 °C, which is called the Leidenfrost temperature (or the minimum film boiling temperature). The Leidenfrost temperature and the film boiling heat transfer were strongly dependent on the jet subcooling. The wall temperature signals also indicated no significant fluctuations in time, which means the vapor film in liquid jet impingement is stable. Bogdanic et al. [7] used a miniaturized optical probe of 1.5 μm tip diameter to investigate the two-phase structures underneath a planar water jet, whose experimental configuration was similar to that used by Robidou et al. [5,6]. The liquid or vapor contact signals obtained from the optical probe showed that the liquid contact frequency was very high in the transition boiling regime whereas the liquid contacts disappeared in the film boiling regime. The measured vapor film thickness was 8 ± 2 μm under the experimental conditions of a jet velocity of 0.4 m/s and a subcooling of 20 °C. Efforts were made to predict the film boiling heat transfer in liquid jet impingement. Nakanish et al. [8] analyzed film boiling in the stagnation region of a planar water jet by solving the similarity equations for the conservation of mass, momentum, and energy in the liquid and vapor phases. They also presented a simplified analytical model to predict the film boiling heat transfer. Applying the potential-flow approximation to the liquid layer and the lubrication approximation to the thin vapor film, they derived the vapor film thickness δv and the wall heat flux qw in the stagnation region as aδ ρv V j hlv v 2W j

qw ¼

λv ΔT w δv

aμ l δ2v 1þ Rel 6μ v W 2j

! ¼

λv ΔT w λl ΔT sub − δv Wj

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a Rel Pr l π

ð1Þ

ð2Þ

W. Lee, G. Son / International Communications in Heat and Mass Transfer 46 (2013) 42–48

Nomenclature a

dimensionless gradient of the free-stream velocity, (u∞/Vj)/(x/Wj) specific heat at constant pressure gravity grid spacing enthalpy of vaporization pressure liquid Prandtl number, cplμl/λl heat flux liquid Reynolds number, ρlVjWj/μl time temperature jet subcooling, Tsat − Tj wall superheat, Tw − Tsat horizontal flow velocity free-stream velocity liquid or vapor velocity at the interface vertical flow velocity jet velocity jet width horizontal and vertical coordinates

cp g h hlv p Prl q Rel t T ΔTsub ΔTw u u∞ uδ v Vj Wj x, y

Greek symbols liquid velocity boundary layer thickness δl liquid thermal boundary layer thickness δt vapor film thickness δv λ thermal conductivity μ viscosity ρ density

43

They used a parabolic profile for the liquid temperature and the lubrication approximation for the vapor film. The wall superheat and heat flux at the Leidenfrost condition were derived for a planar jet as   1=2 λ μv 1=2 Pr l ΔT sub ΔT w ¼ 0:99 l λv μl

ð5Þ

λl ΔT sub 1=2 1=2 Rel Pr l Wj

ð6Þ

qw ¼ 0:7

Direct numerical simulation (DNS) is another way further clarifying the physics of film boiling in liquid jet impingement. Computational efforts were made for film boiling on flat or cylindrical surfaces using a moving-grid method [11], a front-tracking method [12,13], a level-set (LS) method [14–16], and volume-of-fluid method [17,18]. Very recently, Kim and Son [19] presented the LS method for computation of liquid jet impingement without and with film boiling. In this study, DNSs are further extended for film boiling in a planar water jet by using the sharp-interface LS method, which is modified to track the liquid–air and liquid–vapor interfaces and to treat the effect of phase change at the liquid–vapor interface. An analytical model to predict the vapor film thickness and wall heat flux in the stagnation region is also developed by simplifying the momentum and energy equations in the liquid and vapor phases. The effects of jet subcooling, jet velocity and wall temperature on the vapor film thickness and wall heat flux during film boiling are investigated. 2. Analysis 2.1. DNS Fig. 1 shows the configuration used for simulation of film boiling in a planar (or two-dimensional) jet. The flow is assumed to be laminar. The liquid–air and liquid–vapor interfaces are tracked by the LS function, which is defined as a signed distance from the interface. The positive sign is chosen for the liquid phase, and the negative sign for the gas (vapor or air) phase. To solve the conservation equations of mass, momentum and energy in the liquid and gas regions as well as the LS advection and reinitialization equations, we employ the numerical approach developed in the previous study [19].

Subscripts j jet l, v liquid, vapor sat saturation w wall

2.2. Analytical model where the dimensionless constant a is used to express the freestream velocity u∞ outside of the liquid boundary layer, which is defined as u∞ x ¼a Vj Wj

An analytical model to predict the vapor film thickness and wall heat flux in the stagnation region of a planar jet is developed by

ð3Þ

Another theoretical model was developed by Liu and Wang [9] for film boiling in the stagnation region of a circular water jet. They used the cubic polynomial profiles for the velocity and temperature in the liquid layer, the Reynolds analogy for the liquid thermal boundary layer thickness and the lubrication approximation for the velocity and temperature profiles in the vapor film. The model was modified by using an empirical correlation factor, which was determined from their experimental data. The wall heat flux for a highly subcooled liquid, which is also applied to a planar jet, was derived as 1=2

qw ¼ 2Rel

1=6

Pr l

1=2

ðλv λl ΔT w ΔT sub Þ

=W j

ð4Þ

Karwa et al. [10] proposed a theoretical model for determining the Leidenfrost condition in liquid jet impingement. The Leidenfrost condition was assumed as the zero shear stress at the liquid–vapor interface.

Fig. 1. Configuration used for computation of film boiling in a planar liquid jet.

44

W. Lee, G. Son / International Communications in Heat and Mass Transfer 46 (2013) 42–48

extending the analytical model of Karwa et al. [10] to determine the Leidenfrost condition. In this work, we do not assume that the shear stress at the liquid–vapor interface is zero. Applying the boundary layer approximation to the liquid layer, the integral momentum and energy equations can be written as d δl μ ∂u ′ δ ′ du∞ ¼ l ′l j0 ∫ u ðu −ul Þdy þ ∫ l ðu∞ −ul Þdy 0 dx ρl ∂y dx 0 l ∞

ð7Þ

 d δt  λ ∂2 T l ′ j ∫ ul T l −T j dy ¼ l ρl cpl ∂y′ 0 dx 0

ð8Þ

Tv ¼ Tw−

ul −u∞ y′ ¼ 1−2 uδ −u∞ δl

!

T l −T j y′ ¼ 1−2 T sat −T j δt

þ

! þ

y′ δl

!2

y′ δt

ð9Þ

!2 ð10Þ

where uδ is the liquid velocity at the liquid–vapor interface and is written, introducing the constant b for convenience, as uδ ¼ ð1 þ bÞu∞

ð11Þ

Substituting Eqs. (9) and (10) into Eqs. (7) and (8) and assuming that δt b δl, we obtain  −1=2 ð5 þ 2bÞa δl ¼ Rel Wj 10 

δt ¼ Wj

Aa Rel Pr l 2

−1=2

ð12Þ

ð13Þ

where

1

A¼∫

0

  !   ul T l −T j y′ 1 bδ b δt 2  d ¼ ðb þ 1Þ− t þ u∞ T −T δt 3 6 δl 30 δl sat

ð14Þ

j

Applying the lubrication approximation to the thin vapor film, the conservation equations of momentum and energy can be written as 2

0¼−

dpv ∂ uv þ μv dx ∂y2

0 ¼ λv

∂ Tv ∂y2

ð15Þ

2

ð16Þ

Using the boundary conditions (uv = ul, pv = pl, Tv = Tsat) at the liquid–vapor interface and the no-slip conditions (uv = 0, Tv = Tw) at the solid wall, the vapor velocity and temperature profiles are obtained as   2 uv y aμ l δv y y ¼ ð1 þ bÞ þ 1− u∞ δv 2μ v W 2j δv δv

ð17Þ

ð18Þ

The shear-stress condition, μl(∂ ul/∂ y) = μv(∂ uv/∂ y), at the interface yields b¼

!   aμ l δ2v μl μv Re −1 = 2 þ l δl δv 2μ v W 2j

μv δv

ð19Þ

Using the energy balance at the interface, which is expressed as ρv hlv

where y′ = y − δv, δl is the liquid velocity boundary layer thickness, and δt is the thermal boundary layer thickness. The free-stream velocity u∞ is expressed as Eq. (3). We assume the liquid velocity and temperature profiles as

y ðT −T sat Þ δv w

d δv ∂T ∂T ∫ uv dy ¼ −λv v þ λl l dx 0 ∂y ∂y

ð20Þ

we obtain ρv hlv V j

aδv 2W j

bþ1þ

aμ l δ2v Rel 6μ v W 2j

! ¼

λv λ ΔT w −2 l ΔT sub δv δt

ð21Þ

The five unknown variables (δl, δt, A, b, and δv) can be iteratively evaluated from Eqs. (12), (13), (14), (19), and (21). Then, the wall heat flux is obtained as qw ¼ λv

∂T ΔT w ¼ λv δv ∂y 0

ð22Þ

3. Results and discussion The present numerical and analytical study is carried out using the properties of water and air at 1 atm. We choose a nozzle width of Wj = 1 mm and a two-dimensional domain which has a width of 2.56 mm and a height of 6 mm. We parametrically vary the jet velocity Vj, the jet subcooling ΔTsub = Tsat − Tj, and the wall temperature Tw. The initial conditions for film boiling are taken from the steadystate solution for liquid–air flow at Tw = 100 °C without including boiling, which is obtained numerically. The first computation is made under the conditions of Vj = 1.5 m/s, Tj = 80 °C (ΔTsub = 20 °C), and Tw = 450 °C (ΔTw = 350 °C). The wall temperature is higher than the Leidenfrost temperature, which is estimated as ΔTw = 146 °C from Eq. (5). To save computation time, we use non-uniform grid spacing in the y direction with the ratio of two adjacent intervals of 1.03–1.04 except near the wall, y ≤ 0.4 mm, where the grid spacing is uniform as hy = 2.5 μm. The grid spacing in the x direction is uniform as hx = 20 μm in all regions. Initially a vapor film of δv = 0.25hy is placed on the solid wall. The initial interface oscillates due to the imbalance between the liquid and vapor-side heat fluxes and then reaches a steady state as the interfacial oscillation decays with time. The liquid–air and liquid–vapor interfaces at the steady state and the associated velocity fields are plotted in Fig. 2. A stable vapor film is formed between the wall and the impinging liquid. Its thickness is 8.9 μm in the stagnation region (x ≤ Wj/2) and increases in the parallel flow region (x > Wj/2). In the present analytical model, the free-stream velocity gradient a, given by Eq. (3), should be first determined to predict the vapor film thickness and wall heat flux. For its value, Zumbrunnen [20] and Tong [21] used π/4 while Zumbrunnen et al. [22], Timm et al. [23] and Karwa et al. [10] used 0.99 or 1. In this work, a is estimated from the result of DNS. The velocity and temperature profiles predicted from the analytical model are compared with the DNS results in Fig. 3. The analytical predictions using a = 0.85 show much better agreement with the DNS results than using the other values. The analytically predicted vapor film thickness and wall heat flux at the stagnation point (x = 0) are within 3.5% deviation with the DNS results.

W. Lee, G. Son / International Communications in Heat and Mass Transfer 46 (2013) 42–48

(a)

(a)

2.0

0.03

Air

DNS Anal (a=0.99) Anal (a=0.85)

1.5

Liquid

0.02

y (mm)

y (mm)

45

1.0

0.01

0.5

0.0 0.0

0.5

1.0

1.5

2.0

0 0.0

x (mm)

0.5

1.0

1.5

2.0

u/Vj

(b)

(b)

0.04

0.03

0.02

0.02

Liquid 0.01

Vapor 0.00 0.0

DNS Anal (a=0.99) Anal (a=0.85)

y (mm)

y (mm)

0.03

0.5

1.0

1.5

0.01 2.0

x (mm) 0 Fig. 2. Steady-state interfaces and velocity fields for Vj = 1.5 m/s, ΔTsub = 20 °C and Tw = 450 °C: (a) in the liquid–air region and (b) in the expanded region near the vapor film.

0

200

400

600

T (°C) Fig. 3. Comparison of the results obtained from the direct numerical simulation and the analytical model at x = 0.5 mm for Vj = 1.5 m/s, ΔTsub = 20 °C and Tw = 450 °C: (a) velocity profile and (b) temperature profile.

observed to vary as V−0.5 , but its variation obtained from the DNS is j more complicated. In Fig. 8 (a), the vapor velocity is observed to have a parabolic profile for all the cases. As the jet velocity increases, the liquid and vapor velocities normalized by Vj decrease. Based on the DNS results, the dimensionless gradients a′s of the free-stream velocity for Vj = 0.8 m/s, 1.5 m/s and 2.2 m/s are estimated as 1.10, 0.85, and 0.78, respectively. It is seen from Fig. 8 (b) that the temperature

0.03

ΔTsub = 20°C 40°C 60°C

0.02

y (mm)

Fig. 4 presents the effect of jet subcooling (ΔTsub = Tsat − Tj) on the vapor film while keeping Vj = 1.5 m/s and Tw = 450 °C. As the jet subcooling increases from 20 °C to 40 °C and 60 °C, the vapor film thickness at the stagnation point decreases from 8.9 μm to 4.6 μm and 3.0 μm. The vapor film thickness varies as ΔT−1 sub . The analytical predictions for ΔTsub = 20 °C, 40 °C and 60 °C are 8.6 μm, 4.5 μm and 3.0 μm, respectively. The deviations from the DNS results are less than 3.5%. In Fig. 5 (a), as ΔTsub increases and the liquid–vapor generation rate decreases, the vapor velocity is observed to decrease and its parabolic profile changes to a linear profile. This transition in velocity profile is expected from Eq. (17) as the vapor film thickness is reduced and the second term becomes relatively smaller. It is seen from Fig. 5 (b) that the temperature gradient increases as the vapor film becomes thinner with the jet subcooling. The effect of jet subcooling on the wall heat flux is plotted in Fig. 6. The wall heat flux is observed to be linearly proportional to the jet subcooling. The predicted wall heat fluxes from the present analytical model match well with the DNS results whereas the analytical predictions of Liu and Wang [9], given by Eq. (4), differ significantly for higher jet subcoolings. Fig. 7 shows the effect of jet velocity on the vapor film for ΔTsub = 20 °C and Tw = 450 °C. As the jet velocity increases, the vapor film thickness decreases. For Vj = 0.8 m/s, 1.5 m/s and 2.2 m/s, the DNS results of vapor film thickness at the stagnation point are 10.1 μm, 8.9 μm, and 7.4 μm, respectively, while the analytical predictions are 11.8 μm, 8.6 μm, and 7.1 μm, respectively. The deviation is less than 4.2% for Vj ≥ 1.5 m/s, but it is increased to 15% for a lower jet velocity of Vj = 0.8 m/s. The analytical prediction of vapor film thickness is

0.01

0

0

1

2

x (mm) Fig. 4. Effect of jet subcooling on the vapor film for Vj = 1.5 m/s and Tw = 450 °C.

46

W. Lee, G. Son / International Communications in Heat and Mass Transfer 46 (2013) 42–48

0.03

(a)

Vj = 0.8m/s 1.5m/s 2.2m/s

0.03

ΔTsub = 20°C 40°C 60°C

0.02

y (mm)

y (mm)

0.02

0.01

0.01

0 0.0

0.5

1.0

1.5

2.0

0

u/Vj

0

1

2

x (mm)

(b)

Fig. 7. Effect of jet velocity on the vapor film for ΔTsub = 20 °C and Tw = 450 °C.

0.03

ΔTsub = 20°C 40°C 60°C

reduced to less than 3.0%. The present analytical model shows better agreement with the DNS results than the Liu and Wang's model. Fig. 10 presents the effect of wall temperature on the vapor film while keeping Vj = 1.5 m/s and ΔTsub = 20 °C. The vapor film thickness is observed to increase slightly as the wall temperature increases. For Tw = 450 °C, 500 °C, and 550 °C, the DNS results of vapor film

y (mm)

0.02

0.01

(a) 0.03 0

0

200

400

Vj = 0.8m/s 1.5m/s 2.2m/s

600

T (°C)

gradient increases as the jet velocity increases and the vapor film thickness decreases. The effect of jet velocity on the wall heat flux is plotted in Fig. 9. The wall heat flux obtained from the DNS is observed to vary as for Vj b 1.5 m/s and V0.48 for higher jet velocities. Compared V0.23 j j with the DNS results, the analytical predictions using a = 0.85 are within 3.3% deviation for Vj ≥ 1.5 m/s, but the deviation is increased to 15% for a lower jet velocity. When using the dimensionless gradient a estimated from the DNS velocity profile, the deviation is

y (mm)

0.02 Fig. 5. Effect of jet subcooling on the numerical results at x = 0.5 mm for Vj = 1.5 m/s and Tw = 450 °C: (a) velocity profile and (b) temperature profile.

0.01

0 0.0

0.5

1.0

1.5

2.0

u/Vj

(b) 0.03

Vj = 0.8m/s 1.5m/s 2.2m/s

4

DNS Anal (a=0.85) Liu & Wang

0.02

y (mm)

q (MW/m2)

3

2

0.01 1

0

0

20

40

60

80

ΔTsub (°C) Fig. 6. Effect of jet subcooling on the wall heat flux at the stagnation point for Vj = 1.5 m/s and Tw = 450 °C.

0

0

200

400

600

T (°C) Fig. 8. Effect of jet velocity on the numerical results at x = 0.5 mm for ΔTsub = 20 °C and Tw = 450 °C: (a) velocity profile and (b) temperature profile.

W. Lee, G. Son / International Communications in Heat and Mass Transfer 46 (2013) 42–48

(a) DNS Anal (a=0.85) Anal (a=various) Liu & Wang

5

Experiment DNS Anal (a=0.85) Anal (a=1.10)

4

q (MW/m2)

q (MW/m2)

2

1

47

3 2 1 0 400

0 0

1

2

3

450

Vj (m/s)

thickness at the stagnation point are 8.9 μm to 9.9 μm, and 10.6 μm, respectively, while the analytical predictions are 8.6 μm, 9.7 μm, and 10.8 μm, respectively. The deviations are less than 3.5%. Computations are made for Vj = 0.8 m/s and ΔTsub = 16 °C to compare with the experimental data of Robidou et al. [5]. The results are plotted in Fig. 11. The experimental data show that the wall heat flux trend changes substantially near Tw = 450 °C, which can be assumed as the Leidenfrost temperature. The wall heat fluxes are nearly constant for Tw > 450 °C. The DNS results as well as the analytical predictions have a similar trend to the experimental data. The difference between the DNS results and the experimental data are about 29%. It is noted from Fig. 11 (b) that the vapor velocity near the Leidenfrost temperature has a parabolic profile with a nonzero gradient at the liquid–vapor interface. This observation does not support the assumption of Karwa et al. [10] that the Leidenfrost condition occurs when the interfacial shear stress is zero. 4. Conclusions DNSs of film boiling in a planar water jet were performed by employing a sharp-interface LS method to track the liquid–air and liquid–vapor interfaces. The computations demonstrated that a stable vapor film was formed between the wall and the impinging liquid. The magnitude and the parabolic to linear profile transition of the vapor velocity were strongly dependent on the jet subcooling. The wall heat flux was linearly proportional to the jet subcooling, but it was little affected by the wall temperature. The dependence of wall

550

(b) 0.03

Tw = 450°C 500°C 0.02

y(mm)

Fig. 9. Effect of jet velocity on the wall heat flux at the stagnation point for ΔTsub = 20 °C and Tw = 450 °C.

500

Tw (˚C)

0.01

0 0.0

0.5

1.0

1.5

2.0

u/Vj Fig. 11. Numerical results for Vj = 0.8 m/s and ΔTsub = 16 °C: (a) wall heat flux at x = 0 and (b) velocity profile at x = 0.5 mm.

heat flux on the jet velocity was more complicated because the free-stream velocity was not linearly proportional to the jet velocity. In this work, an analytical model was also developed to predict the vapor film thickness and wall heat flux in the stagnation region. The analytical predictions using a = 0.85 for the free-stream velocity gradient were in good agreement with the numerical results except for a low jet velocity of 0.8 m/s. The numerical results as well as the analytical predictions had a similar trend to the experimental data reported in the literature. The difference between the numerical results and the experimental data was about 29%. Acknowledgments

0.03

Tw = 450°C 500°C 550°C

This work was supported by Energy Efficiency & Resources of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (2011T100200316).

y (mm)

0.02

References

0.01

0

0

1

2

x (mm) Fig. 10. Effect of wall temperature on the vapor film for Vj = 1.5 m/s and ΔTsub = 20 °C.

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