III. Nucleate Boiling Model

1)  Active Nucleation Site Density

① Griffith and Wallis(1960) ;  [eqn: rm r_c =2sigma T_sat /rho_g i_fg (T_w -T_sat )]

② Gaertner(1963) ;  [eqn: rm N/A=N_0 exp(- k /T_w^3 )]

③ Shoukri and Judd(1975)

[eqn: rm r_{c,min} CONG 2sigma T_sat /rho_g i_fg (T_w -T_sat )]

[eqn: rm (N/A)=C LEFT [{rho_g i_fg (T_w -T_sat )} over{2sigma T_sat } RIGHT ] ^m]

- This model canont be widely applied due to the dependence on the surface

roughness and the assumption for the fluid property.

④ Kocamustafaogullari and Ishii(1983) ; express the correlation of the active

nucleate site density as measurable physical quantities

[eqn: rm h over{ k_f N_pn^0.5 }=14.0 LEFT [ {rho_f C_pf triangle T_sat} over{rho_g i_fg}RIGHT ] Pr^-0.39 N_pn^*-0.125]

where    [eqn: rm h~ PROPTO ~(triangle T_sat )^0.5 `N_pn^0.375]

And       [eqn: rm N_pn^* =N_pn^* (r_{c,min}^* ,rho_c^* )]

where  [eqn: rm N_pn^* &==& N_pn D_d^2 ~,~~~ r_{c,min}^* &==& r_c / (D_d /2) ~,~~~rho_c^* &==&(rho_f -rho_g ) /rho_g]

From Thomson correlation and Clausius-Clapeyron equation,

[eqn: rm r_{c,min} = { 2 sigma {(1+rho_g /rho_f )} / P_f }over {exp [ {i_fg (T_g -T_sat )/RT_g T_sat} ] }]

For pool boiling,

[eqn: rm N_pn^* ==r_{c,min}^m f(rho^* )]

where [eqn: rm &m&=-4.4# rm &f(rho^* )&=2.157 TIMES 10^-7 rho^*-3.2 (1+0.0049rho^* )^4.13]

For convective boiling, by using vapor superheat temperature instead of wall          temperature in the pool boiling

[eqn: rm N_cn^* ==r_{ce,min}^m f(rho^* )]

⑤ Benjamin and Balakrishnan(1997)

[eqn: rm N over A =218.8 Pr^1.63 LEFT ( 1 over gamma RIGHT ) theta^-0.4 (triangle T)^3]

where, γ is a reciprocal factor between the surface and liquid, and is defined

as,

[eqn: rm gamma == LEFT ( { k_w rho_w C_pw} over {k_l rho_l C_pl } RIGHT ) ^1/2]

where, θ is the nondimensional variable related on the surface roughness, is          defined by the experiment as,

[eqn: rm theta =14.5 -4.5 LEFT ( { R_a P} over {sigma } RIGHT ) +0.4 LEFT ( { R_a P} over {sigma } RIGHT )^2]

Fig. 3-7 Nucleation Site Density vs. Critical Radius

(Zeng et al., 1993)

Fig. 3-8 Nucleation Site Density

vs. Vapor Velocity

(Zeng et al. 1993)

2)  Bubble Departure Size

- Determine the bubble departure size by the balance of forces acting on the

bubble

o buoyancy and hydraulic drag to enhance the detachment

o surface tension and liquid inertia to interfere the detachment

- Liquid inertia is represented by the bubble growth velocity. This is directly           proportional to the superheat of liquid and inversely proportional to the size

of the cavity.

① Fritz and Ende(1935) ; consider the balance between the surface tension and

the buoyancy

[eqn: rm D_d =0.0208 theta LEFT [ sigma over {g(rho_f -rho_g )} RIGHT ] ^1/2]

where θ ; contact angle between the vapor and the heated surface

② Zuber(1959) ; include the terms by the surface tension, the buoyancy and the                      thickness of thermal boundary layer

[eqn: rm D_d = LEFT [ { sigma }over {g(rho_f -rho_g )} RIGHT ] ^1/3 LEFT ( {6k_l (T_w -T_sat )}over q'' RIGHT ) ^1/3]

where, 6kl(Tw-Tsat)/q'' is the thickness of superheated boundary layer.

③ Nishikawa and Urakawa(1960) ; correlate between the bubble departure size

and the pressure,

[eqn: rm D_d =0.672p^-0.572]

④ Ruckenstein(1963) ; Model considering the balance of the buoyancy, the

surface tension ,and the drag,

[eqn: rm D_d = LEFT [ {3 pi^2 rho_f alpha_Tl}over{g(rho_f -rho_g )} RIGHT ] ^1/3 Ja^4/3]

()

where, αTl is thermal diffusivity of the liquid

⑤ Cole and Rohsenow(1968)

[eqn: rm D_d =C LEFT [ sigma over {g(rho_f -rho_g )} RIGHT ] ^1/2 (Ja^* )^5/4]

where,     [eqn: rm Ja^* == {T_c c_pf rho_f} over {rho_g i_fg}]

C = 1.5×10-4 for water  and  4.65×10-4 for other liquids

⑥ Kutateladze and Gogonin(1979) ; consider the dependence of Ja, Pr and                                                   Archimedes number(Ar)

[eqn: rm D_d =0.0208 theta LEFT [ sigma over {g(rho_f -rho_g )} RIGHT ] ^1/2 LEFT [ 1+ LEFT ( Ja over Pr RIGHT )^2 1 over Ar RIGHT ]]

where,

[eqn: rm Pr== nu_f over alpha_f = {nu_f c_pf rho_f }over k_f]

[eqn: rm Ar == g over nu_f^2 LEFT ( sigma over {rho_f g} RIGHT ) ^2/3]

⑦ Levy(1967)

From balance of buoyancy force ; [eqn: rm F_B ={C_B r_b^3 (rho_l -rho_g )g}over g_c]

surface tension ; [eqn: rm F_S =C_S r_b sigma ~~~~~~~~~~]

drag force     ;  [eqn: rm F_D = C_D tau_w over D_h r_b^3 ~~~~~~~],

[eqn: rm r_b = root{ { C_S sigma} over {C_B g over g_c (rho_l -rho_v ) +C_D tau_w over D_h } }]

For spherical bubble, with only drag and surface tension,

[eqn: rm D_b =0.015 LEFT ( {sigma D_h}over{tau_w} RIGHT ) ^0.5]

-  Bubble departue size ～ 2.7μm

⑧ Unal(1976) - heat transfer controlled bubble model

- Since the experimental range of this correlation is very wide, the Unal's               correlation can be applied to the general system.

- Assume the very thin and annular liquid film in the circular or elliptical               bubble contacted with the bubble surface

- Explain the bubble growth with the evaporation process of this liquid film

- Energy equation for the bubble

[eqn: rm q_B { pi D^2} over {4 } LEFT ( 1- D_d^2 over D^2 RIGHT ) =h_c triangle T_"sub" {pi D^2} over 2 + pi over 6 rho_g lambda dD^3 over dt]

The left side of above eqaution is the heat flux from the thin liquid film on           the surface to the bubble. The first term of the right side is the heat flux            emitted to the bubble outside by the condensation in the bubble                      upper-heimsphere, and the second term of the right side is the variation              rate of the thermal capacity according to the change of the bubble volume.

Using the proper models for each term, the above equation is rewritten as            follow.

[eqn: rm dD over dt = a `alpha `t^-1/2 -C Phi bD]

where a, b are the variables expressed by all sorts of thermal-hydraulic              factors, Φ is a variable used for the condensation heat transfer on the                bubble surface, and is defined as follow.

[eqn: &rm Phi &~=~ LEFT ( v over v_0 RIGHT ) ^0.47 & for ~v~ >~ 0.61 m/s``## &rm &~=~~~1&for ~v~ le~ 0.61 m/s``## &rm v_0 &~=~~0.61 m/s]

And, through the experiments, [eqn: rm alpha/C^1/2 =2.0 times 10^-5 p^0.709]

The maximum bubble size is represented by the time to reach the maximum           bubble size,

[eqn: rm D_d ={2.42 times 10^-5 p^0.709 a}over{(b Phi)^1/2}]

Fig. 3-9 Bubble Departure Size (Unal, 1976)

⑨ Farajisarir(1993)

[eqn: rm {D_max sigma} over {rho_l alpha_Tl^2} =10.02 times 10^9 Ja^-1.65 LEFT [ { T_w -T_l} over {T_w -T_s } RIGHT ]^-1.65]

∝ (Tw-Tl)-1.65

Fig. 3-10 Bubble Departure Size (Al-heys et al., 1981)

3)  Bubble Departure Frequency

① Jakob(1949)

[eqn: rm fD_d^n =C]

where n=3     by Rallis and Greenland(1961)

n=1/2    by McFadden and Grassman(1962), Cole(1967)

② Zuber(1959)      [eqn: rm fD_d =0.59 LEFT [{ sigma g (rho_f -rho_g ) } over{rho_f^2}RIGHT ] ^1/4]

③ Cole(1960)   [eqn: rm f^2 D_d = 3 over 4 LEFT [{ g (rho_l -rho_g ) } over{ C_d rho_l^2}RIGHT ] ^1/4]    ,  Cd = 1.0 for water at 1atm

④ Mikic and Rohsenow(1969) [eqn: rm f^{1/2}D_d =0.83 Ja root{pi alpha_l}] (heat transfer controlled

growth region)

⑤ Shoukri and Judd(1977)

[eqn: rm f= 1 over {t_w +t_g}]

()

where  [eqn: &rm r_c &= LEFT ( root {pi alpha_Tl }/2K RIGHT ) LEFT ( {sin beta} over {1+cos beta} RIGHT ) # rm &&times LEFT [ gamma_w +- root{gamma_w^2 -4 LEFT ( {A/(theta_w -theta_sat}over{root {pi alpha_Tl }} RIGHT )(1+cos beta) K} RIGHT ]]

and

[eqn: &rm K&= LEFT [ root{1 over {t_w +t_g}}+ theta_sat over {theta_w -theta_sat} root{1 over t_w}RIGHT ] ## &rm gamma_w &=1- 2 over pi Sum_m=0^inf C_m over {m!(2m+1)2^2m } LEFT ( root{t_g over {t_w +t_g}} RIGHT ) ^2m+1## &rm C_m &= 2(m-1)C_m-1 ~~~and~~~C_0 =1]

And the bubble growth time is

[eqn: rm root t_g = 1 over 2 LEFT [ LEFT ( X over Y RIGHT ) +- root{LEFT ( X over Y RIGHT )^2 +35.4 X} RIGHT ]]

where,

[eqn: rm &X=0.4215 beta root{ { 2sigma g_c} over {g(rho_l -rho_v ) } }## rm &Y= 2 over root pi Ja root alpha_Tl]

IV. Bubble Condensation

The bubble condensate rate in the sub-cooled boiling flow is represented by the     re-condensation ratio of the bubble generated by the boiling, η as follow,

[eqn: rm phi_cond = eta ``phi_wn]

The re-condensation rate is expressed by the change of bubble energy for the        nucleate boiling heat flux as,

[eqn: rm eta=1- d over dz LEFT { G_g LEFT [ i_fg +C_pf (T_sat -T_f ) RIGHT ] RIGHT } /[( dot q ''-{dot q ''}_sp )(xi_h /A_c )]~~~~~]

where, q''sp is a convective heat flux to the single phase.

Assuming the physical properties are constant, and using the mass and energy        balance equations of the mixture, the right side of the above equation is              approximated as follow.

[eqn: rm eta= {GC_pf (dT_f / dz) - {dot q ''}_sp (xi_h /A_c ) } over{ ( dot q ''-{dot q ''}_sp )(xi_h /A_c )}]

If the temperature distribution on the axial-direction is assumed as an exponential      function,

[eqn: rm eta= {(T_sat -T_f )/(T_sat -T_0 ) - {dot q ''}_sp /dot q '' } over{1- {dot q ''}_sp /dot q ''}]

where, T0 is the temperature of water at the onset of the boiling.

When, for fully developed sub-cooled boiling, the convective heat flux to the          single phase is ignored,

[eqn: rm eta= {T_sat -T_f }over{T_sat -T_0}]

∴ [eqn: rm phi_cond = {T_sat -T_f }over{T_sat -T_0} ``phi_wn]     for T0≤Tf≤Tsat