Sub-models for the Wall Boiling Model

As mentioned in the previous section, the two most important parameters governing the heat partitioning model are the nucleation site density n and the bubble departure diameter d_w. In the RPI wall-boiling model they are correlated to the wall superheat ΔT_sup=T_wall−T_sat and to the near-wall liquid sub-cooling ΔT_sub=T_sat−T_liq, respectively.

The following topics will be discussed:

• Wall Nucleation Site Density (p. 143)

• Bubble Departure Diameter (p. 143)

• Bubble Detachment Frequency (p. 144)

• Bubble Waiting Time (p. 144)

• Area Influence Factors (p. 144)

• Convective Heat Transfer (p. 144)

• Quenching Heat Transfer (p. 145)

• Evaporation Rate (p. 145) Wall Nucleation Site Density

The model for wall nucleation site density adopted in the RPI model is that of Lemmert and Chawla (1977) [164 (p. 293)]:

⎡⎣ ⎤

⎦ = = =

n m−²

(

m

(

^ΔT^{sup[ ]}K

) )

^p m ^210, p ^1.805

Note that the wall superheat in the above equation cannot be negative. Its negative value means that the wall temperature drops below the saturation temperature, where there is no boiling and the heat partitioning model is not used.

This model was implemented in CFX-4 with parameter m=185, giving an overall correction factor of 0.8. Egorov and Menter (2004) [163 (p. 293)], conjecturing that this was a deliberate alteration by the RPI group, related to the corresponding factor in the bubble waiting time model. Egorov and Menter also reformulated the correlation as follows:

=

n n^ref

(

^ΔT^sup/ΔT^ref

)

^p

where = × + ⎡⎣ ⎤

− ⎦

n_ref 0.8 9.922E 5 m ² and ΔT_ref=10 [ K ]. This formulation avoids fractional powers of physical dimensions, hence is more amenable to the use of different unit systems.

Alternative correlations have been proposed for nucleation site density by Kocamustafaogullari and Ishii (1983) [160 (p. 293)].

Bubble Departure Diameter

Kurul and Podowski (1991) [159 (p. 293)] adopted the correlation for bubble departure diameter due to Tolubinski and Kostanchuk (1970) [165 (p. 293)]:

The Thermal Phase Change Model

(Eq. 4.255)

= ⎛

⎝⎜ ⋅ ⎛

⎝⎜− ⎞

⎠⎟

⎞

⎠⎟ d_w min d_ref exp ^Δ_Δ^T_T^sub ,d_max

ref

The parameters of the model are dimensional (d_max=1.4 [ mm], d_ref=0.6 [ mm], ΔT_ref=45 [ K ]) and are chosen to fit pressurized water data. Hence the model is clearly not universal. Negative liquid sub-cooling is possible here, where it means onset of the bulk boiling. Limiting the bubble departure diameter applies to this situation and prevents d_w from growing too high.

Note that the model is strongly dependent on a liquid temperature scale. In the original experimental data, this was taken as the pipe center-line temperature. CFX-4 used the cell-center value in near wall cells, but this proved to give mesh-dependent results. Egorov and Menter restored mesh independence by using the logarithmic form of the wall function to estimate the liquid temperature, T_l, at a fixed value of y⁺=250. ANSYS CFX uses this method, with a user-modifiable value of y⁺.

Bubble Detachment Frequency

The computation of the evaporation rate m˙ requires an additional model parameter, namely the frequency of the bubble detachment from the nucleation site. The model adopted by Kurul and Podowski is that due to Cole (1960) [166 (p. 293)]:

= ⁻

f ^4g_{C d}

(

^{ρ ρ}_ρ

)

3 l g D w l

Note that, due to its dependence on gravity, this correlation is taken from pool boiling. It is simply estimated as the bubble rise velocity divided by the bubble departure diameter. The drag coefficient factor C_D was taken to be unity by Ceumern-Lindenstjerna (1977) [169 (p. 294)].

Bubble Waiting Time

Kurul and Podowski (1991) [159 (p. 293)] employed the model of Tolubinski and Kostanchuk (1970) [165 (p. 293)].

This fixes the waiting time between departures of consecutive bubbles at 80% of the bubble detachment period:

(Eq. 4.256)

= t_w ^0.8_f

The numerator in this equation is adjustable in ANSYS CFX.

Area Influence Factors

Recall from Partitioning of the Wall Heat Flux (p. 141) that Kurul and Podowski assumed a diameter of influence of a nucleating bubble equal to twice the bubble departure diameter d_w. Encoding this as a user-modifiable parameter F₂ (default value = 2), the area fraction of the bubble influence is given by:

= ⎛

⎝⎜ ⋅ ⎞

A₂ min ^{πF d}₄^{22 w2} n, 1⎠⎟

The area fraction A₁ subjected to single phase liquid convective heat transfer is limited from below by a small value, so its actual form is:

= ⁻ −

A^{1 max 10 , 1}

(

⁴ A²

)

Convective Heat Transfer

As discussed in Partitioning of the Wall Heat Flux (p. 141), single phase convective heat transfer to the liquid phase is modeled using the turbulent wall function (Egorov and Menter 2004 [163 (p. 293)]).

= −

Q_c A h T1 c

(

w Tl

)

The Thermal Phase Change Model

This replaces the mesh dependent Stanton Number correlation originally employed by Kurul and Podowski (1991) [159 (p. 293)].

Quenching Heat Transfer

As discussed in Partitioning of the Wall Heat Flux (p. 141), quenching heat transfer to the liquid phase in the area of influence of the vapor phase is modeled using a quenching heat transfer coefficient:

= −

Q_q A h T2 q

(

w Tl

)

In order to close the model, the quenching heat transfer coefficient h_q, participating in the quenching heat flux to liquid, must be defined. This value depends on the waiting time between the bubble departure and the next bubble formation.

With this value, the quenching heat transfer coefficient is correlated as:

=

h_q 2λ_lf _π^t_a^w

l

where al=λl^/

(

CPl lρ

)

is the liquid temperature conductivity coefficient. (Mikic and Rohsenow 1969 [167 (p.

293)], Del Valle and Kenning 1985 [168 (p. 294)]).

As for the case of bubble departure diameter, Egorov and Menter (2004) [163 (p. 293)] used the logarithmic form of the wall function to estimate the liquid temperature T_l at a fixed y⁺ value of 250.

Evaporation Rate

Knowing the bubble departure frequency, as well as the bubble size and the nucleation site density, one can obtain the evaporation rate as a product of the bubble mass, the detachment frequency and the site density:

=

m˙ ^πd₆^w3ρ_gfn

This was the form adopted by Kurul and Podowski (1991) [159 (p. 293)].

Egorov and Menter expressed the evaporation rate in terms the non-limited area fraction A₂^′:

= ⋅

A₂′ ^{πF d}₄^{22 w2} n

(unlike A₂, A₂^′ can exceed 1). In this case, the evaporation rate obtains the form:

= ^′⋅

m A˙ ₂ ¹_{6 w g}d ρ f

In the final form of the evaporation rate, the area fraction factor A₂^′ is limited by:

(Eq. 4.257)

= ^{′ ′} ⋅ ^′ =

m^{˙ min}

(

A A2^, 2,max

)

1d ρ f^, A ⁵

6 w g 2,max

The last limiting procedure is not used if the bubble departure diameter is below 1 mμ . In this case the non-limited form Equation 4.257 (p. 145) is used. The estimated value of 5 for A_2,max^′ is the upper limit for the nucleation sites area fraction, taking into account the effect of overlapping neighboring sites, operating out of phase. The functional form of this parameter is given in terms of the departure frequency and waiting time:

′ =

A_{2,max 1}− ⋅_{f t}¹ w

The Thermal Phase Change Model

which gives 5 with the correlation Equation 4.256 (p. 144) for t_w.