IV. Subcooled Boiling Model

√ bubble formation

√ bubble condensation

➡ complicate phenomena

For calculating the void fraction in

subcooled boiling, we have to know

① [eqn: Z_n ~,~~ (h_l )_Z_n]

② [eqn: Z_d ~,~~ (h_l )_Z_d]

③ [eqn: h_l (z) ~~or~~ delta w']

-> thermodynamic nonequilibrium ([eqn: h_l ~ != ~h_f] )

④ recondensation ratio of the detached bubbles

⑤ slip ratio

(1) Determination of void departure point, [eqn: Z_d]

[eqn: rm {CASES{ & ∙~ wall ~voidage ~region ## & ∙ ~ detached ~ voidage ~ region }}]

☞ location of the initial void ejection into the subcooled liquid core

☞ determined by the void departure criterion of Table 5-1

(2) Heat removal mechanism from the heated surface during subcooled boiling

① latent heat to form the vapor, [eqn: q''_evap]

[eqn: q''_evap] = [eqn: {dot V}_B``rho_g `` h_fg] ,        where  [eqn: {dot V}_B ~=~ n``f``B_d]            (4.4.1)

② by convection caused by bubble agitation or pumping by boundary layer,

[eqn: q''_pump]

[eqn: q''_pump] = energy flux to raise from [eqn: <h_l >] to [eqn: h_l]

+ energy flux for saturation

= [eqn: {dot V}_B left[ rho_l ``(h_l - <h_l > ) ~-~ rho_g ``(h_f - h_l ) right]]

(4.4.2)

Bowing(1962) defined

[eqn: epsilon ~==~ q''_pump over q''_evap ~=~ { rho_l ``(h_l - <h_l > ) ~-~ rho_g ``(h_f - h_l ) } over {rho_g h_fg }]

[eqn: APPROX ~{rho_l (h_f - h_l ) } over {rho_g h_fg }]                                  (4.4.3)

③ by condensation at the top of growing bubble while stilled attached to the

wall

④ singe phase convection, [eqn: q''_1phi]

Neglecting the third mechanism,

[eqn: q'' ~=~ q''_1phi ~+~ q''_evap ~+~ q''_pump]                            (4.4.4)

(3) Models to determine the void fraction profile in subcooled boiling region

☞ profile fit model

☞ mechanistic model

① Profile Fit Model by Levy

Assume   [eqn: h_ell ~=~ h_ell (<h>)]

i.e  [eqn: {h_f - h_ell } over {h_f - (h_ell ) _d } ~=~ F left[ {<h> - (h_ell )_d } over {h_f - (h_ell )_d } right ]]                           (4.4.5)

B. C ;  F =  1     at     [eqn: h_ell ~=~ (h_ell )_d]

F → 0     for     [eqn: <h> ~>>h_f]

The function that satisfy these requirements are the form of  F = EXP

∴ [eqn: {h_f - h_ell } over {h_f - (h_ell )_d } ~=~ exp left[ {<h> - <h_ell >_d } over {h_f -(h_ell )_d} right] ~=~ F]                  (4.4.6)

Since

<x(z)> [eqn: =~ {<h(z) > - h_ell (z) } over {h_g - h_ell (z) }]

∴ [eqn: <x> ~=~ {<h> - h_f + [h_f - (h_ell )_d ] F} over {h_fg + [h_f - (h_ell )_d ] F}]                             (4.4.7)

Then,

[eqn: <x> ~=~ { {<h> - h_f } over h_fg }][eqn: ~+~ { {h_f - (h_ell )_d} over h_fg }~ F]

∴    [eqn: <x> ~=~ <x_e > ~-~ <x_e >_d F]

Finally,

[eqn: <x> ~=~ <x_e >][eqn: -~ <x_e >_d ``exp left({<x_e >} over {<x_e >_d} -1 right )]                             (4.4.8)

Now from the flow quality, we can obtain the void fraction from void-quality

correlation.

(例) Zuber-Findley Void-Quality Correlation

[eqn: <alpha> ~=~ <x> over { C_o ``left[ <x> + rho_g over rho_f (1-<x>) right] + {rho_g V_gj } over G}]

Modified Armond model

[eqn: < alpha > ~=~ {(0.833 + 0.167 <x> ) <x> v_g } over {(1. - <x> ) v_f + <x> v_g }]

② Mechnistic Model

1) Determination of the void departure point

- neglect the wall voidage region

- solve the energy balance equation to calculate [eqn: <h_l (z)>]

- find the position which satisfy Zuber-Saha correlation,

[eqn: h_f - (h_l )_d ~=~ {CASES{ & 0.0022 {q'' D_H c_pl } over k_l ~,~~~for ~Pe ~==~ {GD_H c_pl } over k_l ~<~ 70,000 ## & 154~ q''/G ~,~~~~~~~~~~for ~Pe ~>~70,000 }}]   (4.4.9)

2) Void propagation equation

[eqn: {partial <alpha>} over {partial t} + C_K partial<alpha> over partialz = ( rho_f over {TRIANGLE rho} - C_o <alpha> ) OMEGA]                     (4.4.10)

where  [eqn: C_K ~=~ C_o ``<j> ~+~ V_gj]

and the characteristic frequency of the phase change is defined as,

[eqn: OMEGA ~==~ v_fg`` GAMMA]

The net amount of vapor formed per unit volume for constant pressure

is,

[eqn: GAMMA ~=~ {p_H ``q''_b (Z, t) } over {A_x-s ``h_fg`` [1+epsilon (Z,t)]} ~-~ {p_H ``q''_cond (Z,t) } over {A_x-s ``h_fg }]            (4.4.11)

3) Condensation heat transfer

[eqn: p_H ``q''_cond ~=~ {H_o ``h_fg``A_x-s ``<alpha> } over v_fg `` TRIANGLE T_{su b}]                      (4.4.12)

where Ho is a condensation coefficient.

4) Ratio of the heat flux due to microconvection to that causing vapor

formation is given by Rouhani as,

[eqn: epsilon(Z,t) ~=~ {rho_l ``(h_f - h_l``)} over {rho_g``h_fg}]                                (4.4.13)

And the boiling heat flux is approximated as,

[eqn: q''_b ``(Z,t) ~=~ q''(Z,t) ``left[ 1- {h_f - h_l ``(Z,t)} over {h_f - h_l_d ``(t) } right]]                (4.4.14)

Inserting Eqs. (4.4.12), (4.4.13) and (4.4.14) into Eq. (4.4.11),

[eqn: OMEGA (Z,t) ~=~ { TRIANGLE rho ``p_H ``q''} over {A_x-s ``rho_f ``(h_f - h_l_d ``)} ``{h_l - h_l_d } over {rho_g ``h_fg + rho_f``(h_f - h_l ``)} ~-~ {H_o <alpha>} over c_pf (h_f - h_l ``)]

(4.4.15)

5) The energy equation of the two-mixture is given as,

[eqn: {TRIANGLE rho} over rho_g (1-<alpha>) partialh_l over partialt ~+~ v_fg ``G(1-<x>) partialh_l over partialz ~=~ {q'' p_H v_fg } over A_x-s ~-~ OMEGA (h_g - h_f )]

(4.4.16)

where the assumptions used are ;

i) The internal heat generation, q''', is neglected.

ii) The kinetic and potential energy term are neglected.

iii) The state is steady.

iv) The vapor is saturated.

6) The drift-flux parameters

□ system constant parameters ; Dix model, constant values 등

□ void-dependent model ; Okawa/Lahey model

그림 7

그림 6

▶ Finally, we have to solve the each coupled equations, (4.4.10), (4.4.15)             and (4.4.16) with void-dependent drift-flux parameters.

[Example]

A 12ft long and rectangular fuel channel has uniform axial heat flux, " (= 55.56 Btu/ft2-sec), under constant pressure, 1000 psia. the inlet coolant velocity is 12ft/sec. Size and radius of channel and fuel are 0.65 and 0.225 in, repectively.

(a) Determine the void departure point with inlet liquid enthalpy of 511.9

Btu/lbm.

(b) Calculate the exit thermal-equilibrium quality.

(c) Extend your calculation to determination of exit flow quality.

(d) Finally, what is the exit void fraction ?

[Answer]

(a) Void departure point, ZD

[eqn: Pe ~==~ {GD_H c_pl } over k_l]

cpf , kf for 500oF water = 1.176 Btu/lboF , 9.76 x 10-5 Btu/ft-oF-sec

DH = 4A/Pf  = 0.335 ft       G = 47.85x12 = 574.2  lbm/ft2-sec

∴ Pe = 2.32x106 > 70,000

From Saha and Zuber's Eq.,

[eqn: h_f - (h_l )_d ~=~ 154~ q''/G ~,~~~~~for ~Pe ~>~70,000 }}]

= 154 x 55.56 / 574.2 = 14.9 Btu/lbm

∴ (hl)d = hf - 14.9 = 527.5 Btu/lbm

From energy equation ; ROC of energy,

w(hl)d - "PHZd - whin = 0

∴  Zd  = 2.5 ft

(b) thermal-equilibrium quality profile

ROC of energy

w<h(z)> - "PH L  - whin = 0

<h(z)> = (1-xe)hf +xe hg

∴  [eqn: x_e ~==~ {( dot q_o '' P_H ~L) /w ~+~ h_{i n} ~-~ h_f } over h_fg]

=  (74.7+511.9-542.4) / 649.4   =  6.8 %

(c) exit flow quality

[eqn: <x> ~=~ <x_e >][eqn: -~ <x_e >_d ``exp left({<x_e >} over {<x_e >_d} -1 right )]

<xe>d,

[eqn: <x_e >_d ~==~ {( dot q_o '' P_H ~Z_d ) /w ~+~ h_{i n} ~-~ h_f } over h_fg]

=  (15.6 + 511.9 - 542.4) / 649.4   =  - 2.3 %

∴ <x> = 0.068 - (-0.023) exp ( 0.068/(-0.023) - 1 )

= 0.068 + 4.38x10-4  ≅  0.68

(d) exit void fraction

Apply Zuber-Findley correlation.  Or from Modified Armond model,

[eqn: < alpha > ~=~ {(0.833 + 0.167 <x> ) <x> v_g } over {(1. - <x> ) v_f + <x> v_g }]

= { (0.833 + 0.167x0.68) x 0.68 x 0.4456 } / 0.31

= 92.5%