2. Forced Convective Boiling

(1) Single-phase forced convection

o laminar flow

o turbulent flow

- standard form of heat transfer coefficient (h1φ)

[eqn: N u = C_1 Re^n Pr^m]

1) Dittus Boelter equation for the turbulent flow

[eqn: N u_bulk = 0.023 Re_bulk ^0.8 Pr_bulk ^0.4]                      [eqn: <T>& = {1}over{A} int int TdA #&: ~~bulk~ temp.]

2) Sieder-Tate equation : modified D-B equation for high heat flux

[eqn: N u=0.023 Re^0.8 Pr^0.4 left({mu_w over mu}right)^0.14]	at [eqn: T_bulk] and [eqn: mu_w] at [eqn: T_wall]

3) Colburn equation

[eqn: St․Pr^{2/3}=0.023Re^-0.2 ~~~~at~~ T_film ~left(≡ {T_wall +T_bulk}over{2}right)]

where [eqn: St~≡~{N u}over{Re Pr}~=~{H_1phi}over{Gc_p}]

; the ratio of heat added from          q"

⇓	h1φ(Tw-Tbulk)
⇒	Gcp(To-Ti)

wall to the heat transfer by

flow

- good for 0.5 < Pr < 100                ①     ②

- need iteration if h1φ is function of Tw

★ Guess [eqn: T_w] and calculate [eqn: H_1phi] with Colburn Eq using [eqn: T_film],

then [eqn: T_w ^{(n ew)} -T_sat = {q''} over {H_1phi (T_w )}]  if [eqn: T_w ^{(n ew)} ne T_w^(guess)], iteration

4) For superheated steam at high pressure

[eqn: N u =0.0214Re^0.8 Pr^{1/3} left[1+ {2.3}over{L/D_H}right]]	at [eqn: T_film]

5) For multirod bundle [Weisman]

[eqn: N u = cRe^0.8 Pr^1/3]

where c = 0.042 P/D - 0.024      for square pitch lattices

= 0.026 P/D - 0.006      for triangular pitch lattices

(2) Subcoole boiling

1) Onset of subcooled nucleate boiling

The minimum limiting condition for nucleation(Tw > Tsat) is,[eqn: Delta T_sat < dot q" ( 4z over {G c_p D_H} + 1 over h_1phi )][eqn: Delta T_sat < dot q" ( 4z over {G c_p D_H} + 1 over h_1phi )]

[eqn: Delta T_sat < {dot q}{''} left [ 4z over {G c_p D_H} + 1 over h_1phi right]]                              (3.2.1)

● By Bergles and Rohsenow based on Hsu's assumption

[eqn: ({dot q}_ONB )'' = 15.6 p^1.156 left(T_w - T_sat right)^{left(2.3/p^0.0234 right)}]                        (3.2.2)

, for steam-water system in the pressure range of 15-2000 psi

● By Davis and Anderson

[eqn: ({dot q}_ONB )'' = k_f over 4B left(T_w - T_sat right)^2 , ~~~~~~ B = {2 sigma T_sat v_fg} over {J h_fg}]                (3.2.3)

● Empirical correlation by Jens and Lottes

[eqn: left(T_w - T_sat right) = 1.9 e^{-p/900} {dot q ''}^{1/4}]                                (3.2.4)

, for 100<p<2500 psi, [eqn: {dot q}'']>4x106 Btu/hr-ft2, G>7.7x106 lbm/hr-ft2

2) Partial nucleate boiling (Wall voidage region)

[eqn: {dot q}'' = ({dot q}_1phi )'' + ({dot q}_SCB )'']

① Begining of fully developed nucleate boiling is defined as,

[eqn: ({dot q}_FDB )'' = ({dot q}_o )'']  at the point D'     McAdams

[eqn: = (1.4{dot q}_o )'']                   Forster and Grief

② Emperical correlations for convective partial nucleate boiling heat transfer

From the saturated nucleate pool boiling heat transfer suggested by              Rohsenow, which is given as,

[eqn: rm LEFT [ {C_pf ```△T_sat } over h_fg RIGHT ] ~~=~~ C_sf ~LEFT [ q'' over {μ_f h_fg } ``` LEFT ( σ over {g (ρ_f - ρ_g ) } RIGHT ) ^{1over2} RIGHT ]^{n} ~LEFT [ {C_p μ } over k RIGHT ] _f ^m+1]

Author	Geometry	ID(mm)	Csf	Fluid-suface
Rohsenow & Clark	수직 튜브	4.56	0.006	water/Ni
Krieth & Summerfield	수평 튜브	14.9	0.15	Water/SS
Piret & Isbin	수직 튜브	27.1	0.013	Water/Cu
Berges & Rohsenow	수평 튜브	2.39	0.02	Water/SS

Csf is experimentally adjusted for the forced(and natural) convective                  subcooled boiling

Kutateladze estimate the curve in the region of partial nucleate boiling as,

[eqn: {dot q''} over {({dot q}_1phi )''} = left[ 1+ left{{({dot q}_FDB )''/(T_w - T_bi )} over {({dot q}_1phi )''/(T_w - T_b )} right}^2 right]^1/4],

where [eqn: ({dot q}_FDB )''] is taken from saturated pool boiling data

Bergles and Rohsenow observed from experiments that the curve for a flow      boiling must be based on actual flow boiling data, not data for saturated         pool boiling.  Thus,

[eqn: {dot q''} over {({dot q}_1phi )''} = left[ 1+ left{{({dot q}_B )''} over {({dot q}_1phi )''} left(1 - {({dot q}_Bi)''} over {({dot q}_1phi )''} right) right}^2 right]^1/2]

where [eqn: ({dot q}_B )''] : fully developed nucleate boiling

[eqn: ({dot q}_Bi )''] : fully developed nucleate boiling at the onset of subcooled                         boiling

The wall and coolant temperature difference is given by Weatherhead as,

[eqn: T_w - T_sat = 0.015x10^6 sigma (dot q ''/10)^1/4]

3) Fully developed subcooled boiling

① Mechanism of subcooled nucleate boiling

a) sequential rate process model by Bankoff

☞ heat from the wall surface to adjacent

two-phase boundary layer

☞ heat flows through the two-phase

boundary layer

☞ pass into main single-phase core

b) thermocapillarity model by Brown

☞ flow jet in the wake of the bubble

② Heat transfer in fully developed subcooled boiling

[eqn: (T_w )_sub ~=~ T_sat + PSI ( dot q'' )^n][eqn: (T_w )_sat ~=~ T_sat + psi`` (dot q '')^n]

McAtams  ;  n=0.259

ψ=22.62 ～ 28.92 (∝ physical properties of fluid and liquid-

surface combination)

Krieth and Summerfield,

[eqn: TRIANGLE T_sat ~∝~ p^-0.75]

Jens and Lottes

[eqn: TRIANGLE T_sat (^o C) ~=~ 25 {dot q ''}^0.25 (MW/m^2 ) e^{-p(b ar)/62}]

Thom

[eqn: TRIANGLE T_sat ~=~ 22.65 {dot q ''}^0.5 e^{-p/87}]

③ Effect of dissolved gas

☞ increases the heat transfer since the gas bubble agitates the liquid

☞ affects on the position where the first evolution of bubbles occurs

☞ dies out at higher heat flux near CHF

④ Effect of system subcooling

☞ D, lifetime and population of bubble decrease as [eqn: TRIANGLE h_{s ub}] increases

☞ also decrease as the flow velocity increases at constant heat flux

⑤ 기타

For constant velocity and [eqn: TRIANGLE h_{s ub}], as the heat flux increases,

☞ diameter and lifetime of bubble moderately increase.

☞ bubble population increases sharply.

(3) Nucleate boiling

[eqn: {dot q ''} over 10^6 = kappa {(T_w - T_sat )}^m]                                     (3.2.5)

Jens-Lottes   :  m=4,  [eqn: kappa = {exp(4p/900)} over (60)^4]

Thom        :  m=2,   [eqn: kappa = {exp(2p/1260)} over (72)^2]

=> insensitive to flow rate and quality

■ Forced Convective vaporization

For high quality annular flow

o nucleation is suppressed

o heat transfer by conduction through thin film

o evaporation at interface

Thus, dramatically improvement of heat transfer

[eqn: dot q '' = {k delta T} over sigma]   (例 : 35,000 Btu/hr-ft2 for water)

[eqn: H_2phi(FC) over H_{1phi(l``o)} ~=~ A left( 1 over X_tt right)^n]                                (3.2.6)

where   A = 3.5,  n = 0.5    by Dengler and Addams

A = 2.9,  n = 0.66    by Bennett

★ Turbulent Martinelli parameter, [eqn: X_{t``t}]

∝ function of quality and pressure

≡ [eqn: sqrt {( d p / dz)_l ``/``(d p / dz)_v}]

[eqn: 1 over X_tt ~=~ left( <x> over (1-<x>) right)^0.9 left( rho_f over rho_g right)^0.5 left( mu_g over mu_f right)^0.1]                         (3.2.7)

○ criterion for the suppression of nucleate boiling

[eqn: delta ^+ ~=~ 7]

The heat transfer within the liquid film

The turbulent heat transfer is

[eqn: q'' ~=~ -(k_f + epsilon_H c_pf rho_f ) dT over dy]                                  (3.2.8)

B.C ;   T = Tw    at y=0

T = Ti    at y=δ

Integrating Eq.(3.2.8) from y=δ(T=T-I≈Tsat) to y=0,

[eqn: h_Tp ~=~ {c_pf rho_f u^*} over T_i^+]       where [eqn: T^+ ~=~ {c_pf rho_f u^*} over q'' (T_w - T)]

[eqn: u^* ~=~ sqrt{tau_w /rho_f }]

여기서, [eqn: T_i^+] 는 Ref 와 Prf 의 함수이다.

■ Chen correlation for both the fully developed nucleate boiling and forced

convection vaporization region

[eqn: h_2phi ~=~h_NCB ~+~ h_C]                             (3.2.9)

In this equation, the hC is given by a modified Dittus-Boeter equation as,

[eqn: h_C ~=~ 0.023 left[ G(1-<x>)D_H over mu_f right]^0.8 left( {c_pf mu_f} over k_f right)^0.4 left( k_f over D_H right)F]         (3.2.10)

where F ; ratio of an effective [eqn: Re_2phi] to Re used in [eqn: h_1phi]

≡ [eqn: left[ Re_2phi over Re_f right]^0.8] = [eqn: left[ Re_2phi over {G(1-<x>)D_H / mu_f} right]^0.8]                   (3.2.11)

∝ Xtt

And [eqn: h_NCB] is modified Foster-Zuber equation by a nucleation suppression

factor as,

[eqn: h_NCB ~=~ 0.00122 left[ {k_f^0.79 c_pf^0.45 (g_c^* )^0.25} over { sigma^0.5 mu_f^0.29 h_fg^0.24 rho_g^0.24} right] left( h_fg over {T_sat v_fg } right) TRIANGLE T_l^0.24 TRIANGLE p_l^0.75 ``S]        (3.2.12)

where  S ; ratio of mean superheat ([eqn: TRIANGLE T_l] ) in the liquid film to wall

superheat ([eqn: TRIANGLE T_sat] )

≡ [eqn: left[ {TRIANGLE T_l} over { TRIANGLE T_sat} right]^0.99]

= [eqn: ( TRIANGLE T_l ```/ TRIANGLE T_sat )^0.24 ``( TRIANGLE p_l ``/ TRIANGLE p_sat )^0.75]   by Clausius-Clapeyron

[eqn: therefore ~h_NCB ~=~ 0.00122 left[ {k_f^0.79 c_pf^0.45 (g_c^* )^0.25} over { sigma^0.5 mu_f^0.29 h_fg^0.24 rho_g^0.24} right] left( h_fg over {T_sat v_fg } right) TRIANGLE T_sat^0.24 TRIANGLE p_sat^0.75 ``S]    (3.2.13)

By curve fit

F  =  1                              for  1/Xtt ≤ 0.1

=  [eqn: 2.35``(1/X_tt + 0.213)^0.736]           for  1/Xtt > 0.1        (3.2.14)

S  =  [eqn: 1``/``(1+2.53x10 ^{-6} Re _{2 phi } ^{1.17} )]                              (3.2.15)

☞ Method to calculate [eqn: h_2phi] at a known q'', G and <x>

① calculate 1/Xtt          from Eq.(3.2.7)

② evaluate F             from Fig. 7.5 (Collier) or Eq.(3.2.14)

③ calculate [eqn: h_C]           from Eq.(3.2.10)

④ calculate [eqn: Re_2phi]          from [eqn: Re_f]  and F  [See Eq.(3.2.11)]

⑤ evaluate S using [eqn: Re_2phi]  from Fig. 7.6 (Collier) or Eq.(3.2.15)

⑥ calculate [eqn: h_NCB]          from Eq.(3.2.13)

⑦ calculate [eqn: h_2phi]           from Eq.(3.2.9)

⑧ Plot q''([eqn: =~h_2phi TRIANGLE T_sat] ) for a range of [eqn: TRIANGLE T_sat] and interpolate [eqn: h_2phi] at a

given q''

(4) Boiling Transient in Forced Convection

[eqn: rm cases { B`o`i`l`i`n`g~` c`r`i`s`i`s~ : ~l`o`w`~`q`u`a`l`i`t`y~D`N`B # C`H`F~:~l`o`c`a`l~h`e`a`t~ f`l`u`x # D`N`B~:~h`i`g`h~ p`r`e`s`s`u`r`e~ ＆~ h`i`g`h~ f`l`o`w~ (`P`W`R`)`` # B`u`r`n`o`u`t # D`r`y`o`u`t~ :~ h`i`g`h`~`q`u`a`l`i`t`y }]

1) Mechanism of boiling transient

A) Subcooled and low quality region

① Dryout under a vapor dot

; A dry patch on heating surface by micro-layer evaporation can be                rewetted when the bubble departures. However, in the case of high                heat flux, a large bubble makes the surface temperature under the dry              patch increase abruptly before the bubble departure, which meets                   the CHF condition.

② Bubble crowding and vapor blanketing

At high heat flux, bubbles are crowded.

∴ bubbly boundary layer is formed and it prevent the liquid from                     incoming toward the surface  ⇨ vapor blanket

③ Evaporation of liquid surrounding a slug flow bubble

- At low flow rate, slug flow exists.

- For high flux, the liquid film may be completely

evaporated and thus dries out

B) Counterflow CHF

C) Film dryout

due to

① x will be 1 by adding

continuous heat (evaporating)

② entrainment

→ Those phenomena depends on

[eqn: rm cases{mass~ velocity #` su bcooling }]

2) BT Analysis

For bulk boling : "local condition hypothesis"

From the heat balnace

[eqn: dot O`-`dot I``+`dot S`=`0`]      for S.S

[eqn: G``A_{x`-`s}`` left[ `h_f `+ `<x>`h_{f`g}`` right]` - `left[`G``A_{x`-`s}``h_{i`n }+`` {bar q}''`P_H```L_H```right]`=`0`]

[eqn: ∴~`{ bar q}''``=`left[ {G`A_{x`-`s}`` h_{f`g}} over {P_H```L_H`} right]`<x_e`> `+``left[ {G`A_{x`-`s}} over {P_H```L_H`} right]``triangle`h_{s``u`b}``]

For a round tube at low mass flux [eqn: (` G``/`10^{6}`〈``0.5``)`],

McBeth correlation (world's data within ± 5.5 % )

will be

[eqn: {{bar q_c}''} over {10^{6}}`=` {h_{f`g}} over {1`3`5} `sqrt{G`/`10^{6}``} ``left( 1`-`<`x_e`>_c``right)``] (3.2.16)

By Long-Sun Tong, from the heat balance,

[eqn: <`x_e`>_c``=`` { left(`{bar q_c}'' `/` 10^{6}``right)`P_H```L_H` } over {left( `G`/`10^{6} right) ``A_{x`-`s}```h_{f`g}} `-` {triangle` h_{s`u` b} } over {h_{f`g} }``]

(3.2.17)

Thus Eqs. (3.2.16) + (3.2.17),

[eqn: ∴~`{bar q_c}''`/`10^{6}``=``{left( `G``/`10^{6}` right)`A_{x`-`s}```left[ `h_{f`g}`+` triangle h_{s `u`b}``right] } over { 1`3`5``sqrt{G`/`10^{6}```}A_{x`-`s}``+`P_H```L_H }``]

A. Flooding and Flow Reversal

☞ Rewetting from the top or countercurrent flow limiting condition

☞ Motion of molten cladding material

☞ Downcomer flow pattern during ECC

Fig. 3.10 Flooding and

flow reversial

experiment

For ②  ․Liquid become unstable and forms waves of large amplitude

․Entrainment

→ Flooding

For ④  ․Liquid injection dries out and no net liquid downflow

․Churn or annular flow in the upper tube

For ⑤  ․When the gas flow rate decrease, liquid film becomes unstable

and forms waves of large amplitude

․△P increase and the liquid film tends to fall down

→ Flow reversal

Fig. 3.11  Flooding and flow

reversial ; Location of

the liquid phase above

and/or below the liquid

injection zone

For ① in Fig. 3.10,  Nusselt thickness (const. film thickness based on             laminar flow) is

[eqn: e=[{3nu_l `theta _l} over { pi Dg} ]^1/3]     [eqn: nu _l] : liquid kinematic viscosity

[eqn: theta _l] : liquid volumetric flow rate

D : pipe diameter

which is valid for  [eqn: Re_l ~(≜ {4Q_l} over { pi D nu _l} )] > 4,000

Fig. 2.12 wg vs wl for flooding

Generally, Wg is independent on

Wl for the case of flow reversal

while decreases as Wl increases

during flooding.

■  Countercurrent Flow Limit (CCFL)

그림 3.13 CCFL in BWR

fuel bundle[Lahey, 1993]

: Global flooding phenomena in BWR bundles

- During the plant transients, when the inlet mass

flow decreases, the water in the upper plenum

falls down to cover the loss of flow.

But, if the jg of exit vapor is sufficiently large,

CCFL occurs in the upper area restriction, which

prevents the water in upper plenum from falling          down. and thus make fuel rods dry out.

∴ no steady flow solution. is possible

① Empirical CCFL Correlations

☞ Kutateladge CCFL Correlation

[eqn: K_g^1/2 + |K_f ~| ^1/2 ~=~ (3.2)^1/2]         (3.2.18)

where, [eqn: K_i ~=~ {( j_i )_c root{rho_g }} over { [ sigma g g_c (rho_f - rho_g ) ]^1/4 }],  i = g or f

☞ Wallis Correlation

[eqn: (j_g ^* ) ^1/2 +(j_f ^* ) ^1/2 ~=~ C]

(3.2.19)

where C is empirical constant which depend on the entrance conditions             and ranges usually 0.7 ～ 1.0 ([eqn: sim 0.6] by Shiralkar)

[eqn: j _g ^*] (Francle #) =  [eqn: G_g over { [ g D_H rho_g (rho_f ~-~ rho_g ) ]^1/2 }]; dimensionless flux of gas

[eqn: j _f ^* ~=~ G_f over { [ g D_H rho_f (rho_f ~-~ rho_g ) ]^1/2 }]

② CCFL Curve

For steady state, a mass balance in Fig 3.13 is              [eqn: G_g A_o ~=~ |G_f | A_o ~+~ G_i A_x-s]   (3.2.20)          where Ao is a restrict area.

And the energy balance (assuming saturated         fluid in the upper plenum) is given as,

[eqn: G_i A_x-s h_i ~+~ |G_f | A_o h_f ~+~ q ~=~ G_g A_o h _g]  (3.2.21)

From Eqs. (3.2.20) and  (3.2.21)

[eqn: G_g ~=~ { ( q -G _i A _x-s triangle h _{s u b} ) } over { A_o h _fg }]                                                                   (3.2.22)

Now Eqs. (3.2.20) and (3.2.22) will give,

[eqn: |G_f | ~=~ { (q -G_i A_x-s triangle h_{s u b }) } over {A_o h_fg} ~-~ G_i left( A_x-s over A_o right)]                    (3.2.23)

Then, inserting Eqs.  (3.2.22) and (3.2.23)  into Eq. (3.2.19)

=> [eqn: q_c] (critical bundle power)

Now if [eqn: q'' (z,t)] >[eqn: q''_c (z,t)], dryout and overheating occurs