B. nonuniform axial heat flux treatment

BT data ∝ [eqn: rm cases {fluid#complex~``g eometry#nonuniform~`axial~ and~ local~ heat~ flux}]

① local cond. hypothesis

: It is only the local heat and local quality that control the BT

∴ the upstream history is not important

② Integral concept

→ upstream history is quite important

→ strong dependence on axial heat flux profile

⇒ "upstream memory effect"

→ valid for the high quality condition

(A) F factor method ; treat the effect of upstream history.

; divide uniform axial CHF by upstream weighting factor, F, that                yield the appropriate thermal limit for nonuniform axial heat flux              profile.

[eqn: rm THEREFORE ``F`≙` {bar{q} _{EQ}} over {q '' _{c} ``(L _{B _{c}} `+` lambda `)`}]  : Tong F-factor                    (3.2.24)

[eqn: =` {OMEGA _{c}} over {q '' _{c} ``(L _{B _{c}} `+` lambda `)`[`1`-`e ^{-` OMEGA `L _{B _{c}} `} `]} `` int _{lambda } ^{L _{B _{c}} `+` lambda } {} q '' (q)`exp` LEFT { -` OMEGA _{c} `[`(L _{B _{c}} `+` lambda `)`-`g]` RIGHT } ``dg]

where [eqn: OMEGA_c`=`C over {D_H}`` APPROX ``0.135 over {D_H}``]

As [eqn: <`x_e`>_c``] increase, [eqn: OMEGA_c`] decrease.

★ Derivation of Tong F-factor

->  analyze the boiling crisis of non-uniform heat flux distribution in

a subcooled or low quality flow [Tong]

; consider an overheating of surface due to a hot patch under bubble            layer

[eqn: cases{#`#}]               big bubble at low flow rate and low pressure.

bubble layer at high flow rate and high pressure.

o suggested the highly viscous bubble layer of crowded bubbles which           covers a layer of superheated liquid near the wall

o The liquid superheat and

그림 3.14 Superheated liquid layer between bubble

layer and heating surface

the local heat flux

determine the wall

temperature and thus

the DNB condition.

o In this model, the fluid

enthalpy from an

energy balance of the

superheated liquid layer

is a representative

quantity for the onset of DNB

[eqn: dot O``-``dot I``+`dot S`=`0]      At low quality, the energy balance through the bubble layer is,

[eqn: {Partial } over {Partial t} `` LEFT [ ` LEFT ( `h`-` {P} over {rho } ` RIGHT ) `` rho ``P _{H} `G`` RIGHT ] ` TRIANGLE `z```+`]

[eqn: H_{1`varphi`}`(T&-`T_b`)`P_H``triangle`z``+``h`G_c`theta`P_H``]     즉,

[eqn: +`` {Partial } over {Partial `z} ` LEFT [ h`G _{c} ` theta `P _{H} `` RIGHT ] ` TRIANGLE `z`-`h``G _{c} `` theta ``P _{H} ``-``q '' `P _{H} `` TRIANGLE `z```=`0]

Dividing [eqn: P_H`triangle`z`] and [eqn: T`-`T_p`=`1 over {C_{p`l}} `left[ `h`-`h_p`right]`]

[eqn: ∴~``{H_{1`varphi}} over {c_{p`l}`} `(`h`-`h_b`)``+``partial over {partial z}[`h`G_c``theta`]`-`q''``+``partial over partialt`[`h`rho`theta`-`P`theta``]`=`0`]

For steady-state ( ÷[eqn: G`theta`] )

[eqn: d over dz`[`h`-`h_b`]``+``OMEGA`(`h`-`h_b`)``=``OMEGA``{C_{p`l}} over {H_{1`varphi}}`q'']

where [eqn: OMEGA``≙``{H_{1`varphi}} over {G_o``theta`C_{p`l}}``=``St over theta `=`const``]

and  [eqn: B.C````````` h(`z`=`lambda`)``=```h_b``=``h_f``]

Introducing the integrating factor, [eqn: e^ OMEGAz],

[eqn: rm (h(z) - h_b`)_{non``uniform}~=~ {OMEGA ```C_pl } over H_1phi``` INT _{ lambda }^{ z}``q″`(z)``e^{- OMEGA ``(z-j)}``dz]

For uniform axial heat flux,

[eqn: rm (h(z) - h_b`)_{uniform}~=~ {C_pl```bar q ″} over H_1phi``` [``1-``e^{- OMEGA ``z`}``]]

At the critical pt. (DNB),  assume,

[eqn: rm (h(z) - h_b`)_{non``uniform}~=~ (h(z) - h_b`)_{uniform}]

Then,

[eqn: rm {bar q″}_c(EQ) ~=~ OMEGA over {[1-e^{-OMEGA`z_c }]}``INT _{ lambda }^{ z_c}``q″`(z)``e^{- OMEGA ``(z_c - z)}``dz]     (3.2.25)

: equivalent uniform axial CHF

Thus, Tong-F factor will be

[eqn: rm F == {bar q″}_c(EQ) over {q″_c (Z_c ) }``=``OMEGA over {q″_c (Z_c )``[1-e^{-OMEGA`Z_c }]}``INT _{ 0}^{ Z _c}``q″`(z)``e^{- OMEGA ``(Z_c - z)}``dz]

for nonuniform heat flux in PWR

Empirically,

[eqn: rm OMEGA ``=``0.44` {[`1-x(Z_c ) ]^7.9} over {(G/10^6 )^1.72}~~~~~["in"^-1 ]]

PWR : low quality → high Ω

BWR : high quality → low Ω

Consider [eqn: rm q″(z)```e^{- OMEGA``(z_c- z)}]

a) For PWR having large Ω,

Eq.(3.2.25) dies out.

∴ little upstream "memory" effect

i.e. local heat flux is important

b) for BWR → small Ω

∴ upstream history is important  (∵ [eqn: rm e^{- OMEGA``(z_c- z)}]    ≫ )

For the case of very small Ω ( limΩ → 0 ),  Eq. (3.2.25) will be

[eqn: rm lim from { OMEGA ->0 }``F`` == {bar q″}_c(EQ) over {q″_c (L_B + lambda ) }``=``OMEGA over {q″_c (L_B + lambda )``L_B}``INT _{ lambda }^{L_B + lambda }``q″`(z)```dz]

where [eqn: rm][eqn: rm {bar q″}_c(EQ) =``1 over z_c``INT _{ 0 }^{z_c }``q″`(z)```dz]

■ Drawbacks of Tong F-factor

a) Computer-based calculation are expressed (numerical integration of

exponential integral)

b) Complex graphic interpretation is required

([eqn: BECAUSE][eqn: ∝] inlet subcooling, axial heat flux profile)

c) Exponential weighting of the upstream history is not nearly important for             BWR (∵high quality)

(B) The critical quality-boiling length([eqn: <X_e >_c -L_Bc]) approach

- CISE in Milan, Italy

Note:  o local condition, hypothesis : [eqn: ddotq_c]  ↔[eqn: <X_e >_c]

o Tong F-factor         : local parameter <-> modification of [eqn: CHF_uniform]

o [eqn: <X_e >_c -L_Bc] approach     : [eqn: <X_e >_c ↔L_Bc]

[eqn: Dot O -Dot I +dot S =0]

[eqn: rmGA_x-s [h_f + <X_e >h_fg ]-GA_x-s h_f - INT _{ lambda }^{ L_Bc + lambda }P_H ddotq (z)dz ~=~0]

where λ is given as

[eqn: rm INT _{ 0}^{ lambda }P_H ddotq (z)dz=GA_x-s DELTA h_{s ub}]

CISE correlation

[eqn: rm <X_e >_c ={a(p,G)L_Bc} over {b(p,G,D_H )+L_Bc }]

Since BT data, Fig 4-25, for uniform axial heat flux profiles are trended to         be linear in the flux-quality plane,

[eqn: rm &{bar q}_c '' =n(G,p)-m(G,p)<X_e >_c]

From

[eqn: {bar q}''_c(EQ) ~≡~ 1 over L_Bc INT _{ lambda }^{ lambda +L_Bc }ddot q_c (z)dz]  and   [eqn: <X_e >_c ~=~{ INT _{ lambda }^{ lambda +L_Bc }ddotq (z)P_H dz} over {GA_x-s h_fg} ,]

we can obtain

[eqn: rm<X_e >_c ={(n/m)L_Bc} over {[(GA_x-s h_fg )/(mP_H )+L_Bc ]}]

Thus, if we can determine n,m for uniform heat flux, BT for nonuniform axial        heat flux can be obtained

→ inadequate for cases of extreme axial heat flux profile.

∴ the ratio of  peak-to-avg. axial flux for BWR is <2.0

3) Parametric effects on CHF

a) Inlet subcooling

- linearly increases with the inlet subcooling

- However, as G decreases, the effect becomes smaller and finally nearly               no effect for the case of experiments at low p and G.

b) Mass flux, G

- increases as G increases, but depends on the pressure

- At low p, it exists a region where CHF doesn't increase as G increases

c) System pressure, p

- At low G, CHF for fixed <x>exit increases as p increases until p<100

but decreases from p>100 bar

d) Tube diameter, D

- increases as D increases due to decrease of <x>exit

- Effect of D increases as the inlet subcooling increases

e) Heated length, L

- decreases as L increases due to increase of <x>exit

f) Exit quality, <x>exit

- strongly depends on flow geometry and conditions

- At low quality, the increase of <x>exit decreases CHF.

g) Surface roughness

- no effect, but shift the boiling curve

h) Effect of Acceleration

; cause the fluctuations of void, the decrease of velocity and the variance of           flow parameters

- CHF decreases with increasing acceleration.

[eqn: & CHF) prime `/~CHF`=~(1-a/g) ^{1/4} ~~~~~~~~~~~~~~~~~~# # & CHF) prime `/~CHF`=~1.0-0.5*a/g~~~~~~~~~~~~~~~~~# # & CHF) prime `/~CHF`=~0.85~~~~~~~~~~~~~~~~~~~~~~~~~~]

㉠ heaving ; When the marine heaves, the coolant flow velocity would                           decrease, and will bring down the CHF.

㉡ listing ; As the marine lists, the hot fluid will concentrate on the side of                    core and the adverse flow will take place near the side. Thus,                        the local values of CHF will decrease.

i) Other Parameters

◆ higher lateral crossflow - increase

◆ mixing vane on spacer  - increase

◆ greater axial spacing    - increase

◆ axail offset

4) CHF correlations

5) Measurement of CHF

a) measurement the temperature excursion with thermocouple

b) Bridge circuit : local resistance increase due to temperature increase

c) Infra-red burnout detector

6) Improvement of CHF

a) For boiling in internal flow

① Generation of Swirl Flow

☞ twisted tape (or ribbon) ; △p and thus pump capacity increase

- Tokamak cooling

☞ ribbed tube

- large once-through boiler in fossil plant 에 사용,  100% enhancement

② Increase of heat transfer area using fin

③ Increase of turbulence or surface roughness control

☞ helical equipment inside tube

b) For rod bundle

① spacing of grid

regime	mechanism	CHF
general	enthaly mixing increase and uniform flow distribution	increase
low <X>	scattering bubble near heated surface	increase
high <X>	break of liquid film	decrease

② mixing vane

regime	mechanism	CHF
general	enthaly mixing increase	increase
low <X>	scattering bubble near heated surface, centrifugal acceleration	increase
hign <X>	break of liquid film	decrease
	droplet dispacement from channel center to wall	increase

c) For pool

☞ oxidation or selective fouling of the heated surface

☞ extened surface by fin

☞ valatile liquid additives

☞ rotation or vibration of heater

☞ fluid vibration

☞ use of electric fields

-> enhancement from several tens % to 200%

7) Design limit for CHF

● Condition I

● Condition II

----------- protection from DNB

● Condition III

● Condition IV

----------- some level of fuel damage is permissible

DNBR Limit  ;

DNBR ≡  [eqn: {{dot q}_CHF (correlation)} over {{dot q}_actual}]

The correlation for[eqn: {{dot q}_CHF }] should have 95% probability with 95% confidence.

a) Establishment of CHF 상관식

① perform multi-rods CHF experiment

; CHF data

② perform Subchannel Analysis

; determine the local parameters

③ Regression Analysis

; establish CHF correlation as a function of key thermal hydraulic

parameters

☞ F-factor to account the nonuniformity of heat flux

☞ grid correlation for mixing vane grid

=> [eqn: CHF(``x,`` p,`` D_H ``,``T_{i n} ``, ``G``)] with 95% probability at 95% confidence level

b) Design limit for CHF correlations, MDNBR

●  W-3 (W)          MDNBR = 1.3

- W-3  L-grid     MDNBR = 1.3    (14x14, 15x15)

- W-3  R-grid     MDNBR = 1.3    (17x17)

●  WRB,              MDNBR = 1.49   for typical cell

MDNBR = 1.47   for thimble cell

●  B&W-2            MDNBR = 1.3

●  CE-1              MDNBR = 1.19

c) reactor thermal-hydraulic design

; assure that minimun DNBR keep above Design Limit DNBR during normal

operation and anticipated transients

① determinstic limit DNBR

☞ DNBR whose possibility is less than

5%  =>  Design limit

☞ conservative input at core minimum

DNBR calculation

② Statistical approach for limit DNBR

; Instead of using input parameters at

their nominal or best estimate values,

□□ operating parameters           ⌍

□  nuclear and thermal parameters □  are considered statistical to obtain

□□ fuel fabrication parameters     □□  DNBR uncertainty factors

A. ITDP(Improved Thermal Design Procedure, Westinghouse)

B. Monte Carlo Method

(5) Post Boiling Transient Convective Heat Transfer

- depends on the quality and flow rate at which BT occurs

① for low-quality/high-flow BT

o severe temperature excursion at DNB point

o inverse annular flow

② for high-quality/low-flow BT

o milder temperature excursion

o mist flow

o typical in BWR technology

그림   Post-Boiling Transition Heat Transfer Regimes

1) Post-Dryout Mechanism

그림    A mechanistic Model of PDO Convective Heat Transfer

①  Wall → Vapor  : heat transfer from surface to vapor under forced

convection

o thermal equillibrium model

o frozen droplet model

②  Wall → Droplet

Droplets fallen on the wall are evaporated if the wall temperature is not          too high, and then cool the wall

③ Wall → Vapor → Droplet

At high suface temperature, droplets come close to the wall and enter           the vapor surface layer.  The side of droplets faces the high temperature           of surface and so there is more evaporation on that side.  This forces the          droplet back from the surface.

2) PDO Heat Transfer Coefficient

▶ Dermination of vapor temperature

- important to determine PDO heat transfer coefficient

[eqn: {dot q ''}_PDO = H_PDO (z)[T_w (z) - <T_v (z)>]]

- Two bounding models

① Thermal Equilibrium

Model

Tv = Tsat , <x(z)> = <xe(z)>

즉, no vapor superheat and all  energy contribute only to

evaporate droplets

② Frozen Droplet Model

<x(z)> = <xe(z)>c

즉, no further liquid droplet evaporation, but superheating

the vapor

▶ PDO heat transfer coefficient

For <xe> ≤ 1.0   apply the thermal equillibrium model

[eqn: Nu_PDO (z) = 0.023 left[G<x(z)>D_H over <alpha(z)>mu_g right ]^0.8 Pr_g ^0.4]

<x>

-> similar with Dittus-Boelter Eq

Tv : superheated

1.0

Tv=Tsat

For <xe> ≥ 1.0   apply the frozen           1.0        <xe>

droplet model

Here, the liquid droplets are assumed not to participate in the heat               transfer process.

[eqn: Nu_film(z) = 0.023 left[{G<x(z)>_c D_H} over <alpha(z)>mu_v right ]_film ^0.8 Pr_film ^0.4 left({c_p ^bulk} over {c_p ^film} right) _v]

☞ Grobal empirical correlations

1) Groeneveld correlation

2) Dougall-Rohsenow correlation

3) Bishop correlation for PWR