22.06 Engineering of Nuclear Syste m s

MIT Department of Nuclear Scienc e and Engineering

NOTES ON TWO ‐ PHASE FLOW, BOILING HEA T TR ANSFER, AN D BOILING CRISES IN PWRs AN D BWRs

Jacopo Buongiorno

Associate Professor of Nuclear Sc ie nc e and Engi neering

Defini tio n of the basic two ‐ phase fl ow par a me ters

In this docum e nt the subscripts “v” and “ ℓ ” indica te the vapor and liquid phase, respectively . The subscripts “g” and “f” indi cate the vapor and li quid ph ase at satur a tion, respectively.

Void fraction:  

A v

A v  A ℓ

Static quality: x  M v 

A v  v

st

v

 M ℓ

A v  v

 A ℓ  ℓ

Flow quality : x  m ˙ v 

v v A v  v

m ˙ v  m ˙ ℓ v v A v  v  v ℓ A ℓ  ℓ

Slip ratio: S  v v v ℓ

Mix t ure dens ity:  m    v  ( 1   )  ℓ

Flow quality, void fraction and slip ratio are generally related as follows:

   1

1  v  S  1  x

 ℓ x

Figures 1 and 2 show the voi d fraction vs flow quality for various values of th e s lip ratio and pressure, respectively, for steam/water mi xtures.

Figure 1. Effect of S on  vs x for water at 7 MPa. Figure 2. Effect of wa ter pressure on  vs x for S= 1.

In 22.06 we assume homo gen e ous flow, i.e. v v =v ℓ or S=1. This ass u mption ma k e s it rela tively simple to treat two ‐ ph ase mix t ures effectively as single ‐ phase fluids with va riable proper ties. If S= 1, it follows

1 x 1  x

immed i ately that x st =x and, after some trivial algebra,      .

Note that the assumption of homogene ous flow is very restrictive, and in fact a ccurat e only under limite d condi tions (e .g. dis p ersed b ubbl y flow, mist flow). In gene ral, a significa nt slip be twe e n the two phases is pre s ent, which requires the use of more re alistic models in w h ich v v  v ℓ . Su ch mod e ls will be discussed in 22.312 and 2 2 .313.

Two ‐ ph a s e fl ow regime s

With ref e rence to upflow in vertical cha nnel, one can ( l oosely) ide n tify several fl ow regimes, or patterns, whose occurr ence, for a gi ven flui d, pressure and cha nnel geome t r y , depe nds on th e flow quality and flow rate. Th e main flow re g i m e s are reported in Ta b l e 1 and sho w n in Fig u re 3. No te that what values of flow quality and flow rate are “low”, “interme diate” or “high” depen d on th e fluid and pressure. In horizontal flo w , in addition to the above flow regimes, there can also be stratified flow, typi ca l of low flow rates at which th e tw o phases separate un der the effect of gravity.

Table 1. Qual itative classification of two ‐ phase flow re g i m e s.

a

b

c

d

e

f

g

T ypical configuration of (a) bubbly flow , (b) dispersed bubbly (i.e. fine bubbles dispersed in the continuous liquid phase), (c) plug/slug flow , (d) churn flow ,

(e) annular flow , (f) mist flow (i.e. fine droplets dispersed in the continuous vapor phase) and (g) stratified flow . Note: mist flow is possible only in a heated channel; stratified flow is possible only in a horizontal channel.

Image by MIT OpenCourseWare.

Mo re q u anti t a tively, one can de termine which flow re g i m e is pres ent in a parti c ular situatio n of interest resorting to an empirical flow map (see Figure 4) . Note that such maps de pend on the fl uid, pressure and channel geometry, i. e. , ther e is no “universal” map for two ‐ ph ase flow regimes. However, there exist method s to ge nerate a flow map for a particula r fluid, pressure and geom etry. Th ese methods will be covere d in 22.312 and 2 2.313.

Image by MIT OpenCourseWare.

Figure 4. A typical flow ma p obtained fr om data for low ‐ pressure air ‐ water mi x t ures and hi g h ‐ pressure water ‐ steam mixtures in small (1 ‐ 3 cm ID) adiabati c tubes (adapte d from He wi tt and Rober t s 1969).

Ho w to calculate pr essur e changes in two ‐ ph ase fl ow channels

For a straight chan nel of flow area A, w e tte d perim e t e r P w , equiva lent (hy d rauli c ) diame t er D e (  4A/P w ), inclina t ion angle  , and length L, con n ec ted to inl et an d outlet ple n a, the total pressure dro p (i nlet plenu m pressure mi nus outlet plen um pressure) can be obtained fr om the steady ‐ state mo mentum equation:

P 0  P L   P tot   P acc   P fric   P grav    P form (1)

j

where the subscripts “0” an d “L” refer to inlet and outlet ple n u m , respectively. Th e last ter m on the RHS of Eq . 1 r epresents th e sum of all for m losses in the chan nel, i n cluding thos e du e to the in l e t , the outlet and all other ab r u pt flow area ch anges in th e channel (e .g. valves, orifices, spacer gri d s).

Under the as sumption of homogenous flow (S=1 or v v =v ℓ ):

 P  G 2 [ 1  1 ] (2)

acc

 m , L

 m , 0

 L 1 G 2

P fric   0 f D 2  dz (3)

 P grav   0  m g cos  dz (4)

Note that the above equati ons are formally identical to the singl e ‐ phase case wi t h the mi xtur e de nsity,

 m =  v +( 1 ‐  )  ℓ , used in st ead of th e single ‐ phase den s ity. The friction factor in Eq. 3 ca n be calc ulated from a single ‐ phase correla tion and using th e “liquid ‐ only” Reynol ds nu mber, Re ℓ o =GD e /  ℓ .

For each abr upt fl ow area chan ge in th e cha nnel (ag a in in cludi ng the i n let and outlet) , the associated pressure loss is the sum of an acceleration term and an irreversible loss term:

 1 2 2 G

P form   P form , acc   P form , irr  2 

( G 2  G 1 )  K

m m

(5)

where the subscripts “1” an d “2” refer to the locatio n immedia t el y upstrea m an d downs t re am of th e abrupt area change, respe c tively.

Boiling heat transfer

Boiling is the transition fro m liquid to va por via formation (o r nu cle a tion) of bub bles. It typi cally requires heat addition . Wh e n the boili ng process occurs at constant pressure (e.g . in the BW R fuel assemblies, PWR steam generators, and practi cally all other heat ex c h ang e rs in i ndustrial a ppl ications), the heat required to vaporize a unit mas s of liquid is h fg  h g ‐ h f , which can be found in th e steam tab l es.

Writing th e (Young ‐ Lapla c e) e quation for the me cha n ical e quili bri u m of a b u bble surrounde d by liqui d, it can be shown tha t the vapor pressure within th e b u bble must be somewhat hi gher than the pressure of the surrou ndi ng li quid . It f o llows that the vapor (an d liquid) temperat ure mus t be somewhat hig h er than the satu ration te mpe r ature, T sat , corresponding to the l i qui d (o r system) pressure. Using th e Clausius ‐ Clapeyron equa ti on to relate s a turation te mperatur es an d pressures, it can the n be shown that the magni tud e of th e superheat (T n ucl eati o n ‐ T sat ) requir ed to sustain the bub ble is inversely pro p ortional to the radius of the bub bles, r bubbl e :

T nucleation

 T sat  r

(6)

bubble

Equation 6 suggests that th e smaller the bubbl e, th e h i gher is the s uperhea t required for nucl e ation. Therefore, if one wants to initiate bu bbl e nu cleation near th e satu ration te mpe r ature, one needs relatively large bu bbles (o r d er of mi cron s) to begin with. It just so happens that the hea t tran sfer surface of engineering sys t ems (e. g . cla dding of a BW R fuel rod, wall of a PWR steam gen e rat o r tube,

etc.) is full of micro ‐ cavities, naturally pr esent due to roughness, corrosion, etc. Therefore, ai r or vapor trapped in th ose cavities se r v e as nucl e ation sites for the bu bbles and ensur e initiation of the boiling process. The bo tto m li ne is that one ne eds to rai s e the temp era t ure of th e surface a li ttl e bit ab ove T sa t if boili ng is to be in i t iat e d and su st ai n e d . For a rigorous derivation of Eq. 6 and more de tails about bubbl e nu c l eat i on , refer to any boiling heat transfer textbook (e.g. Boiling, Condensation an d Gas ‐ Liquid Flow by P. B. Whalley, Ox ford Scie nce P ublica t ions, 1987).

Pool Boiling

Consider a simple experi ment in which water (o r ot h e r fluid) boi l s off the u pper surface of a flat pla t e. The plate is connected to an ele c tri c po wer supply, and thus is he ated via Joule effect. Th is situation is referred to as pool boiling (vs flow boiling) beca use the pool of fluid above the heat er is st agnan t . In this experi ment we con t rol th e heat flux q" (via the pow e r supply ) and we measur e th e wall te mperatur e T w (via a thermo couple ) . Th en a q" vs T w ‐ T sat curve can be cons tructe d from the da ta. This curv e is kn own as the boiling curve and is shown in Fig 5.

q  ( W / m 2 )

 10 6 B

C

A

O  5

 50

 1 500

T w  T sat (  C )

Figure 5. Qualitative boili ng curve for water at atm o spheric pressure (T sat =100  C) on a flat plat e.

As one progressively incre a ses the heat flux, several regions (o r heat transfer regimes ) ca n be identifi ed from this cur v e. A de picti o n of the ph ysic al situation associated wi t h these hea t transfer regimes is shown in Fig 6.

O → A

The wall tem p erature is not sufficiently beyond T sat to initiate bub ble nu cleatio n . In this region the he a t from the wall is transferred by single ‐ ph ase free con v ection.

Point A

The first b ubb les appear on th e surface. Th is point is called the onset of nu clea te boiling. A → B

Many bub ble s are fo r m ed and grow at the wall, and detach from the wall und er the effect of buoyancy. This heat tra n sfer regime is called nu cl eate boiling. The bu bbles create a lot of agitation and mixin g near the wall, w h ich e nhan c es heat transf er. As a r e sult, nu cleate boiling is a much more effective mecha n ism than free conv ection, as sugg est e d by the high er slope of the boiling curve in this region.

B → C

At a criti c al (high) value of the heat flux, ther e are so many bu bbles crowdi ng th e wall that th ey can merge and form a conti n u o us stable vapor film. T hus, there is a sudden transit i on from nucl eate boiling to film boiling. This sudden transition is called Departure from Nuclea te Boiling (DNB ), or Critical Heat Flux (CHF), or burnout, or boiling crisis , and is assoc i ated with a drastic reduction of the heat transfer coefficie n t bec a u se vapor is a poor conductor of heat . Conseq uently, the wal l temp erat ure incr eases abruptly and dramatically (to >1000  C), which may resu l t in physi cal destructio n of the heater.

C and beyond. This is the film boiling region. Heat transfer from the wall occurs mostly by convection within th e va por film and r a diation acro ss the vapor f ilm.

Free convection (O A)

Onset of nucleate boiling (A)

Nucleate boiling (A B)

Film boiling (C and beyond)

The heat transfer r egimes in pool boiling.

Image by MIT OpenCourseWare.

In th e absence of an experim e nt , the various segments of the boiling curve can also be predicted by Newton’s la w of cooling, q "  h ( T w  T sat ) , with th e heat transfer coefficient, h, given by th e following empiri cal cor r elations:

O → A. Singl e ‐ phase free convec tion from a flat plate can be pre d icted by th e Fishende n and Saund ers correlation:

hL  0 . 15 Ra 1 / 3

g  ( T  T ) L 3

where th e Rayleigh num b er is Ra L  w sa t

k f   f k f 

 2 

 p , f f 

In this correlation (valid for 10 7 <Ra L <10 10 ), L is the repr esentative di mens ion of th e plat e and is calcula t ed as A/P, where A is the plate area and P is the plate perimeter,  is the therm a l expansio n coefficie n t fo r the liquid (a property ).

Point A. The onset of nu cleat e boili ng can be cal c ul ated as the i n tersection of the correlations for free convect i on an d nu c l eat e boiling.

A → B. Th e Rosenhow correlation is a popular tool to predict nucl e ate boiling heat transfer:

 g (    )  0 . 5  c  3

h   h

f g

 p , f  ( T  T ) 2

f fg 



   C

s , f

h fg

P r f  

sat

In this correlation Pr f is th e Prandtl nu m b er for th e liquid and C s,f is an empirical coeffi cien t which depen d s on the fl uid/surf ace co mbina t ion. For example, for wat e r on polis hed stainless steel the recomme nde d value for C s,f is 0.013. C s,f is meant to capture th e effect of su r f a c e (mi c ro ‐ cavities) on nuclea t e boili ng.

B → C. The he at flux corres p onding to DN B can be estimated by th e correlation of Zuber:

q   0 . 13  h

 g  (  f



  g

)  1 / 4



DNB g fg



2  

C and beyon d . In film boili ng, the correlation of Berenson expr esses the hea t tr ansfer coefficient as the sum of a con v ection term and a radiation term:

h  h c  0 . 75 h rad

With:

 g ( 

  ) 

k 3 h 

 0 . 25

h  0 . 425  f g g g f g 

   g

( T w

 T sat

)  T  

4 4

h rad   SB w sa t

T w  T sat

h  fg  h fg  0 . 5 c p , g ( T w  T sat )

 T 

Flow boili ng

S i n g l e p h a s e v a p o u r

x = 1

D r y o u t

T w

T s a t

P l u g f l o w

z

B u b b l y f l o w

T b

S i n g l e p h a s e l i q u i d

O n s e t o f n u c l e a t e b o i l i n g ( x = 0 )

The situation of interest in nuclear reactors is flow boiling. Consider a vertical channel of arbitrary cross ‐ sectional shape (i.e. not necessarily rou nd), flow area A, equivale n t diameter D e , uniformly heated (axially as well as cir c umf e rentially) by a heat flux q" . Let T in be the inle t temperat ure (T in <T sat ), m ˙ the mass flow rate and P th e o p erating pres sure of th e fluid (e. g . wa ter) flowing in the channe l. We wish to describe th e various flow and hea t tran sfer regimes present in th e cha nnel as well as the axial variation of the bulk an d wall te mperatures. Th e physi cal situation is sho wn in Fig 7.

Image by MIT OpenCourseWare.

Figure 7. Hea t transfer and flow regimes in a vertical heated channel. (T hermal non ‐ equil i bri u m eff ects have bee n neg l ect e d in sketching the bu lk te mpera t ure)

At axial locati ons below the onset of nu cleate boiling, the flow regime is single ‐ phase liq uid. As th e fluid marches up the channe l, more and mo re steam is gen e rat e d be cause of th e heat addi tion. As a result, the flow regime goes fro m b ubbly flow (fo r relatively low values of the flow quality ) to pl ug (int ermed i at e quali ty) and annular (hi g h quali ty). Ev entually, the l iquid film in contac t with the wall dries out. In the re g i on be yond th e poi n t of dryout, the flow regime is mist flo w and fi nally, when all droplets have evaporated, single ‐ phase vapor flow.

To calculate the bu lk te mp erature in the cha nnel, sta r t from the st e a d y ‐ state en e r g y e quati on:

G dh  q " P h dz A

where P h is the chan nel heat ed peri m e ter and G= m/A is the mass flux. Sin c e q" is assumed axiall y constant, this equa tion is r e adily integra t ed:

h ( z )  h in

 q " P h z (7)

GA

where h in is the en thal p y corresp onding to T in . The li quid beco m e s saturate d (h=h f ) at

GA ( h f  h in )

z sat =

q " P h

. In the subcooled liquid region (z<z sat ), we also have dh=c p, ℓ dT b where c p, ℓ is the liquid

specific heat . Therefore, Eq 7 can be re ‐ writte n as T ( z )  T

 q " P h z . Thus, the bu lk tem p eratur e

b in

GAc

p , ℓ

distribu tion in th e subcooled liquid region is linear (u nder the assumption of axi a lly uniform heat flux).

For z>z sat , T b remains equal to T sat until all liquid ha s evaporated. This locati on in the ch annel can be calcula t ed by setting h=h g in Eq. 7 and solving for z. For z beyo nd this loca ti on, i.e. the si ngle ‐ ph ase vapor region, we have again a linearly in c r e a si n g bu lk tem p eratur e but with a different slope equal to

q " P h where c

is the vapor spe c ific heat.

GAc

p , v

p, v

The flow qual i ty is zero in the liquid pha se region. In the two ‐ phas e region, the flow quality is related to enthalpy: h= (1 ‐ x)h f +x h g . Th en Eq. 7 suggests that in this region th e flow quality increases (fro m 0 to 1) linearly with z. For locatio n s at which h> h g (single ‐ ph ase vapor), x stays equal to 1.

To calculate the wall te mp erature distri bution in the chann e l, one can use New t on’s law of co oling:

q "  h ( T w  T b )  T w ( z )  T b ( z )  q " / h

where the he at transfer co efficien t h is given by the f o llowing empirical correlations:

Single ‐ phase liquid region: use a single ‐ phase heat transfer correlation appropriate for th e geometry , fluid and flow regime (tu r bulen t vs laminar) prese n t in th e cha nnel. For example, the D i ttus ‐ Boelter correlation could be used, if one has fully ‐ developed turbul ent flow of a non ‐ metallic fluid.

Two ‐ phase region: for this region (fro m the onset of nuclea t e boili ng to the poi n t of dryou t ), we ca n use Klimenko’s correlation. Th e Kli m en ko’s correlation distingu ishes betw een tw o subregions: a nu cleate boiling domi nated subreg ion (ro u g h ly corresponding to bub bly an d pl ug flow) and a forced evaporation dominate d s ubregion (ro ughly corresp onding to annular flow). In the nu c l eat e boiling su bregion, hea t

transfer mostly occurs thr o ugh nu c l eation and det a c h m e n t of b ubbles at the wall, while in th e forced evaporation subregion th ere are typically no bubbl es, and heat transfer occu rs mainly thru convect i on within the li quid film and evaporation at the li quid film/vapor core inte rface. The heat transfer coefficie n t is high in both s ubregions. Th e ph ysic al si tuation is shown in Fig 8.

q" q"

(a) (b)

Figure 8. (a ) Nuclea te boiling (typi c al of bub bly and plu g flow), (b) Forced evaporation (typical of annular flow).

Klimenko’s correlation co mprises the f o llowing equat i on s: Nuclea te boiling do minate d subre g ion:

hL  PL

 0 . 54

c  4 . 9  10  3 Pe 0 . 6 Pr  0 . 33  c 

(8)

m f   

Forced evaporation subreg ion:

hL  

 0 . 2

c  0 . 087 Re 0 . 6 Pr 1 / 6  g 

(9)

k m f   

f  f 

Where in Eq. 8, P is the operating pressure, Pr f is the liquid Prandtl nu mber an d the following definitions are used for Pe m , Re m and L c :

Pe  q " L c  f c p , f

fg g k f