Heat Transfer With Phase Change 2014

HEAT TRANSFER WITH PHASE CHANGE

1.Nuclear Heat Transport by El-Wakil2.Nuclear Systems 1 & 2 by Kazimi

BASIC CONCEPTS & TERMINOLOGIES -1

Why heat transfer with phase change is important in nuclear reactors ?

Liquids are subjected to high heat fluxes. Resulting in 1. Liquids attaining saturation or superheated temperatures

2. Causing liquids to boil due to phase change

3. Causing dissolved gases to come out of the liquid forming bubbles

4. Causing other phenomena which generate bubbles

1. This bubble formation cause agitation and turbulence, resulting in1. High heat transfer rates a benefit

2. Void are formed, undesirable in nuclear reactors, affects moderation

3. Heat fluxes, limited due to burnout (leading to structural instability)

2. Hence, an in-depth knowledge of phase-change behavior of working fluid can result in better performance of nuclear reactors1. Two-phase coolant thermal conditions in accidental loss of coolant

2. Provision of sufficient safety margin between anticipated transient heat fluxes and critical boiling heat fluxes


TERMS USED TO DESCRIBE PHASE CHANGE PROCESSES

VAPORIZATION

Conversion of liquid into vapors

EVAPORATION

Conversion of liquid into vapors below boiling point

BOILING

Formation of vapors within a liquid phase at and above boiling point

TWO-PHASE FLOW

Where both the vapor and liquid move together in a channel

CONDENSATION

Reverse of evaporation


CLASSIFICATION OF BOILING

BASED ON THE LOCATION OF BOILING PHENOMENON

BASED ON THE MECHANISM OF BOILING

BASED ON THE TEMPERATURE CONDITIONS OF FLUID


POOL BOILINGNON FLOW

VOLUME OR BULK BOILINGCAN BE NON FLOW OR FLOW BOILING

BASED ON THE LOCATION OF BOILING PHENOMENON

Boiling due to heat added to the liquid by a surface in contact with or submerged within the liquid

Boiling due to heat generation within the liquid by chemical or nuclear reaction


NUCLEATE BOILING FILM BOILING

BASED ON THE MECHANISM OF BOILING

bubbles are formed around a small nucleus of vapors or gas

there is a formation of a continuous film of vapor that blankets the heating surface

Partial film boiling, Partial nucleate boiling, Transition film boiling, Unstable nucleate film boiling etc.

pool nucleate boiling or volume nucleate boiling

there is only pool film boiling rather than volume film boiling

Under certain conditions nucleate and film boiling coexist


SATURATED BOILING SUBCOOLED BOILING

BASED ON THE TEMPERATURE OF LIQUID

bulk of the liquid is at saturation temperature

bulk of the liquid is subcooled

In saturated boiling, the bubbles rise to the liquid surface where they are detached

In sub-cooled boiling, the bubbles begin to rise but may collapse before they reach the liquid surface

.

An important point

to generate bubbles the heating surface must be attemperature > saturation, consequently some of the liquid immediately adjacent to that surface, is superheated


HENCE

BOTH SATURATED AND SUBCOOLED BOILING CAN BE

NUCLEATE OR FILM BOILING

WHEREAS VOLUME OR BULK BOILING CAN ONLY BE

SATURATED BOILING

TWO-PHASE FLOW CLASSIFICATION

DIABATICWITH BOILING

ADIABATICWITHOUT BOILING

BASED ON HEAT TRANSFER


Single component flowWater and steam

Two component flowWater and air

BASED ON COMPONENTS

IN BOTH THESE TYPES OF TWO PHASE FLOW1. Vapor and liquid flow at different velocities2. Vapors normally flow faster than liquid3. The ratio of their velocities is called slip ratio

BUBBLE STATICS AND DYNAMICS -1

BUBBLE FORMATION REQUIRES LIQUID SUPERHEAT

(to what degree later)

Bubble Formation Aided By So Called Nucleation Aids

1. DISSOLVED GASES OR VAPORS PRESENT IN LIQUID Prominent in nuclear reactors Charged particles presence in bubbles aid bubble motion ‘h’ in BWR > ‘h’ in conventional boilers

1. CAVITIES OR CREVICES ON THE SOLID SURFACE Never completely filled due to surface tension Retain gases Centers of high temperature

the presence of ionization radiations


1. WETTING CHARACTERISTICS OF SOLID SURFACE

Depends on the interaction of solid, liquid and gas at interface

For a bubble resting on a solid surface there are three interfaces

1. Solid-liquid interface, function of liquid and solid surface properties

2. Liquid-vapor interface, function of liquid and vapor phase properties

3. Solid-vapor interface, function of vapor and solid surface properties

Making a force balance along the surface reveals

cosgs fs fg

Term Wetting Always Refer To The Liquid Phase


If = 90, borderline wetted and unwetted surface

If > 90, unwetted surface

If < 90, wetted surface


SURFACES DESIRABLE FOR BUBBLE FORMATION ???

First study Bubble Growth Depends on

If > 90, facilitates

bubble grows / larger bubbles

> 90,

lesser superheat required

> 90, larger bubbles, film is readily formed,

Film, lesser heat transfer, not desired

If < 90,

More superheat required

< 90,

Smaller Bubbles

< 90Bubble more easily detached

No Film, higher heat transfer, desired


To form a bubble some degree of superheat is required

(ODD isn’t it)Consider a stable floating bubble of radius ‘r’ in a saturated liquid

2

4 b f fg

Dp p D

4 fg

b fp pD

Bubble is at thermal equilibriumTb = Tf

But pb > pf to overcome the surface tension force

Since pb corresponds to at least Tsat,

Hence, Tf > Tsat

Liquid is superheated

A force balance reveals


What degree of superheat is required

*

4 2fg fgb fp p

D r

pg = vapor pressure of

liquid inside bubble

pg = pb & Tg = Tb vapor pressure and vapor temperature can be related using Clausius Clapeyron eq.

fg

sat sat g f

hdp

dT T v v

Assume vapors exist at Tsat (pg/b ) inside the

bubble & vg >> vf

fgb

b b g

hdp

dT T v

using Ideal Gas relation to get2

fgbb

b b

hdpdT

p RTIntegrate between limits

of pb to pl & Tb to Tsat


1 1ln fgb

l b sat

hp

p R T T

lnb sat b

b satfg l

RT T pT T

h p

*

4 2fg fgb fp p

D r

Combine to get

*

2ln 1 fgb sat

b satfg f

RT TT T

h p r

*

2 fg satb sat

fg b

TT T

h r

If rc the order of molecular dimension Tg – Tsat quite large,

* 2 fg sat

fg g b sat

Tr

h T T

BUBBLE STATICS AND DYNAMICS -7 For water, predicted Tg – Tsat 220oC,

Measured Tg – Tsat 16oC

Thanks to dissolved gases, which reduce required vapor pressure for bubble mechanical equilibrium, i.e.

4 2fg fgg vap f

c

p p pD r

Bubble DetachmentProcess


• Nucleation at solid surfaces/on suspended bodies , micro-cavities 10-3 mm at the surfaces act as gas storage volumes

• This arrangement allows the vapor to exist in contact with sub-cooled liquid, provided the angular opening of the crack is small (micro-cavity)

• A solid surface contains a large number of micro-cavities with a distribution in sizes

• Boiling at the surface can begin if the Tcoolant near the surface is high enough that the preexisting vapor at the cavity site may attain sufficient pressure to initiate the growth of a vapor bubble at that site.

The various stages in the pool boiling curve

Jets and Columns

Film Boiling

BOILING REGIMESBOILING HEAT TRANSFER - History1. Nukiyama (1934)Performed an experiment using an electrically heated platinum wireimmersed in water – BOILING CURVE2. Gaertner (1965)Vapour structures in nucleate boiling

HOW DO YOU THINK BOILING PHENOMENA GOES AS TEMPEATURE OF THE HEATING SURFACE CHANGES

SATURATED POOL BOILING CURVE -1

Reveals range of conditions associated with saturated pool boiling

Water at Atmospheric Pressure

Little vapor formation. Liquid motion is due principally to single-phase natural convection.

• Free Convection

5eT C

• Onset of Nucleate Boiling 5eONB T C

SATURATED POOL BOILING CURVE -2

• Nucleate Boiling 5 30eT C

Isolated Vapor Bubbles 5 10eT C

- Liquid motion is strongly influenced by nucleation of bubbles at the surface.

- h and qs" increase sharply with increasing Te

- Heat transfer is principally due to contact of liquid with the surface (single-phase convection) and not due to vaporization.

Jets and Columns 10 30eT C

- Increasing number of nucleation sites causes bubble interactions and coalescence into jets and slugs.

- Liquid/surface contact is impaired- qs

" continues to increase with Te while h begins to decrease.

SATURATED POOL BOILING CURVE -3• Critical Heat Flux - CHF, max 30eq T C

Maximum attainable heat flux in nucleate boiling. q"

max 1 MW/m2 for water at atmospheric pressure

• Potential Burnout for Power-Controlled Heating

• Film Boiling Heat transfer is by conduction and radiation across the vapor blanketA reduction in q"

s follows the cooling curve continuously to Leidenfrost point corresponding to minimum heat flux q"

min for film boiling

An increase in q"s beyond q"

max causes the surface to be blanketed by vapors and the surface temperature can spontaneously achieve a value that potentially exceeds its melting point (Ts > 1000 oC)

If the surface survives the temperature shock, conditions are characterized by film boiling

Saturated Pool Boiling Curve (SPBC) -4

A reduction in q"s below q"

min causes an abrupt reduction in surface temperature to the nucleate boiling regime.

• Transition Boiling for Temperature-Controlled Heating

Characterized by a continuous decay of q"s (from q"

max to q"min

with increasing Te

Surface conditions oscillate between nucleate and film boiling, but portion of surface experiencing film boiling increases with Te

Also termed unstable or partial film boiling.

SPBC IN HEAT FLUX CONTROLLED MODE

SATURATED POOL BOILING CURVE - 5


1. As many as three different heating surface temperatures can be attained at the same heat flux

2. Safer to operate at the lowest possible temperature against the highest heat flux

3. Thus normal operation is in the region of NUCLEATE BOILING

4. In actual practice ‘q’ is the independent variable and Ts is the dependent variable

5. When qc is reached, any further increase in ‘q’ results in a temperature jump from C to E

6. Temperature at E normally exceeds the safe surface temperature limits.

7. The condition is referred as BURNOUT and heat flux is referred as BURNOUT OR CRITICAL HEAT FLUX

THE FLOW BOILING CURVE - 1

1. Curve a is for the liquid bulk temperature2. Curve b is for the heating surface temperature at low heat flux3. Curve b is for the heating surface temperature at high heat flux4. Rise in Ts at point of DNB is sudden5. Height and width of this rise and fall of Ts depends on many things

BOILING INCIPIENCE

• As bulk of coolant heats up, bubbles grow larger, possibility of detachment from wall surface into flow stream increases region II Z > ZD

• Bubbles detach regularly. Condense slowly as they move through fluid, vapor voidage penetrates to the fluid bulk Void fraction increases significantly

• Bulk liquid becomes saturated at Z = ZB, region III, void fraction continues to increase, approaching thermal equilibrium condition at Z = ZE

• Beginning of region IV, where the thermodynamic non-equilibrium history is completely lost

• Z = ZNB is called the point of boiling incipience, where boiling starts

• Sub-cooled boiling begins as nucleate boiling starts, Z = ZNB and Tbulk< Tsat

• For nucleation to occur, Tf near the wall must be somewhat > Tsat

• Most of liquid is still sub-cooled, bubbles do not detach but grow & collapse, attached to the wall, giving a small nonzero void fraction (region I)

III III IV

BOILING INCIPIENCE CRITERIONA criterion for boiling inception (Z = ZNB) in forced flow was developed byBergles and RosehsenowMain Idea: Tliquid due to the heat flux near the wall = temperature associated with the required superheat for bubble stability (developed earlier)Both these temperature change as distance from wall changesHence this occurs when the two temperatures tangentially make contactLiquid temperature & near wall q// are related

liqliq

T T qq k

r r k

This gradient = gradient of required superheat

2 fg satg sat

fg g c

TT T

h r

Take derivative T/r and put this to get the following

2

8l fg w sat

ifg sat fg

k h T Tq

T v

BOILING INCIPIENCE CRITERION

At low heat fluxes the point of tangency may be at a radius larger than the available size cavities in the wall. In this case , as in natural convection above equation would under predict the required superheat and (q")I

Bjorge et al. recommended that a maximum cavity for most surfaces in contact with water be rmax = 10-6 m. When the liquid has a good surface wetting ability , the apparent cavity size is smaller than the actual size .

c w sati iq h T T Combine 2

8l fg w sat

ifg sat fg

k h T Tq

T v

21

8l fgw sat

w bulk fg sat fg ci

k hT T

T T T v h

THE BURNOUT HEAT FLUX CORRELATIONS - 2

Pool saturated boiling in Nucleate Boiling Region: Clean heating surfaces

THE ROHSENOW CORRELATION

Where, Csg

1. Dimensionless constant determined experimentally for various surfaces and fluids

2. 0.0133 (water stainless steel system)

3. Depends on surface wettability

4. Determined by noting q" and T in a single experiment

Pr’s exponent varies from 0.8 to 2 due to the presence of contaminants

0.33

1Prpf c fg

sgmfg f fg f g

c T gqC

h h g

11Re Prn mL

f sg

hNu

k C

w w

w sat

q qh

T T T

L

f gg

Values of constants in the Rohsenow’s correlation

SATURATED NON FOW BOILING

Rohsenow and Griffith proposed

CHF CORRELATIONS -1

Based on theoretical and experimental analysis

hfg in Btu/lbm at system pressure

g and f in lbm/ft3 saturated densities at system pressures

gc conversion factor 4.17108 lbm ft /lbf hr2

Effect of ‘g’ is to increase qc" due to increased separation of phases

Optimum pressure to give highest qc" 1.2 106 Btu/lbm ft2 at 1200 psia

0.6 0.25

143 f gc fg g

g c

gq h

g

SUBCOOLED NON FOW POOL BOILING CHF CORRELATIONS

Zuber etal. correlation

Ivey and Morris, correlation

0.25

( )1 0.1

( )g pf f wall bulkc

c f fg g

o

c T Tq subcooled

q saturated h

where temperatues are in F

0.25

0.5

1 8

2

( ) 5.31

( )

/

f gcf pf f

c fg g c

f g c

sat bulk

g

o

of

gq subcooledk c

q saturated h g

ggT T

where temperatues are in F

k thermal conductivity of liquid Btu hr ft F

THE PARAMETRIC EFFECT ON POOL CHF -1

EFFECT OF PRESSURE

1. Increases the TB, g, hfg and

2. Shifts the Nucleate Boiling regime

3. Shifts the CHF

4. CHF increases to a maximum corresponding to an optimum pressure and then decreases

5. Popt is 33 to 40 % of Pc of liquid in question

6. CHF (at increased P) increases by 4 to 10 times as compared to CHF at 1 atm

THE PARAMETRIC EFFECT ON POOL CHF -2

Advantage:

1. Operate the System at or near the Optimum Pressure

2. This corresponds to maximum CHF

3. Which in turn gives a much higher safety margin of operation as well as higher nucleate boiling heat flux

OTHER PARAMETRIC EFFECTS

4. Dirty surfaces give 15% higher CHF as compared to the clean surfaces corresponding to the same pressure

5. Certain additives (having heavier than liquid molecules) increase the CHF

6. CHF also increase in the presence of ultrasonic and electrostatic fields

7. CHF decreases with presence of dissolved gases

8. CHF decreases with agents which reduce surface tension

THE PARAMETRIC EFFECT ON FLOW BOILING -2

Effect of Pressure

1. Rather little effect

2. Effect is same for upward or downward flow

3. Compare with the pool boiling curve


Effect of Velocity & Sub-cooling

Degassed distilled water, 4.13 bar (60 psia), Tsat = 145°C (293°F) (McAdams 1946)

1. No effect of velocity on the nucleate boiling heat flux for the same degree of sub-cooling

2. For same velocity increase in degree of sub-cooling increases the onset of boiling and CHF

3. For the same degree of sub-cooling increase in velocity also increases the onset of boiling and CHF


Inlet enthalpy Weatherhead (1963) for a tube

ID of 7.7 mm, length 45.7 cm & a system pressure of 2000 psia

Increases approx linearly with the inlet sub-cooling

This relationship occurs over fairly wide ranges

Has no fundamental significance, except to indicate that more energy goes into saturating the fluid.

If a very wide range of inlet sub-cooling is used, then departures from linearity are observed.

difference in enthalpy between saturated liquid and inlet liquid


Geometry of the flow passage (For fixed P, G, Δisub & L)

CHF increases with tube diameter, DH

the rate of increase, decreases as the diameter increases


Effect of tube Diameter• As the tube diameter is increased, so the critical

heat flux increases at constant inlet sub-cooling. The relationship between critical heat flux and

• Inlet sub-cooling effect is linear for small tube diameters (D 1 2.8 mm)

• For larger diameters show a marked curvature.


Effect of tube Diameter• Same data plotted against exit quality

• Where the data for different tube diameters

• There is an overlap of data for ID > 23.6 mm

• It can be seen that the critical heat flux decreases with increasing tube diameter for a given exit quality


Effect of heating length (For fixed P, Δisub , G, & DH )

1. CHF decreases with increasing tube length L

2. The power input required for burnout, Pcr , decreases first rapidly, and then less rapidly

3. For very long tubes, the critical power appears to asymptote to a constant value independent of tube length in some cases.

4. This only applies over a limited range of length


Effect of heating length

1. Lee and Obertelli (1963) and Lee (1965) reported data for a pressure of 69 bar (1000 psia), a mass velocity of 2000 kg/m2s (1.5 x 106 lb/hr.ft2) and a tube diameter in the range of 10.75-10.85 mm (0.424-0.426 in)

2. Data shows the variation of CHF with length and sub-cooling for fixed mass velocity & tube diameter

3. CHF increases as tube length is decreased for constant inlet sub-cooling

4. For longer lengths (z > 1 m) the relationship between heat flux and sub-cooling is linear. This linearity breaks down for short tubes.


Effect of heating length

1. If their data are re-plotted as CHF vs exit quality it is seen that the data points for different lengths all fall on one curve

2. It can be concluded from this observation that the effect of tube length on the critical heat flux for fixed exit quality is small.

3. So the observation that CHF is independent of tube length for a given exit quality can be interpreted as the tube may be divided into two lengths;

4. Over the first length the liquid is raised to the saturation condition

5. Over the second length the quality is raised from zero to the outlet quality corresponding to critical condition.

6. The lengths over which these temperature conditions occur can be found by heat balance


T NB BH H H

''4sub i

NBCHF

DG hH

q

''4

fgB

CHF

x H DGhH

q

'' , , ,CHFq f x H G p DFor a fixed CHF & exit quality

'' , , ,CHF Bq f H G p DOne can also write for if CHF occurs at the exit location,

, , , ,BH CHF Bx f H G p D

2 independent equations, very useful as data of CHF can also be represented independently in terms of xHB,CHF

A local relationship between CHF and quality

Mass fraction of the liquid which can be evaporated (xHB,CHF) in the channel before the onset of the critical condition is a function of the length over which the evaporation takes place, (HB)

THE PARAMETRIC EFFECT ON FLOW BOILING -6 '' , , ,CHF Bq f H G p D , , , ,

BH CHF Bx f H G p D

Neither of these two formulations of CHF phenomenon is completely correctFor the critical condition in a uniformly heated channel it is impossible to distinguish between the two independent parameters

THE PARAMETRIC EFFECT ON FLOW BOILING -6 Effect of mass flux G (For fixed P, DH , L and Δisub)

CHF rises approximately linearly with G

OTHER PARAMETRIC EFFECTS' ON FLOW BOILING

Type of heat flux

Little effect

Dissolved gasses and additives

Little Effect

Surface roughness

Increases CHF in subcooled flow boiling

Decreases CHF in saturated flow boiling

EFFECT OF PRESENCE OF HEATING WALLS - 1

• Presence of unheated walls Near the CHF point, Increases CHF.

• A wide variety of data has been obtained for burnout in annuli and rod bundles.

1.In the nuclear industry, the main interest is CHF in rod bundle geometries. The annulus might be regarded as a "single rod bundle" or "sub-channel.“

EFFECT OF PRESENCE OF HEATING WALLS - 2Most of the data is where the inner surface is

heated.

Data has also been reported where both surfaces are heated and the fraction of the power input to the outer surface is varied.

This useful data is reported by Jensen and Mannov (1974).

For a fixed inlet subcooling, the critical quality (quality at burnout) initially increases as the fraction of power on the outer surface is increased.

In this region, CHF occurs first on the inner surface.

As the fraction of power on the outer surface is further increased, a maximum burnout quality is reached, and beyond this point burnout begins to occur first on the outer surface and the critical quality decreases with increasing fractional power on that surface.

Empirical Correlation Of Experimental Data• The technical importance of the critical heat flux condition has led to the

• development of a huge variety of correlations

• Milioti (1 964) catalogues fifty-nine correlations and detailed comparisons of a wide range of correlations reveal considerable differences

• Many of the more recent correlations do, however, show fairly reasonable agreement despite wide variations in their functional form.

• Three types of correlating procedure have bee adopted

1. Correlations of an empirical nature which make no assumptions whatever about the mechanisms involved in the critical heat flux condition, but solely attempt a functional relationship between the critical heat flux and the independent variables.

2. Correlations where attempts have been made to look at and write down equations for the hydrodynamic and heat transfer processes occurring in the heated channel and to relate these to the critical heat flux condition

3. The USSR Academy of Sciences (Kirillov et al. 1991) has produced a series of standard tables of CHF as a function of the local bulk mean water condition and for various pressures and mass velocities for a fixed tube diameter of 0.315 in (8mm). These tables are valid for z/D 20. For tube diameters other than 8 mm the critical heat flux is given by the approximate relationship

Empirical Correlation Of Experimental Data1. The USSR Academy of Sciences (Kirillov et al. 1991) has produced a series of

standard tables of CHF as a function of the local bulk mean water condition and for various pressures and mass velocities for a fixed tube diameter of 0.315 in (8mm). These tables are valid for z/D 20. For tube diameters other than 8 mm the critical heat flux is given by the approximate relationship

,8 0.008

k

CHF CHF mm

Dq q

Empirical LOOK-UP TABLES

The ANL correlation, for sub-cooled flow boiling

Represented a large number of experimental data of sub-cooled water flowing upward through a vertical tube under various operating conditions, including high pressures.

Range of G:

0.96×106 to 7.8×106

Degree of subcooling:

5.5 to 163 oF

Not suitable for saturated boiling, why

0.22

2 610

M

CHF sat f

Btu Gq C T T

hr ft

P C M

500 psia 0.827 0.16

1000 psia 0.626 0.275

2000 psia 0.445 0.5

3000 psia 0.25 0.73

Limitations of use:1. Steady State2. Uniform heat flux3. Vertical up flow in tubes

4. Sub-cooled water in oF5. Interpolate between pressure values6. Mass flux in lbm/hr ft2

The Westinghouse Correlation for subcooled flow boiling-1

For uniform heat flux conditions, the Westinghouse, W-3 Correlation

Pressure effect

G effect

Diam effect

Sub-cool effect

Range

of Data

Limitations of Use:

1. Steady State

2. Uniform Heat Flux

3. Tubes or Rod Bundles

1. 0.5(106) G 5.0(106) lbm/hr ft2

2. 800 P 2000 psia

3. 10 L 79 inches

4. 0.2 De 0.7 inches

5. ii 400 Btu/lbm

6. -0.15 xe 0.15

6 18.171 0.004129

6

3.151

2.022 0.0004302

10 0.1722 0.0000984

0.1484 0.1596 0.1729 10 1.037

1.157 0.869 0.2664 0.8357

0.8258 0.000794

e

CHFp x

D

f i

pq

p e

x x G

x e

h h

The Westinghouse Correlation for subcooled flow boiling-2

For non-uniform heat flux conditions, divide the previous qc″ by the factor ‘F’

Where

C = 0.44(1 — xc)7.9/(G x l0-6)1.72, in-i

lc=channel length at which DNB takes place, inches

q″lc= heat flux at lc, Btu/hr ft2

z = variable distance from channel entrance, inches

xc=quality at DNB

01''

c

c

c

l c

lC l z

C l

CF q z e dz

q e

The above correlation correlates most existing heat flux data to within ± 20 percent.

General Electric Correlation for Saturated Flow Boiling

GE Correlation

(El-Wakil, 1979)

System pressure

= 1000 psia

where

Range of Data

For system pressure 1000 but < 1450 psia

1000 440 1000CHF CHFq q at psia p

61

6 61 2

6 62

0 705 10 0 237

1 634 10 0 27 4 71 10

0 605 10 0 164 0 654 10

. .

. . .

. . .

e

CHF e e

e e

G x x

q G x x x x

G x x x

61

62

0 197 0 108 10

0 254 0 026 10

. .

. .

x G

x G

6 2

0 1 0 15

600 1450

0 4 6 10

29 108

0 245 1 25

. .

. /

. .

ex

P psia

G lbm hrft

L inches

D inches

Westinghouse Correlation for Saturated Flow Boiling

Predicts the steam-water enthalpy, hc at which burnout is likely to occur in terms of the inlet enthalpy, hi instead of q″CHF

• flow in circular, rectangular channels and along rod bundles,

• uniform and non-uniform heat flux

• a pressure range of 800 to 2,750 psia,

• hi = 400 Btu /lbm to saturation

• G= 0.4 x 106 to 2.5 x 106 lbm/hr ft2

• local heat flux = 0.1 x 106 to 1.8 x 106 Btu/hr ft2

• De = 0.1 to 054 in., L = 9 to 76 in

• heated-to-wetted-perimeter ratio = 0.88 to 1.0

• exit quality = 0 to 0.9.

517 1 5 10

0 0048

0 529 0 825 2 3

0 41 1 12 0 548

.

.

. . .

. . .

e

e

D Gc i f i fg

gL Dfg fg fg

f

h h h h e h e

h e h h

Correlates most data to

within ± 25 percent.

Applies for

OTHER CORRELATIONS FOR SATURATERD FLOW BOILING

BIASI Correlation (El-Wakil, 1979)

7

0 032

1 6 1 6

1 883 100 724 0 99 ... . pp

CHF e pnH

fq x where f pe

D G G

Low Quality

Region

7

0 019

0 6 2

3 78 10 8 991 1 159 0 149

10.

.

. ., . . p

CHF p e pnH

pq f x f pe

D G p

High Quality Region

2

1

0 4 1 0 6 1

0 3 3 75 20 600

2 7 140 10 600

1 1

. , .

. . ,

. ,

H H

HH

e f g

n D cm n D cm

D cm z cm

P bar G g cm s

R x R

Range of DataLimitations of use

1. Steady State

2. Uniform heat flux

3. Water/Up flow or down flow

4. Vertical tubes or Rod Bundles

OTHER CORRELATIONS FOR SATURATED FLOW BOILING

1 36

0 4

0 4 6

4

1

101 11

62 1 10.

.

.

fgCHF exit

cr

re

G iq x

where

PP

G

PD G

P

CISE Correlation (El-Wakil, 1979)

2

2

fg

e

q BTU hrft

i BTU lbm

G lbm hrft

D inches

Limitations of use

1. Steady State

2. Uniform heat flux

3. Rod or tube bundles

Some More Subcooled Flow Boiling CHF Correlations• CHF very important in the safety analysis of PWRs’

• PWRs’ are designed to operate such that their primary coolant systems contain pressurized and subcooled water everywhere and at all times

• Rules for safe normal operation require that the CHF conditions never be approached anywhere in the reactor core.

• The criterion is represented in terms of a maximum DNBR, (DNB ratio)

max max 1CHF wDNBR q q DNBR DNBR

This condition should apply everywhere in the core

Oldest correlation for PWR is that of Tong (1969)

0.4 0.6

0.6

fCHF

fg

GqC

h D

21.76 7.433 12.222eq eqC x x

Range of Data

2

2

4 0 60 6

15 190

0 1 5 5

2 2 40

12 40

2 5 8 0

CHF

sub exit

heat

q MW m

T K

P MPa

G Mg m s

L D

D mm

,

. . /

. .

. /

/

. .

Some More Subcooled Flow Boiling CHF Correlations• Celata et al. (1994) improved the accuracy of Tong (1969) correlation

21

1

0.216 4.74 10

0.825 0.987 0.1 0

1 0.1

1 2 30 0

eq eq

eq

eq eq

C C P

x for x

C for x

x for x

Range of Data3 3 2

90 230

0 1 8 4

2 10 90 10

0 1 0 61

0 3 25 4

sub in

heat

T K

P MPa

G kg m s

L m

D mm

,

. .

/

. .

. .

Some More Subcooled Flow Boiling CHF CorrelationsThe correlations of Hall and Mudawar (2000a,b)

Correlation based on inlet conditions

3 52

3 52

*1 4

1 4

1

1 4

c ccD f g f g in

c ccD f g heat

cWe c xBo

c c We L D

Correlation based on exit conditions

2

*, ,

, ,

1 2

3 4

5

and are exit properties

0.0722, 0.132,

0.644, 0.90,

0.724

D f

in in f out fg out

f out fg out

We G D

x h h h

h h

c c

c c

c

3 52

1 4 ,1c cc

D f g f g eq outBo cWe c x

Parameter ranges both correlations 0.25mm < D < 1.5 cm, 300 ≤ G < 30,000 kg/m2·s 1 ≤ P ≤ 200 bars.For their inlet-conditions correlation, ,−2 < xeq,in < 0.0, −1 < xeq,out < 0.0, 2 ≤ Lheat/D ≤ 200.

BOILING MAPS• Vertical up flow, preferred configuration for boiling channels

• As buoyancy helps mixture flow & slip velocity; improved heat transfer

• Flow boiling in horizontal, vertical down flow are also of interest

• Horizontal boiling channels are not uncommon

• Flow boiling in a vertical, downward configuration occur in accident conditions in PWR type systems, where otherwise liquid up flow occurs under normal conditions

• Under typical boiler liquid in is sub-cooled, heat addition takes it to either

• Higher temperatures but sub-cooled (with and without boiling)

• Higher by Saturated temperatures with boiling

• Even superheated vapour temperatures

• Boiling behaviour is all these cases depend on the magnitude of heat flux

• Let us discuss these behaviour qualitatively in order to be able to apply various correlations to predict either heat transfer coefficients or CHF

BOILING MAPS

Development of 2-phase flow patterns in flow boiling. (Hsu and Graham, 1986.)

Heat transfer, two-phase flow, and boiling regimes in a vertical tube (up flow), steady state, constant G and uniform and moderate heat flux.

• Inlet fluid is highly sub-cooled, • Sub-cooled liquid in entire channel• With , boiling occurs in part of the

channel, exit flow regime = f()• If is high or low inlet sub-cooling a

complete sequence of boiling and related two-phase flow regimes take place in the channel

• Boiling starts at ONB point• If inlet fluid is saturated or a

saturated liquid–vapor mixture, boiling and two-phase flow patterns will be similar to the one shown

BOILING MAPSUniformly heated vertical channel with upward flow and moderate heat flux, withSub-cooled fluid at inletTw & Tf are shown as a function of height• Near inlet, liquid too sub-cooled, bubble

nucleation not permitted, single phase flow & forced convection heat transfer

• Once boiling is initiated, a sequence of flow regimes develop

• Bubbly, Slug, Annular, Dispersed droplet flow, finally single-phase vapor flow

• Nucleate boiling in bubbly & slug flow• Forced convective evaporation in

annular flow • Extremely efficient heat transfer regime,

heated wall covered by a thin liquid film. Liquid film is cooled by evaporation at its surface, hence unable to sustain a sufficiently large superheat for bubble nucleation

Two-phase flow and boiling regimes in a vertical pipe with moderate wall heat flux. (From Collier and Thome, 1994.)

BOILING MAPS• Droplet entrainment at high vapor flow

rate, leading to dispersed-droplet flow• Liquid film eventually completely

evaporate and lead to dry-out (equivalent to CHF in pool boiling)

• Sustained macroscopic contact between heated surface & liquid does not occur downstream from the dry-out point (A liquid-deficient region)

• Sporadic deposition of droplets onto the surface may take place.

• Finally entrained droplets evaporate completely, pure vapor single-phase flow

• ‘h’ in liquid-deficient region << nucleate boiling or forced convective evaporation regimes

• Result, occurrence of dry-out with large temperature rise for the heated surface

• Dry-out phenomenon is similar to CHF discussed for pool boiling

Two-phase flow and boiling regimes in a vertical pipe with moderate wall heat flux. (From Collier and Thome, 1994.)

BOILING MAPS

• Flow patterns differ from described earlier• Due to high , ONB occurs bulk liquid is almost

at inlet temperature• Significant Nucleate boiling takes place

downstream of ONB point, leading to increased voidage

• Growing bubbly layer adjacent to wall eventually form a continuous layer thus initiating the DNB phenomenon (Departure from Nucleate Boiling)

• Another mechanism similar to pool boiling CHF

• Very high ‘h’ in sub-cooled boiling regime, deteriorates very significantly beyond the DNB point, even though bulk flow in heated channel may still be highly sub-cooled

Two-phase flow & boiling regimes in a vertical pipe with high wall

heat flux

Flow & heat transfer regimes in a vertical heated channel at very high heat flux

BOILING MAPS

• Moderate , follow line (II)• Flow/heat transfer regimes• Liquid forced convection, sub-

cooled boiling, saturated boiling, forced convective evaporation, dryout, and post dryout (post-CHF; liquid-deficient)

• Very high , follow line (IV)• Flow/heat transfer regimes• ONB occurs in highly subcooled

bulk liquid• DNB instead of dryout• Even higher line VI or VII ONB

and DNB both occur while the bulk fluid is highly subcooled

A qualitative picture of evolution of heat transfer regimes as one moves along a heated vertical channel with constant and G

BOILING MAPS

• (I), (II)…(VII) same as before• ‘h’ is very high in nucleate boiling

and forced-convective evaporation regimes,

• Drops dramatically once CHF (dryout of ONB) is reached

• Increases in post-CHF regime but still low in comparison with nucleate boiling & convective evaporation regimes

• At xeq is high, ‘h’ slightly increases with increasing xeq. Heat transfer is essentially by nucleate boiling at low xeq ,by forced convective evaporation at high xeq

A qualitative picture of variation of local heat transfer coefficients alonga uniformly heated vertical channel with constant and G

• Kandlikar (1998), at high xeq ‘h’ may actually decrease with increasing xeq in some circumstances

• This implies contribution from both nucleate boiling & forced convection

BOILING MAPS

• At high pressure for water H/Hfo decreases with xeq

• At low pressure the opposite trend is observed.

• Refrigerants with relatively low ρf/ρg at normal refrigeration operating conditions also show a decreasing H/Hfo trend

Flow boiling map showing the variation of H/Hfo with xeq

Saturated flow boiling map Kandlikar, 1998

Variation of H/Hfo with xeq with boiling number & density ratio as parameters

w

fg

qBoiling Number Bo

Gh

TWO PHASE FLOW AN INTODUCTION. -2

In gas-liquid flow the two phases can adopt various geometric configurations: known as flow patterns or flow regimes.

Important physical parameters effecting the flow pattern are:– surface tension, which tries to keep the channel walls always wet and

which tends to make small liquid drops and small gas bubbles spherical

– gravity, which (in a non-vertical channel) tends to pull the liquid to the bottom of the channel,

Importance of 2 phase flow in liquid cooled nuclear reactors, already discussed

Also commonly occurs in pipelines which nominally carry oil or gas alone

GAS LIQUID GENERAL FLOW REGIMES

DISPERSED

FLOW

STRATIFIED

FLOW

2 PHASE FLOW PATTERNS EXPERIMENTAL SETUPS

Adiabatic Vertical 2 Phase Flow Patterns

ADIABATIC FLOW

Bubbly Slug Churn Annular

Flow is• co-current and steady-state in• a long tube with low or moderate

but constant liquid volumetric flow rate QL

• The gas volumetric flow rate QG is started from a very low value and is gradually increased

Adiabatic Vertical 2 Phase Flow Patterns. -1

Bubbly flow pattern characteristics

Liquid

Gas

• Distorted-spherical and discrete bubbles• move in a continuous liquid phase• Bubbles have little interaction at very low QL

• Increase in number density as QG is increased

• At higher QG rates, bubbles interact, leading to their coalescence and breakup

Adiabatic Vertical 2 Phase Flow Patterns. - 2

Liquid

Gas

• Large QG discrete bubbles coalesce to form very large bubbles

• The slug / plug flow regime then develops; dominated by bullet-shaped bubbles (Taylor bubbles)

• Approximately hemispherical caps and are separated from one another by liquid slugs.

• The liquid slug often contains small bubbles• A Taylor bubble approximately occupies the entire cross

section and is separated from the wall by a thin liquid film

• Taylor bubbles coalesce and grow in length until a relative equilibrium liquid slug length (Ls/D 16) in ∼common vertical channels is reached

Slug / Plug flow pattern characteristics


• At further higher QG large Taylor bubbles are disrupted to form churn (froth) flow

• A chaotic motion of the irregular-shaped gas pockets takes place, where the interface shape cannot be defined

• Both phases • May appear to be contiguous• Have incessant churning and oscillatory

backflow• The liquid near the tube wall continually pulses up and

down• Churn flow also occurs at the entrance of a vertical

channel, before slug flow develops.

Churn flow pattern characteristics


• Annular-dispersed (annular-mist) flow replaces churn flow at even higher QG

• A thin liquid film, often wavy, sticks to the wall while gas occupies the core often with entrained droplets

• In common pipe scales, the droplets are typically 10–100 μm in diameter

• The annular-dispersed flow regime is usually characterized by continuous impingement of droplets onto the liquid film and simultaneously an incessant process of entrainment of liquid droplets from the liquid film surface

Annular flow pattern characteristics

Dispersed Bubbly flow pattern characteristics


Flow regimes associated with very high QL

• All flow regimes where the 2 phases are not separated • Due to very large liquid and mixture velocities the slip

velocity between the two phases is often small in comparison with the average velocity of either phase

• The effect of gravity is relatively small• As long as the void fraction is small enough to allow

the existence of a continuous liquid phase• Highly turbulent liquid flow does not allow existence

of large gas chunks & shatters gas into small bubbles• The bubbles are quite small and nearly spherical• No churn flow take place• Transition from slug to annular-mist flow may only

involve churn flow due to the oscillatory flow caused by the intermittent passing of large waves through a wavy annular-like base flow pattern. Dispersed Bubbly Flow

FLOW REGIME MAPS. -1• Flow pattern maps are an attempt, on a 2-D graph, to separate the

space into areas corresponding to the various flow patterns

• Simple flow pattern maps use the same axes for all flow patterns

• and transitions

• Complex maps use different axes for different transitions

• There are flow maps for vertical as well as for horizontal flow

• Several maps are available in the literature

• There are quite a few limitations in the use of these maps as will be pointed out at the end of this topic

VERTICAL FLOW REGIME MAPS. -3The Hewitt and Roberts (1969) map for vertical up flow in a tube

2

2

2

21

,

superficial velocity

mass flow rate

tube cross-sectional area

G GG

L LL

Gxj

G xj

where

j

22 21( )

G LG L

G L

G xG Gx Q QQj j j

A A A

FLOW REGIME MAPS. -4 (vertical flow)

Gover and Aziz 1972, vertical two phase flow map, 2.6 cm ID

HORIZANTLE TWO PHASE FLOW PATTERNS. - 1

Bubbly Flow

Increasing Gas flow Rate

Plug Flow Stratified Flow

Wavy Flow Slug Flow Annular Flow

HORIZANTLE TWO PHASE FLOW PATTERNS. - 1

1. Bubbly flow, in which the gas bubbles tend to flow along the top of the tube;

2. Plug flow, in which the individual small gas bubbles have coalesced to produce long plugs;

3. Stratified flow, in which the liquid—gas interface is smooth. Note that this flow pattern does not usually occur, the interface is almost always wavy as in wavy flow.

4. Wavy flow, in which the wave amplitude increases as the gas velocity increases;

5. Slug flow, in which the wave amplitude is so large that the wave touches the top of the tube;

6. Annular flow, which is similar to vertical annular flow except that the liquid film is much thicker at the bottom of the tube than at the top.

FLOW REGIME MAPS. -2

1

2g l

air water

12 3

water l water

water l

Baker’s map for horizontal 2 phase flow in a tube

FLOW REGIME MAPS. -5 (horizontal flow)

Gover and Aziz 1972 Horizontal two phase flow map, 2.6 cm ID

QUALITY AND VOID FRACTION. -1

FOR A STATIONARY TWO PHASE MIXTURE;

mixture of mass Total

mixture in vapors of M assQuality x

mix ture o f v o lume To ta l

mix ture in v a po rs o f Vo lumeFra ct io n Vo id

Relationship between ‘x’ and ‘’ can be established by assuming a unit mass. From thermodynamics the volume of this unit mass is;

fgf xvvmass

volume

f

g

g

ffgf

g

xx

v

v

xxxvv

xv

1

1

1

11

1

For x=2% steam water system =97.1% at 1 atmosphere

Quality

Voi

d f

ract

ion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(air-water system)

(water-steam system, 100C, 1 bar)

(water-steam system, 300C, 86 bar )

(water-steam system, 350C, 165 bar)

water-steam system, 374.15C, 221.2 bar)


Important Inferences:

1. For constant x, decreases with pressure. x = at critical pressure

2. For a constant pressure d/dx decreases with quality

3. For low ‘x’ as in BWRs, d/dx increases with decreasing pressure,

becomes exceptionally large affecting reactor stability


FOR A FLOWING TWO PHASE MIXTURE

mix ture o f ra te f lo w ma ss To ta l

mix ture in v a po rs o f ra te f lo w M a ssQua lity x

Using eq. of continuity one can find the velocities of vapor or liquid

g

tgg

g

gggt A

mxvV

v

VAmmx

Similarly

(1 )(1 ) f f f t

t f ff f

A V v x mx m m V

v A

In flowing systems the vapor phase tends to move faster especially in

vertical systems. The ratio of velocities of the two phases is Slip Ratio

ft

gt

mmx

mmx

liquid of rate flow mass

vapor of rate flow mass

1

SLIP RATIO, -1

Hence SLIP RATIO is;f

g

g

f

f

g

v

v

A

A

x

x

V

VS

1

1

g

f

gf

g

A

A

AA

A

SLIP RATIO, QUALITY & VOID FRACTION can be related as;

1(1)

1g g

f f

V vxS

V x v

1 1(2)

11 11 f

g

xvxS xx v

1 11 1 (3)x

If S=1,

there can be two cases

1. A non flow system

2. A flow system with

1. Bubbly flow

2. Dispersed bubbly flow

SLIP RATIO, -2

Important Inferences from the developed equations:

1. For constant pressure and quality, Eq(1), S (1-)/2. Hence for constant pressure and quality decreases with S

3. will be less for a flow with slip as compared to a no slip system

4. Thus a high S system will have less voids, so better heat transfer and moderation

SLIP RATIO. -3Effect of other operating parameters on S, experimental facts only

• S decreases with system pressure and volumetric flow rate

• S increases with quality

Slip ratio correlations -1

Several correlations for calculation the slip, S, and the void fraction are presented in literature.

A few of the most recommended in order of decreasing accuracy.

1 22

0 220 19

1

0 080 512

2

11

1

1 578

0 0273

..

..

.

.

surface tension

L

G

H L

L G

H H L

L L G

yS E yE

yE

where

xy

x

GDE

G D GDE

CISE correlation

Slip ratio correlations -2

0.10.67

1 11 1

g fv vg

f

v

x v

Von Glahn empirical correlation based on data of lot of workers

1

1

1 11 1

11

G

G G L

L L

xK

xx xK K

xx xK

x

Smith’s void correlation

Where K = 0.4 in order to achieve good agreement with experimental data

1 1 L

G

S x

Chisholm’s slip correlation

3L

G

S

Zivi’s slip correlation

1S

Homogeneous flow model

SLIP RATIO. -4

Simplified procedure to design a BWR core channel

Estimate a maximum value of at the channel exit

usually fixed by the nuclear moderation constraints

Corresponding value of ‘x’ is then determined from the earlier mentioned correlations

This ‘x’ then determines the amount of heat generated in the channel

Since depends on S, generally it is assumed that S is constant throughout the channel

This simplification introduces some errors but consider the change in S along a particular channel

SLIP RATIO. -6

1. S increases rapidly in the initial portion of fuel channel

2. S increases quite slowly along the major part of fuel channel

3. S increases sharply at the exit of fuel channel due to turbulence

Flow regimes not found in adiabatic flowsOnly found in boiling channels1. Inverted-annular regime2. Dispersed-droplet regime

• Inverted-annular flow regime a vapor film separates predominantly liquid flow from the wall

• Liquid flow contain entrained bubbles• Flow regime takes place in channels

subject to high wall heat fluxes, leads to DNB

• Dispersed-droplet regime superheated vapor containing entrained droplets flows in an otherwise dry channel

• Regime occurs in boiling channels when massive evaporation has already caused the depletion of most of the liquid

1. These flow regime maps generally address “developed” flow conditions and are not very accurate for short flow passages

2. Empirical flow regime maps often attempt to specify parameter ranges for various flow regimes using a common set of coordinates.

1. Mechanisms that cause various regime transitions are different, a common set of coordinates may not be appropriate for the entire flow regime map

3. Most flow regime maps are based on data obtained with water, or liquids whose properties are not significantly different than water, in channels with diameters in the 1- to 10-cm range. The maps may not be useful for significantly different channel sizes or fluid properties.

4. Hence whatever data reported in literature depends on certain specific conditions of apparatus as well as the choice of the fluids. Any significant variation from the reported conditions lead to significant errors as the basic physics of the multiphase flow is still developing.

5. Change of one flow regime into the other changes the physics of the problem, hence regime dependent data is available which cannot be safely applied to the transition region, unless the physics of the transition is well understood

SHORTCOMMINGS IN FLOW REGIME MAPS

CALCULATION OF BOILING HEIGHTS IN BWR-1

Why we need to calculate Boiling & Non-boiling heights in a BWR

Flow in a BWR channel encounters various types of

1. Flow through restrictions

2. Flow across baffles, spacers, tie plates etc

1. Flow behavior changes as phase changes

2. An increase in the volume of fluid causes acceleration effects

All these lead to pressure drop, which in turn depends on the properties (, etc) of the working fluid

We need to know the height of channel where 2-phase flow is occurring in order to calculate the total pressure drop across the core

1. Flow geometries 1. Flow conditions


1. Heat added to the fluid from subcooled to saturation occur in NBH.

2. Rest of the height is the boiling height

3. Hence the ratio of amount of heat added in the entire channel and in the non-boiling height is related to the ratio of the two heights

f is

t f e fg i

h hq

q h x h h

Total heat added

Sensible heat added

If the heat added is uniform along the height then

f is o

t f e fg i

h hq H

q Hh x h h

1 oB HH

H H


If the heat added is sinusoidal then

For other types of heat additions either simple functions are needed otherwise stepwise graphical solutions are needed

0

0

11

2

sin

cos

sin

oH

c

s oH

tc

zq dz

Hq H

q Hzq dz

H

Pressure Drop in 2 Phase Flow - 1

Total ac fr st

dP dP dP dP

dz dz dz dz

21

sinw f

Total

d AG v pdp Gg

dz A dz t A

Acceleration pressure drop due to change in momentum as phase

change occurs

2

,8w TP

Gwhere f

Wetted perimeter

The Friction Pressure Drop in 2 Phase Flow - 1

• 2 phase flow P is > the single phase pressure drop for same L & G• The difference depends on the flow patterns discussed earlier due to

increased speeds of various phases• The difference in the two pressure drops can be found by

,2 exp, ,2 ,1 ,( )

fr P Total P ac others fr P ONLYP P P P P A

• We need a concept to find the (P)fr,2P from the (P)fr,1P,ONLY

• The concept is the development of the two phase multipliers 2

• By definition • 2 are always greater than 1• There is one hidden problem in equation A• What fluid you use to determine the (P)fr,1P,ONLY liquid / liquid only

or a gas/gas only• In fact researchers have reported 2 for all cases• Hence, 2 is reported as • Generally the liquid only or gas/vapor only is adapted

2

,2 ,1 ,fr P fr P ONLYP P

The Friction Pressure Drop in 2 Phase Flow - 2• While using this concept of 2 one needs to know the thermodynamic

condition of the fluid (for Pfr,1P,ONLY) as well as its flow properties• The thermodynamic state of the single phase liquid or vapor is

always taken as the saturated state• The flow properties are calculated in terms of the superficial velocity

of the liquid only or vapor/gas only conditions for the same mass flow rate through the same cross-sectional area

• This concept is very good but there are some further issues that need to be addressed

• Hence in general the 2 depends on• The phasic flow rates which in turn depends on the flow

regimes. Imagine the liquid vapor interaction in various flow regimes. Each flow regime interaction has its own physics

• The thermodynamic conditions of the multi-phase flow• The phasic thermodynamic properties and how they will be

calculated

The Friction Pressure Drop in 2 Phase Flow - 3• Therefore, the two phase multipliers cannot be reported as a

function of Re number or other parameters as in the case of single phase flow

• The experimental data has been reported for various thermodynamic and geometric conditions as a function of various parameters such as quality, void fraction, pressure, mass flux etc.

• This data can only be used by others only if dynamic similarity exists between the experimental setups used and the equipment that need to be designed by others

• The only way one can feel that the similarities exist is the matching of the flow regimes, which in turn are not yet fully understood

WHAT IS THE WAY OUT• Researchers have developed theoretical models which are regime

dependent for 2 and then by comparing the experimental data with these models researchers have suggested various correction factors to better predict the 2

Two Phase Flow Modelling• Rigorous modelling of gas–liquid 2P flow based on the solution of local

and instantaneous conservation principles is generally very complicated• Simplified models that are based on idealization and time and volume

averaging are usually used instead. • Simplified multiphase flow conservation eqs can be obtained in many waysa. Assuming that each point in the mixture is simultaneously occupied by

both phases and deriving a mixture modelb. Developing control-volume-based balance equationsc. Performing some form of averaging (time, volume, flow area OR

composite) on local and instantaneous conservation equationsd. Postulating a set of conservation equations based on physical and

mathematical insight.

Most widely used is the averaging method, Lead to flow parameters that are 1. Measurable with available instrumentation2. Continuous3. In case of double averaging have continuous first derivatives

Two Phase Flow Models -1Various two–phase flow models

1. Homogeneous mixture model: 1. The simplest two-phase flow model, and it essentially treats the

two-phase mixture as a single fluid2. The two phases are assumed to be well mixed and have the same

velocity at any location, S = 13. Thus, only one momentum equation is needed4. If in a single-component flow, thermodynamic equilibrium is also

assumed between the two phases everywhere, the homogeneous–equilibrium mixture model results

5. The two phases do not need to be at thermodynamic equilibrium, however. Examples include flashing liquids and condensation of vapour bubbles surrounded by sub-cooled liquid

6. The solution of conservation equations is more complicated than single-phase flow

7. The fluid mixture is compressible, with thermo-physical properties can vary significantly with time and position.

Two Phase Flow Models -2

2. Multi-fluid models:

1. The flow field is divided into at least two (liquid and gas) domains, and each domain is represented by one momentum equation. Example: A two–fluid model (2FM), which is currently the most widely used two–phase flow model. (The Separated Flow Model)

2. In 2FM, gas and liquid phases are each represented by one complete set of differential conservation equations (for mass, momentum, and energy).

3. The assumptions of thermodynamic equilibrium between the two phases or saturation state for one of the phases are sometimes made. Either of these assumptions will lead to the redundancy and elimination of one of the energy equations.

Two Phase Flow Models -3

3. Diffusion models:

1. In these models the liquid and gas phases constitute the two domains

2. However, only a single momentum equation is used

3. This is made possible by obtaining the relative (slip) velocity between the two phases, or the relative velocity of one phase with respect to the mixture, from a model or correlation

4. The slip velocity relation is usually algebraic (rather than a differential equation).

5. The drift flux model (DFM) is the most widely used diffusion model. The DFM (the Zuber–Findlay model) is more often used for void fraction calculations

2-P Frictional Pressure Drop in Homogeneous Flow-1

Assumptions;• The two phases are assumed to remain well mixed• The two phases move with identical velocities everywhere S = 1• A homogeneous mixture thus acts essentially as a singe-phase fluid that is

compressible and has variable properties• Two-phase pressure drop is developed by analogy with single-phase flow

For a turbulent single-phase flow

2 21

1 1,1 1

1 1

2 2P

P Pfr P H H P

Vdp Gf f

dz D D

Assuming that the friction factor may be expressed in terms of theReynolds number by the Blasius equation

0.25

0.25

11

0.079 Re 0.079 HP

P

GDf


By analogy for a turbulent two-phase flow, one can write2 2

22 2

,2 2

1 1

2 2P

P Pfr P H H P

Vdp Gf f

dz D D

14

14

2 22

0.079 Re 0.079 HP P

P

GDf

1

2 2

11P g f P

g f

xxx x

A mean two-phase viscosity 2P of the homogenized fluid is needed, which must satisfy the following limiting conditions

2 20, ; 1,P f P gx x


Various forms of such relationships quoted in literature

1

2

(1942)

1P

g f

McAdams et al

xx

2

(1960)

1P g f

Cicchitti et al

x x 2 2

(1964)

1P P g g f f

Dukler et al

xv x v

Substitute the relation of 2P density and f2P into the 2P Darcy equation

2

,2LO

fr P Lo

dp dp

dz dz

14

1 1 1 L GL

G G

x x

2

,2GO

fr P Go

dp dp

dz dz

14

1 1G G

L L

x x x x

2

,2G

fr P G

dp dp

dz dz

14

741 1 1G G

L L

x x x x

2

,2L

fr P L

dp dp

dz dz

14

741 1 1 1 L GL

G G

x x x


Values of the two-phase frictional multiplier for liquid only for thehomogeneous model steam-water system

14

2 1 1 1 L GLLO

G G

x x

The two-phase friction factor -HEM-8

Application of the homogeneous theory to experimental observations

Friction factor for use in the HEM can be calculated, by either

From single-phase flow correlations

OR, estimate directly from measured two-phase pressure drops

values of fTP in the range 0.0029-0.0033 have been suggested for low-pressure flashing steam-water flow (Benjamin and Miller 1942; Bottomley 1936-37; Allen 1951)

Values of about 0.005 for analysis of circulation in high-pressure boilers

(Lewis & Robertson 1940; Markson et al. 1942) and for petroleum pipe stills(Dittus & Hildebrand 1942)


1. The experimental two-phase friction factor fTP was plotted against the

Reynolds number for all-liquid flow.

2. Large discrepancies from the single-phase friction factor were

observed at Reynolds numbers less than 2 x 105

3. Considerably better agreement with the normal single-phase flow

relationship if the experimental friction factor is plotted against the all-

liquid Reynolds number multiplied by the ratio of the inlet to outlet

mean specific volumes. This is equivalent to defining a new average viscosity

1

1 Re 1fg fgf

f f f

v vGD GDx x

v v

• The HM model performs reasonably well for well-mixed configuration (e.g., dispersed bubbly).

• It deviates from experimental data for flow patterns such as annular, slug, and stratified flows

• Using empirical correlations remain the most widely applied method to date

• Most empirical correlations use the concept of two-phase flow multipliers that are applicable to all flow regimes (i.e., flow regime transition effects are implicitly included in them).

• Original concept was proposed by Lockhart and Martinelli (1949) based on a simple separated-flow model

• The separated flow model in general it indicates that

• A number of empirical methods are available• We will discuss quite a few of these empirical methods

2-P Frictional Pressure Drop in Separated Flow-1

The Lockhart-Martinelli Method -1Martinelli and his co-workers argued that the two-phase friction multipliers ϕ2

L and ϕ2G can be correlated uniquely as a function of a parameter X, where

,2

,

fr L

fr G

dpdz

Xdpdz

This was verified using their experimental data. The resulting graphical correlation is shown where friction factor multipliers are is plotted against X for all four flow regimes

The Lockhart-Martinelli Method -2

2 2 22 2 1 2f f f f ff

f

f x G v f u

D D

2 2 22 2 2g g g g gg

g

f x G v f u

D D

,2

,

f fr

g fr

dpdz

Xdpdz

2 2

, ,g f

fr g fr f fr

dP dP dP

dz dz dz

2,2

2

,

f fr g

f

g fr

dpdz

Xdpdz

Combine


2 2

2

2 2

2 1

2

f f

g g

f x G v

DX

f x G v

D

n

f ff

f

u Df K

n

g gg

f

u Df K

Simplification leads to

For non circular geometry D = De

Using Blasius correlation where

n= 0.25

22 1

n nf g

g f

xX

x

0.25 1.752 1f g

g f

xX

x


1. Turbulent (L)-Turbulent (G) ϕtt

C = 202. Viscous (L)-Viscous (G) ϕvv

C = 53. Turbulent (L)-Viscous (G) ϕtv C

= 104. Viscous (L)-Turbulent (G) ϕvt C

= 12

The curves are well represented by these when C has the following values in the following equations

• To use the Lockhart-Martinelli correlation to calculate the two-phase friction pressure gradient,

• calculate the friction pressure gradients for each phase flowing in the channel and then use the Figure OR use the correlations in terms of X

• The correlation was developed for horizontal two-phase flow of two-component systems at low pressures (close to atmospheric) and its application to situations outside this range of conditions is not recommended.

22

2 2

11

1

L

G

C

X X

CX X


It follows that if the parameter ϕf is a function of the parameter X then

the void fraction α must also be a function of X. The correlation between α and X can be derived

2 2

, ,fo f

fr fo fr f fr

dP dP dP

dz dz dz

22 22 2

,

2 2 1fo f f ffo f

frf fr

f G v f G x vdP

dz D D

0.25

10.079f

f

G x Df

0.25

0.079fof

GDf

0.251

1f

o

f

f x

22 2 1ffo f

fo

fx

f

1.752 2 1fo f x

The Martinelli-Nelson ( 1 948) correlation -1

The Lockhart-Martinelli correlation was related

1. To the adiabatic flow of low pressure air-liquid mixtures,

2. However, the information was purposely presented in a generalized manner to enable the application of the model to single component systems and to steam-water mixtures.

1. For the prediction of pressure drops during forced circulation boilingMartinelli and Nelson (1948) assumed the flow regime would alwaysbe ' turbulent-turbulent '.

Features of Martinelli-Nelson correlation

2. Correlation of frictional pressure gradient is worked in terms of the parameter which is more convenient for boiling and

condensation problems than

2fo

2f


3. Thermodynamic equilibrium was assumed to exist at all points in the flow and the curve correlating was arbitrarily applied to atmospheric pressure steam-water flow.

2,f tt

4. A relationship between ϕf and Xtt was established for the critical pressure level by noting that as the pressure is increased towards the critical point, the densities and viscosities of the phases become similar. The relationship may be represented by

with the value of C=1.36

22

11f

C

X X

5. Knowing the CURVES for critical and atmospheric pressure, curves at intermediate pressures were established by trial and error using the experimental data of Davidson et al. (1943) as a guide.

6. With this knowledge of f and Xtt for a number of pressures a plot of 2

fo and mass quality x was made using 22 2 1fo f f fof f x


HEM correlation


In order to calculate the total two phase pressure drop let us revisit the separated flow model equation

22

0

2 1 xfo f

fo acc grav

f G v Lp dx p p

D x

Need to solve this term, an average value of friction factor multiplier

Martinelli & Nelson evaluated this integral


Martinelli and Nelsonused the Lockhart-Martinelli curve for steam-water flow at Patm showed that, at the. critical pressure α = β = x. Knowing the α-Xtt curves for both atmospheric and critical pressures, curves atintermediate pressures were interpolated. These curves were then transposedto give values of α as a function of mass quality x with pressure as parameter

The Thom correlation -1An alternative set of consistent values for the terms 2

0

1, and

x

fodx xx

Why there was a need Values were interpolated between Patm and Pcr

Thom ( 1964) revised values were derived

using an extensive set of

experimental data for steam-water pressure drops obtained at Cambridge, England, on heated and unheated

horizontal and vertical tubes.

Application of the SFM to experimental observations -1

1. The Lockhart-Martinelli-Nelson model has been used extensively for the correlation of experimental pressure gradients and void fraction measurements for both single- and two-component gas-liquid flow.

2. Generally, it is found that the separated flow model is capable of more accurate predictions than the homogeneous model.

3. Two general observations can be made concerning the application of the Lockhart -Martinelli correlation.

1. It has been widely recognized that the curves of experimental data plotted as ϕ2

f or ϕ2g versus X are not smooth as shown.

There are discontinuities of slope which are associated with changes of flow pattern

2. An effect of mass velocity upon the curves of ϕ2f versus X has

also been widely reported for steam-water flow at high pressures. The original Martinelli-Nelson correlation corresponds to a mass velocity of 500- 1000 kg/m2s. The homogeneous model yields values close to those obtained experimentally for mass velocities greater than 2000 kg/m2s.

Application of the SFM to experimental observations -2

Quantitative data which illustrate the effect have been reported by Zuber et al. (1967) for the case of Freon at elevated pressures and by Hughmark and Pressburg (1961) for low-pressure air-liquid flow.

Correlations for use with the HEM or SFM -1

Attempts to correct existing models for the influence of mass velocity on the frictional multiplier, have been published by Baroczy (1965), by

Chisholm (1968), and by Friedel (1979).

2,fo tt

THE BAROCZY CORRELATION

This method employs the use of two separate set of curves

The first of these is a plot of the two-phase frictional multiplier as a function of a physical property index with mass quality ‘x’ as parameter for a reference mass velocity of 1356 kg/m2 s (1x106 1b/hr ft2)

The second is a plot of correction factor Ω expressed as a function of the same physical property index for mass velocities of 339, 678, 2712, and 4068 kg/m2 s with mass quality as parameter. This plot serves to correct the value of obtained from the first plot to the appropriate value of mass velocity.2

,fo tt

The method proposed by Baroczy was tested against data from a wide range of systems including both liquid metals and refrigerants with satisfactory

agreement between the measured and calculated values.

THE BAROCZY CORRELATION plot-1

THE BAROCZY CORRELATION DATA-1

THE BAROCZY CORRELATION plot-2

22

( 1356),

2 fo ffo G

fo fr

f G vp dP

L dz D

CHISHOLM'S METHOD -1

22

11f

C

X X

Chisholm and Sutherland proposed the following procedure to account for effects of G for steam-water flow in pressure tubes at pressures above 3 MPa (435 psia/30 bar) .

For Gm ≤ G*; (G* is a reference mass flux)

0.5 0.5 0.5

2fg g f

g f g

v v vC C

v v v

where

20.5 2 2 power coeff of Re in the ' ' relationn n f *

2

GC

G

22

11f

C

X X

For Gm > G*; (G* is a reference mass flux)

0.5 0.5

g f

f g

v vC

v v

where

2 2

1 11 1

C CT T T T

2 2 1 2

1

n nn f f

g g

vxT

x v

CHISHOLM'S METHOD -2

C is obtained as in the case of G ≤ G*

For rough tubes;G*=1500kg/m3s, λ = 0.75

and n = 0.2

For smooth tubes;G*=2000kg/m3s, λ = 1.0

and n = 0

COMPARISON OF METHOD -4

For G > G*, ψ <1

2

2( )

fo

fo HEM

THE FRIEDEL CORRELATION -1

It was obtained by optimizing an equation for to using a large data base of two-phase pressure drop measurements

2fo

Friedel (1979), One of the most accurate two-phase Δp correlations

2 2 31 0.045 0.035

3.24fo

A AA

Fr We

where

2 21 1 f go

g fo

fA x x

f

0.2240.78

2 1A x x

0.91 0.19 0.7

3 1f g g

g f f

A

2 2

2

G G DFr We

gD

0.91 0.19 0.7

0.242 0.78 0.045 0.0351 3.24 1 1f g g

fog f f

A x x Fr We

Substituting A2 and A3 to get the following relation

THE FRIEDEL CORRELATION -2where fgo and ffo are the friction factors defined for the total mass velocity G as all vapour and all liquid, respectively by the equation

The correlation is valid for vertical upwards flow and for horizontal flow

2/ /

/

2 fo go f g

fo go

f G vdp

dz D

D is the equivalent diameter σ is the surface tension the homogeneous density given by eq.

Standard deviations are about 40-50 per cent, which is large with respect to single-phase flows but quite good for two-phase flows.

1

f g fv x v v

For vertical, downward flow, Friedel’s correlation gives

0.9 0.73 0.74

0.292 0.8 0.03 0.121 48.6 1 1f g g

fog f f

A x x Fr We

2

Re0.25 0.86859ln

1.964ln Re 3.8215fo

fo

fo

f

THE FRIEDEL CORRELATION -3

Whalley (1980) has evaluated separated flow models against a large

proprietary data bank and gives the following recommendation:

f g For most fluids and operating conditions, ( ) is less than 1000 and the Friedel correlation will be the preferred method.

f g (c) For ( ) > 1000 and G < 100 kg/m2 s: Utilize the correlations ofLockhart and Martinelli (1949) and Martinelli and Nelson (1948)

f g (b) For ( ) > 1000 and G > 100 kg/m2 s: Utilize the most recent refinement of the Chisholm (1973) correlation;

f g (a) For ( ) < 1000: Utilize the Friedel (1979) correlation;

THE ACCELERATION Δp FOR TWO PHASE FLOWThe net force acting due to acceleration pressure drop is the product of

acceleration pressure drop and cross-sectional area

a c f fe g ge t iF p A m V m V mV

1

1e t fe e t ge t ia e fe e ge i

c c c

x mV x mV mVp G x V x V V

A A A

1 1 1

1 1f f e t f e t f e f

fefe fe e c e

m V x m v x m v x GvV

A A A

e gge i i

e

x GvV and V Gv

Finally 2 2

2 21

1e e

a f g ie e

x xp G v v v r G

Acceleration multiplier

THE MINOR LOSSES FOR TWO PHASE FLOWSimilar to frictional pressure drop, often a two–phase multiplier is

used

The two–phase pressure drop in a sudden expansion USING THE DEFINITIONS OF HEM

The two–phase pressure drop in a sudden contraction

The homogeneous flow model has been found to do well in predicting experimental data (Guglielmini, 1986)

THE MINOR LOSSES FOR TWO PHASE FLOWEmpirical correlations for two phase pressure drop and it is assumed

that the pressure drop associated with single-phase flow is known

For flow through orifices, Beattie (1973) proposed

For flow through spacer grids in rod bundles, Beattie (1973) proposed

Chisholm (1967, 1981) for two-phase pressure drop in a bend Martinelli’s factor

defined for the bend

where KL0 is the bend’s single–phase loss coefficient for the conditions when all the mixture is pure liquid, R = bend curvature radius and D = pipe diam

Po

43

1

2

CRITICAL FLOW -1V

eloc

ity

Length

5

000

1

2

V*

3, 4, 5

Pback

Pexit

Po

12

Pe*

3, 4, 5

0 0

1

2

345

PbPre

ssur

e

Length

1. This phenomenon occurs in both single- and two- phase flow.

2. The phenomenon has long been observed in boiler and turbine systems, flow of refrigerants and rocket propellants, and many others.

3. In nuclear reactors, the phenomenon is of utmost importance in safety considerations of both boiling and pressurized systems.

4. A break in a primary coolant pipe causes two-phase critical flow in either system since even in a pressurized reactor, the reduction of pressure of the hot coolant from about 10 MPa to near atmospheric causes flashing and two-phase flow.

5. This kind of break results in a rapid loss of coolant and is considered to be the maximum credible accident in power reactors built to date.

6. An evaluation of the rate of flow in critical 2P systems is important for the design of emergency cooling and for the determination of the extent and causes of damage in accidents.

CRITICAL FLOW -2

Steady, 1-D Isentropic Flow With Area Change

Governing equation for Steady 1-D Flow With Area Change;Assumptions:1. No body forces, gdz = 02. No friction, Ff = 03. No heat Transfer, adiabatic flow, Q = 04. No drag force, D = 05. No work done, W = 0

Continuity Equation

constantm AV

Momentum Equation

0dp VdV

Energy Equation

2

constant2

Vh H

Entropy Equation constants

Effect of Area Change on Flow Properties

Continuity Eq.

constantm AV

Differentiate logarithmically

0d dA dV

A V

Momentum Eq.

0dp VdV

Can be written as

2 0dp dV

VV

Speed of Sound

2

s

pa

For isentropic flow

2a d dp

Combine the three equations and use the definition of Mach Number to get

2 1dA dV

MA V

22

1dA p dp

MA V p

These equations suggest the manner in which area should change to accomplish the required expansion or compression of a compressible fluid

Effect of Area Change on Flow Properties, contd. -2

(a) Nozzle flow/action

(b) Diffuser flow/action

The area change which shows the effect of increase in velocity with decrease in pressure

The area change which shows the effect of decrease in velocity with increase in pressure

Relationship between dA & dM for Steady 1-D Isentropic flow

dAM

< 1.0subsonic

> 1.0supersonic

dA < 0 dM > 0 dM < 0dA > 0 dM < 0 dM > 0

Tabl

e

Effect of Area Change on Flow Properties, Choking

WHAT WOULD HAPPEN ONCE THE

SONIC CONDITIONS ARE REACHED & AREA CHANGE CONTINUES

For subsonic/supersonic flowOnce M=1 at a particular section of converging section then What would be M2, whereas dA < 0 Table

There can be two possibilities/assumptions1. M2 < M1 where M2 = subsonic2. M2 > M1 where M2 = supersonic

For: M2 < M1

Table suggests, that for dA < 0 dM > 0

Contradiction with basic assumption, M2 < M1

For: M2 > M1

Table suggests, that for dA < 0 dM < 0

Contradiction with basic assumption, M2 > M1

Effect of Area Change, Choking, contd. -2

The Table Is In Fact A Representation Of The Conservation Laws Under Isentropic Conditions

THESE CONSERVATIONS ARE NEVER VOILATED BY NATURE

CRITICAL MASS FLOW RATE-1The energy equation (no heat transfer or work)

where h and ho are the specific enthalpy & stagnation enthalpy of the fluid.

Thus

For an ideal gas dh = CpdT, so

for an ideal gas (reversible)

applying the continuity equation, equation becomes

1

o o

T p

T p

2 2 1P o o P oo

TV gC T T T gC T

T

2 oV g h h

0c

VdVdh

g

2

2 oc

Vh h H

g

1

o

o o

pp p

RT RT p

2 1

2oc P o

o o o

Ap p pm g C T

RT p p

Effect of pressure ratio on mass flow rate

The value of rp* can be obtained by differentiating m in the above equation with respect to p and equating to zero.

1* 2

1pr

Effect of pressure ratio on mass flow rate

A representative value of rp* is 0.53 for air at low temperatures, = 1.4

Steam is not a perfect gas

rp* for single-phase steam flow is approximated from the perfect gas relationship by replacing with 1.3 for superheated & supersaturated

rp* value of 0.545.

Two Phase Critical Flow -1

22

, 2

cos2

1

TP mm g m

e m

Total HEM gm

f G dxG v g

D dzdpvdz

G xp

For a 2P critical flow condition, the requirement is that dp/ dZ ∞

2 1m cr

g

dpG

x dv

122

,SFM

22 2

2 22

2 2

1

2 2 1

2 1

1cos

1

gm

Total

g flo mlo m

e f

m f g m

vdp xG

dz p

xv x vf G dxG

D dz

x x dG v v g

dz

22m cr

g

dpG

x dv

The total pressure drop using the HEM, (Reference KAZMI, P=489)

The total pressure drop using the SFM, (Reference KAZMI, P=491)

Two Phase Critical Flow -2• The different results for the above 2 models different slip ratio • In flashing fluids, a certain length of flow in a valve or pipe is needed

before thermal equilibrium is achieved. • Before that particular length there should be non equilibrium

conditions in terms of both the velocity and the temperature differences between the two fluids

• Hence length of the flowing section plays a critical role in determining the flow rate at the exit

• In the absence of sub cooling and non condensable gases the length to achieve equilibrium appears to be on the order of 0.1 m

Source D(mm) L/D L (mm)Fauske (water) 6.35 6 100Sozzi & Sutherland (water) 12.7 10 127Flinta (water) 35 3 100Uchida & Nariai (water) 4 25 100Fletcher (freon II) 3. 2 33 105Van Den Akker et al . (freon 12) 4 22 90Marviken data (water) 500 > 0.33 < 166

Relaxation length observed in various critical flow experiments with flashing liquids

Two Phase Critical Flow -3

In long channels, 1. Residence time is sufficiently long and thermodynamic equilibrium

between the phases is attained2. The liquid partially flashes into vapor as the pressure drops along

the channel, and the specific volume of the mixture v attains a maximum value at the exit.

3. As v is a function of x & , it must be a function of the slip ratio S

In the absence of sub cooling and non condensable gases the length to achieve equilibrium appears to be of the order of 0.1 m

• If flow lengths < 0.1m,• Discharge rate increases strongly with decreasing length as the

degree of non-equilibrium increases• More of the fluid remains in a liquid state

CRITICAL FLOW IN LONG CHANNELS -4

1. Having L/D, ratios between 0 (an orifice) and 40

2. Believed to be independent of diameter alone.

3. Critical pressure ratio was found to be approx, 0.55 for long channels in which the L/D ratio exceeds 12, region III

4. This is the region in which the Fauske slip-equilibrium model is applicable.

Data obtained on 0.25 in. ID channels with sharp-edged entrances

The critical pressure ratio varies with L/D, for shorter channels, but independent of the initial pressure in all cases

Equilibrium Models For Two Phase Critical FlowTotal enthalpy of 2P mixture

under thermal equilibrium conditions undergoing isentropic expansion can be written as:

2 2

1 12 2g f

o g f

V Vh xh x h x x

Total entropy of the two-phase mixture can also be written as:

1 o fo g f

g f

s ss xs x s x

s s

Enthalpy of the two-phase mixture undergoing isentropic expansion under thermal equilibrium conditions can be written in terms of G as:

2 1cr o g fG h xh x h

1

1 2

2

1 1

g f

x Sx xwhere x

S

, , ,gcr o o cr

f

VS and G G h p p S

V


• If the critical pressure is known, ρg , ρf , hg , s and hf are known• x can be determined provided the slip ratio (S) is known• There are three models available to use the value of slip ratio S

Moody model is based on maximizing the specific kinetic energy of the mixture with respect to the slip ratio

HEM model

1S 1 3

Moody model

f gS 1 2

Fauske model

f gS

2 210

2 2g fxV x V

S

Fauske model is based on maximizing the flow momentum of the mixture with respect to the slip ratio

1 0g fxV x VS


Solutions for the set of equations defining the Fauske slip-equilibrium model

Critical flow is described by local conditions at the

channel exit. Flow is seen to increase with increasing pressure &

with decreasing quality at the exit.

The Fauske model assumes thermodynamic equilibrium a case which due to the duration of flow, applies to long flow channels.

Experimental data by many investigators showed the applicability of the Fauske model to L/D ratios above 12


A comparison of the predicted flow rate from various models• HEM model prediction is good for pipe lengths greater than 300 mm

and at pressures higher than 2.0 Mpa• Moody' s model over predicts the data by a factor of 2,• Fauske's model falls in between• When the length of the tube is such that L/D > 40

• HEM model appears to do better than the other models• Generally, the predictability of critical two-phase flow remains

uncertain• Results of one model appear superior for one set of experiments

but not others

CRITICAL FLOW IN SHORT CHANNELS - 1

Liquid flashing into vapor occurs when the liquid moves into a region at pressure lower than psat, if thermal equilibrium is maintained,

Flashing can be delayed due to1. lack of nuclei about which vapor bubbles may form2. surface tension which retards their formation3. heat-transfer problems

Such a situation is called a case of metastability

Metastability occurs in rapid expansions, particularly in short flow channels, nozzles, and orifices.

The case of short channels has not been completely investigated analytically

For 0 < L/D <12 the critical pressure ratios depend on L/D, unlike long channels

The experimental data covers both long and short tubes, 0 < L/D < 40

Thermal non-equilibrium cases


For orifices (L/D = 0) the experimental data showed that because residence time is short flashing occurred outside the orifice and no critical pressure existed.

The flow is determined from the incompressible flow orifice equation

For Region 1, 0 < L/D < 3, the liquid immediately speeds up and becomes a metastable liquid core jet where evaporation occurs from its surface

0.61 2 o bG p p

0.61 2 o crG p p

The flow is determined from the incompressible flow orifice equation but pb changes to pc which can be obtained from


For 3 < L/D < 12, the metastable liquid core breaks up, resulting in high-pressure fluctuations. The flow is less than would be predicted.

experimental critical flows for region II.


It will have some effect on rounded-entrance channels. The existence of gases or vapor bubbles will affect the flow also, since they will act as

nucleation centres.

All the above data were obtained on sharp entrance channels

In rounded entrance channels, the metastable liquid remains more in contact with the walls and flow restriction requires less vapor.

For 0 < L/D < 3 channels, such as nozzles, the rounded entrances result in higher critical pressure ratios than indicated earlier as well as greater flows.

The effect of rounded entrances is negligible for long channels (L/D > 12) and the slip equilibrium model can be used

The effect of L/D ratio on flow diminishes between 3 and 12.

The condition of the wall surface does not affect critical flow in sharp entrance channels, since the liquid core is not in touch with the walls and

evaporation occurs at the core surface or by core break up


In the absence of significant frictional losses, Fauske proposed

1fgcr

fg f

hG

v NTc N, a non-equilibrium

parameter

2

2 210

2fg

f fg f

hN L

p K v Tc

where Δp = po - pb K = discharge coefficient (0.61 for sharp edge)L = length of tube, ranging from 0 to 0.1 m

For large values of L (L 0.1 m) , N =1.0 and above eq. reduces to

1fgcr

fg f

hG

v Tc

When the properties are evaluated at po the value of Gcr predicted by this eq. is called the equilibrium rate model (ERM)


A comparison of ERM model with experimental data and other models


• The effect of sub-cooling on the discharge rate is simply obtained by accounting for the increased single-phase pressure drop [po - p(To)] resulting from the sub-cooling

• For flow geometries where equilibrium rate conditions prevail for saturated inlet conditions (L 0.1 m) , the critical flow rate is

Good agreement between this prediction and various data including the large-scale Marviken data

If sub-cooling is zero [p(To) = po] , the critical flow rate is approximated by ERM model

22cr o o l ERMG p p T G Eq. A


Comparison of Marviken test 4 (D = 509 mm and L/D = 3.1) and calculated values based on Eq. A

Comparison of typical Marviken data (D ranging from 200 to 509 mm and L ranging from 290 to 1809 mm) and calculated values using Eq. A

Axial Temperatures in fuel rods-1

A typical radial temperature distribution in a fuel channel

The radial temperature distribution across a fuel rod already done in 1st semester.

Now we study the axial distribution of the coolant, clad and fuel temperatures in steady state conditions. Both the PWR and BWR configurations

Steady-state temperature distribution in a PWR core

As a reasonable assumption, that the variation of the axial heat generation rate, q′(z), along the fuel rod has a cosine trend:

( ) cosc eq z q z H

In terms of per unit length ( ) cosc eq z q z H

Axial Temperatures in fuel rods-2Assumptions:

• coolant remains always in the liquid phase

• physical properties of the coolant, fuel and cladding are constant.

COOLANT TEMPERATURE

Energy balance in the coolant for a volume having infinitesimal height is:

Heat generated in infinitesimal volume = sensible heat gained by coolant

( ) ( ) ( )p coolant f coolantmc dt q z dz If cross-sectional area of fuel is constant

OR

( ) ( ) ( )p coolant f coolant cmc dt q z A dz

Temperature rise in fluid when it interacts with fuel rod height dz

Volume of infinitesimal height of fuel rod

Axial Temperatures in fuel rods-3Total temperature increase in fluid when it passes through the flow channel while interact with fuel rod of height –H/2 to z

( ) ( )

2 2

( ) cosf

fin

t z z

p coolant f coolant c c cet H H

zmc dt A q z dz A q dz

H

( )

sin sin2in

c c efluid f

p coolant e e

A q H z Ht t

mc H H

This equation gives temperature rise of fluid as function of channel height z

( )

2sin

2out in

c c ef f

p coolant e

A q H Ht t

mc H

( )

2out in

c cf f

p coolant

A q Ht t

mc

To get the fluid exit temperature put z= H/2

In case H = He


To get the cladding temperature tc make the heat balance across the heat transfer surface between cladding and coolant at steady state

Heat convected through cladding surface = heat generated in volume of height dz

cosc c c c fe

zA q dz A q dz h t t Cdz

H

Solve this equation in combination with the equation giving the fluid temperature at any height z

( )

1sin sin cos

2in

ec f c c

p coolant e e e

H z H zt t A q

mc H H hC H

CLADDING TEMPERATURE

heat transfer coeff.assumed constant

Surface area of heat transfer

Temperature rise across the liquid film


( )

1sin sin cos

2in

ec f c c

p coolant e e e

H z H zt t A q

mc H H hC H

hence

cosc cc f

e

A q zt t

hC H

hence

(tc – tf ) has also a cosine distribution as q′′′

tf

A constant = (tc – tf )c at z = 0

Axial Temperatures in fuel rods-6How to get the height where maximum cladding surface temperature will occur

0cdt

dz

Solve to get

1

( )

tane ec

p coolant

H hCHz

mc

If R = radius of the fuel pellet and c = cladding thickness with no gas gap;

1

( )

tan1 12

e ec

p coolant

H Hz

mch R c

Axial Temperatures in fuel rods-7Similarly one can find other temperatures and their locations

1,

( )

tan1 1 1 1

ln4 2

e em fuel

p coolantf c

H Hz

R cmc

k k R h R c

1,

(coolant )

tan1 1 1 1

ln4 2 2 2

e em fuel

cop

f c ci g g co

H Hz

Rmc

k k R R h R h

( )/,CL

sin sin2

1 1 1 1ln cos

2 2 2 4

in

e

p coolant e e

f f c c

co

co c ci g g f e

H z H

mc H Ht t A q

R z

R h k R R h k H

Axial Temperatures in fuel rods-7Similarly one can find other temperatures and their locations

HOT SPOT FACTORS• How the reactor design on paper differ from the actually built reactor.• Does the following parameters are exactly manufactured as per the

drawings: • Fuel elements dimensions• Cladding thickness• Fuel Enrichments• Physical of the materials used such as kfuel, kclad and kgap

• Coolant flow areas and the pressure drop• Neutron flux distribution exactly as predicted in the reactor physics

• The performance limiting parameters are the various maximum temperatures that occur in a power reactor core

• Calculated strictly on the basis of the above mechanical and nuclear parameters mentioned on the reactor design sheets and diagrams

• In practice, deviations occur on both sides• Only those matter that cause the temperatures to exceed the nominal

temperatures• Nominal values that are based on no deviations• Hence deviated values are called hot values. They exceed the nominal

values by factor called hot spot or hot-channel factor

HOT SPOT FACTORS

• Classification of Hot-Spot Factors:• Nuclear Hot-Spot Factors

• Those which occur due to variations in from the core average value• Due to valleys and peaks in fuel and control rods (partial or full

insertion)• Non homogeneous moderator, boiling and structural materials• Maximum effect of inhomogeneity is when neutron mean free path for

thermalization is small as in PWR• Less effect in fast or heavy water or graphite moderated cores• Presence of good infinite reflectors causing peaks at ends of core

• Engineering Hot-Spot Factors (have two sub factors)• Mechanical Factors• Flow distribution factors

The individual sub-factors are usually determined from statistical data

HOT SPOT SUB-FACTORS• Consider a 99.865 % confidence level in deviation of actual to nominal

flow areas• Let Ah be the hot spot flow area as compared to An the nominal low area

• Should An > Ah will worry us or An < Ah

• For An < Ah there will be reduced flow rate hence reduced velocity hence low heat transfer coefficient

• As heat generated is independent of flow rate and remains constant hence for low flow rate the coolant temperature rise will be more.

• Consequently all the other fuel temperatures will increase• Therefore Δtc,h (z) > Δtc,n (z)

• Inlet temperature of coolant is same for both cases tc,h (z) > tc,n (z)

• For low coolant velocity in hot channel Reh will decrease and therefore Δtf,h (z) > Δtf,n (z). The temperature rise across the film (Twall - Tbulk)

• This will lead to a higher outside clad temperature due to two factors originating from one physical deviation i.e flow area

• The two different factors are flow rate decease and heat transfer coefficient decrease

HOT SPOT SUB-FACTORS• The hot spot sub factor for the coolant temperature rise fc for the deviation

in flow area• Pressure drop will be same for all channels irrespective of their flow area

being parallel channels• Assuming turbulent flow and using Darcy equation. Write Darcy equation

for hot and for nominal cases and take their ratio as ΔP is same

0.5 0.5

,

,

e hh n

n e n h

DV f

V D f

0.2

ef D V

0.5 0.1

, ,

, ,

e h e h hh

n e n e n n

D D VV

V D D V

23

,

,

e hh

n e n

DV

V D

4e

AD

P

Flow area different for hot and nominal case

Wetted perimeter is independent of flow area

eD A

23

h h

n n

V A

V A

HOT SPOT SUB-FACTORS• The hot-spot sub factor are ratio of hot to nominal temperature rise• Coolant temperature rise is obtained from the heat generated

( ) ( )p cq z AVc t z ,

,

c h n nc

c n h h

t A Vf

t A V

23

h h

n n

V A

V A

1.667

,

,

c h nc

c n h

t Af

t A

• Similarly the hot-spot sub factor for the film temperature rise0.8

0.40.023 PreVDNu

0.8

0.2e

Vh

D

Film temperature rise is obtained from heat generated and convected away

( ) ( )fq z h t z 0.2 0.8

, ,

, ,

f h e h nf

f n e n h

t D Vf

t D V

1( )ft z

h

HOT SPOT SUB-FACTORS

23

0.2 0.8

, , ,

, , ,

f h e h e nf

f n e n e h

t D Df

t D D

eD A

23

23,

,

e hhe

n e n

DVV D

V D

13

,

,

f h nf

f n h

t Af

t A

OVEL ALL HOT SPOT FACTORS


• Two approaches to estimate the over all hot spot factors• Multiplicative approach, conservative• Statistical approach, realistic

Multiplicative approach = 1134 + 277 = 1411oFStatistical approach = 1134 + 101 = 1235 oF ( for 3σ confidence)Statistical approach = 1134 + 68 = 1202 oF ( for 2σ confidence)Statistical approach = 1134 + 34 = 1168 oF ( for 1σ confidence)

Maximum center line temperature occurs at 70% of the fuel rodCoolant inlet temperature = 556.2 oFNominal temperature rise of coolant = 285+41.5+15.5+39.8+196 = 577.8 oFNominal centerline fuel temperature calculated = 556.2+577.8 = 1134oF

POST CHF HEAT TRANSFERWhy it is important?

1. LOCA or an overpower accident in LWRs may result in the exposure of part of the fuel elements to CHF and post CHF conditions

2. Once through steam generators routinely operate with part of the length of their tubes in CHF and post CHF conditions

General Observations:

3. Post CHF behavior in flow boiling is somewhat dependent on the controlling of wall heat flux and wall temperature

4. In vertical tubes with uniformly applied heat flux there are no hysteresis as in the case of pool boiling

5. If on a short length of tube heat flux is controlled independently of the rest of the tube length then significant hysteresis are observed

6. If wall temperature is controlled rather than wall heat flux then transition boiling is observed

POST CHF HEAT TRANSFER w/o HYSTERESIS

Significant hysteresis, observed in pool boiling once CHF is reached

BUT NO hysteresis, in flow boiling in vertical tubes with uniform heat flux

POST CHF HEAT TRANSFER with HYSTERESIS

q" is controlled over a short length of tube independently of the rest of the tube

CHF is initiated at the junction of the two sections, where the q" is same

Post CHF is observed by decreasing the q" over the short length, whereas over the rest q" remains unchanged

Tw decreases smoothly until rewetting occurs and Tw drops sharply and film boinling occurs

POST CHF HEAT TRANSFER with HYSTERESIS. -2

Suggested Physics of this phenomenon

If the surface is partially wet then rewetting normally occurs once q" is decreased below CHF.

Time taken to rewet the surface depends on the V of the fluid (quench front), where heat transfer through the liquid is conduction controlled

If the surface has completely dried out due to CHF the surface q" must fall below CHF by 10 to 20 % before rewetting occurs

The Tw does not drops with decreasing q" the way it has risen.

The advancing liquid at Tsat encounters a surface at high temperature and heat transfer to the liquid is through the vapors instead of the direct contact with the surface. Thus the liquid is supposed to slip over the hot surface. (INVERTED ANNULAR FLOW)

Once the Tw falls to such a level that the advancing liquid is able to wet the surface only then the Tw falls sharply

POST CHF HEAT TRANSFER with HYSTERESIS. -2

Compare this with the NORMAL ANNULAR FLOW pattern

Tw > Tsat and

Tv Tsat

Tw > Tsat and

Tv > Tsat

High Tw at low quality

INVERTED ANNULAR FLOW

POST CHF TRANSITION BOILING

If Tw is controlled rather than q" over a short length then transition boiling occurs

CHF is initiated at the end of the long section due to high temperature attained in the short section.

If Tw in the short section is lowered to Tsat a smooth cooling curve is obtained rather than a rapid jump in Tw as in case of controlled q".

Point X can be about 80 to 100 percent of the CHF


Martinelli & Nelson similarly provided the quantitative data for the term

22

0

2 22

2

0

2 1

(1 )1

(1 )

sin1

xfo f

fo

gf

f

x

g f

f G v Lp dx

D x

vx xG v

v

Lgdx

x

The Thom correlation -2

The Thom correlation -3

Heat Transfer With Phase Change 2014

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Transcript of Heat Transfer With Phase Change 2014