227

Two-Phase Friction Experimental data indicates that the frictional pressure drop in a boiling channel is substantially higher than that for a single-phase channel with the same length and mass flow rate. Explanations for this include an apparent increased surface roughness due to bubble formation on the heated surface and increased flow velocities. The standard approach to correlating two-phase frictional losses is to assume the total system mass flow rate is due to a saturated liquid and then multiply by an empirical correction factor Φlo

2 called the two-phase multiplier. The local frictional loss would then be

− =∂∂ ρPz

fD

Ggfriction

f

e f co

22

2Φl (1)

A simple functional form for Φlo

2 can be derived by assuming homogeneous flow. For a constant mass flux, velocity increases in a boiling channel as density decreases. If we assume the increase in the frictional loss is due solely to the increase in velocity, then

− =∂∂ ρ φ

Pz

fD

Ggfriction

f

e c

2

22 (2)

or

− =⎛

⎝⎜⎜

⎞

⎠⎟⎟

∂∂ ρ

ρρ φ

Pz

fD

Ggfriction

f

e f c

f2

22 (3)

such that the two phase multiplier is

Φlof2

2=ρρ φ

(4)

The two-phase density is defined as ρ α ρ α ρφ2 = +l f g g (5)

where we have assumed the phases are at equilibrium. The volume fraction under the assumption of homogeneous flow can be obtained from the Fundamental Void-Quality-Slip Relation

α ρρ

gg

f

xx

=+

−1

1 1 (6)

and α α

ρρρρ

l = − =

−

+−

1

1

1 1g

g

f

g

f

xx

xx

( )

( ) (7)

The two-phase density is then

228

ρρ

ρρ

ρ

ρρ

ρρρ

φ2

1 1

1

1 1 1 1=

+−

+

−

+−

=+

−g

g

f

g

g

f

g

g

f

xx

xx

xx

xx

x( )

( )

( )/

( ) (8)

ρρ

ρρ

φ2

1=

+ −

g

g

fx x( )

(9)

1 1

2ρ ρ ρφ= +

−x x

g f

( ) (10)

Note, that Equation 10 implies υ υ υ= + −x xg f( )1 (11) which is a familiar result. Substituting Equation 10 into Equation 4, gives

Φlof

g

f

gx x x2 1 1 1= − + = + −

⎛

⎝⎜⎜

⎞

⎠⎟⎟

ρρ

ρρ

(12)

giving for the Homogeneous Multiplier

Φlofg

fx2 1= +

υυ

(13)

This simple model suggests that the two-phase multiplier varies with quality and therefore position along the channel. As an example, at 1000 psia υ υfg f = 19.6, which would imply Φlo

2 increases rapidly with flow quality. Typical flow qualities in steam generators and BWR cores are on the order of 10 to 20 %. The corresponding two-phase frictional loss would then be 2 - 4 times that in an equivalent single-phase system. In practice, the two-phase multiplier is given empirically as a function of pressure, flow and quality with a number of correlations available in the literature. An alternate approach by Martinelli and Nelson correlates an average value of the two-phase multiplier over the boiling height,

Φ Φl lo o

x

xdx2 2

0

1= ∫ (14)

in terms of pressure and exit quality. The Martinelli-Nelson two-phase multiplier is given in Figure 1 below. The total frictional drop in a boiling channel would then be the sum of the single-phase and two-phase components.

Forms or Local Losses The two-phase pressure loss due to local flow obstructions is treated in a manner similar to the frictional losses. We define a two-phase, local loss multiplier Ψ such that

∆ ΨP K Gglocal

f c=

2

2ρ (15)

229

For design purposes it has been found that the Homogeneous Multiplier given as Equation 13 does an adequate job of correlating local two-phase losses.

Figure 1: Martinelli-Nelson Average Two-Phase Friction Multiplier (Taken from Figure 11-16, Todreas and Kazimi)

230

Pressure Drop in a Two-Phase Channel Consider a boiling channel of constant cross sectional area. We assume the fluid enters the channel subcooled, with two phase flow beginning at some point Ho up the channel. The point Ho is called the non boiling height and is equivalent to the bubble departure point. The total steady-state pressure drop in the channel can be obtained by integrating the single and two-phase conservation equations up the channel. Mixture Mass Equation

( )∂∂z

GA GA mx x= ⇒ = =0 & constant (1)

which for a uniform area channel implies G is constant. Mixture Energy Equation

& ( )m hz

q z∂∂

= ′ (2)

which may be integrated to give for the enthalpy distribution up the channel

h z hm

q z dzz

( ) ( )&

( )= + ′ ′ ′∫0 1

0

(3)

Mixture Momentum Equation

1 12 2 2 22

g zG x G x P

z

P

Aggc g g

w w

x c

∂∂ α ρ α ρ

∂∂

τρϕ( )−

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪= − − −

l l

(4)

Integrate over the channel height

− =−

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+ +∫ ∫ ∫ ∫∂

∂∂∂ α ρ α ρ

τρϕP

z dz g zG x G x

dzP

A dzgg dz

H

c g g

Hw w

x

H

c

H

0

2 2 2 2

0 0 0

1 1 2( )

l l

(5)

and examine the integrals one at a time

Channel Pressure Drop

− = − − =∫ ∂∂Pz

dz P H P PH

channel0

0[ ( ) ( )] ∆ (6)

Acceleration Pressure Drop

1 12 2 2 2

0g z

G x G x dz Pc g g

H

acceleration∂∂ α ρ α ρ

( )−+

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪≡∫ l l

∆ (7)

The integral of the acceleration term is broken up into integrals over the boiling and non boiling heights, such that

231

∆Pg z

G x G x dzg z

G dzaccelerationc g gH

H

c

H

o

=−

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+

⎧⎨⎩

⎫⎬⎭∫ ∫1 1 12 2 2 2 2

0

0∂∂ α ρ α ρ

∂∂ ρ

( )

l l l

(8)

{ }

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=∆

inexitggc

indc

dexitggc

onaccelerati

xxgG

gGxx

gGP

υραρα

υυυραρα

222

2222

)1(

)1(

ll

llll

(9)

Friction and Forms Losses The integrals of the friction and forms loss terms are also broken up into integrals over the boiling and non boiling heights giving

τ τ τ

ϕ ϕ ϕ

w w

x

H

frictionw w

xH

Hw w

x

HP

A dz PP

A dzP

A dzo

o

0 2 2 0 1∫ ∫ ∫≡ = +∆ (10)

Single Phase Component

τ ρδ

ρw w

x

H

e cj

j

jc

HPA dz

fD

vg K z z

vg dz

o o

0

2 2

0 2 2∫ ∑∫= + −

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

( ) (11)

cHz

jce

oH

x

ww

gGK

gG

DHfdz

AP

oj

o

ρρτ

22

22

0∑∫∈

+≅ (12)

where the average fluid properties are over the non boiling height Ho . Two-Phase Component

τ ρ δ ρw w

xH

H

e co j

j

jcH

HPA

dz fD

vg

K z z vg

dzo o

∫ ∑∫= + −⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

22

2

2 2Φ Ψl ( ) (13)

Assuming the fluid properties are equal to the saturation properties and constant over the boiling height and letting H H HB o= −

jcfHz

j

H

Ho

cfe

fH

H x

ww

gGKdz

gG

Df

dzAP

Bjoo

Ψ+Φ≅ ∑∫∫∈

ρρτ

22

22

2

l (14)

232

fD

Gg

dzf H

DG

g Hdz

f HD

Gg

f

e f co

H

Hf B

e f c Bo

H

Hf B

e f co

o o

22

22

22

2 21

2ρ ρ ρΦ Φ Φl l l∫ ∫= = (15)

Note, the average two-phase multiplier in this development is the average over the boiling height and as such is not strictly equal to the Martinelli-Nelson Multiplier. However, the Martinelli-Nelson multiplier can be assumed a reasonable approximation. It can be shown, that if the quality varies linearly over the boiling height, the Martinelli-Nelson multiplier is equivalent to that obtained by averaging over the height.

cHz

jce

oj

cfHz

jocfe

Bffriction g

GKg

GDHf

gGK

gG

DHf

PojBj

ρρρρ 2222

2222

2

∑∑∈∈

++Ψ+Φ=∆ l (16)

If the inlet subcooling is small, the friction and forms losses may be approximated as

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+Ψ+⎟

⎠⎞⎜

⎝⎛ Φ+≅∆ ∑∑

∈∈ oB Hj

jj

Hj

jcf

oBocfe

ffriction KK

gGHH

gG

Df

Pρρ 22

22

2

l (17)

Elevation Losses

∆P gg

Helevc

core= ρ (18)

The elevation losses require knowledge of the void distribution and typically require numerical integration.

The total channel pressure drop is the sum of the acceleration, friction, local (forms) and the elevation terms, i.e. ∆ ∆ ∆ ∆ ∆P P P P Pchannel acceleration fricition local elev= + + + (19)

and requires knowledge of the boiling and non boiling heights. In a simple single channel analysis under the equilibrium model assumptions, the non boiling height is obtained directly from the energy balance

h hm

q z dzf

Ho

= + ′ ′ ′∫( )&

( )0 1

0

. (20)

Otherwise, the non boiling height is taken to be the bubble departure point and determined as described in earlier sections.

233

Example: A Boiling Water Reactor has operating characteristics given below. For the given data, determine the total core pressure drop. Assume the heat flux is uniform axially, and the void fraction varies linearly over the boiling height. You may also assume an equilibrium model for the flow quality. Core Averaged Heat Flux 144,032 Btu/hr-ft2 Active Fuel Height 150 inches Bundle Height 176 inches Rod Diameter 0.493 inches Rod Pitch 0.640 inches Mass Flux 1.42 x 106 lbm/hr-ft2 Core Inlet Temperature 532 F System Pressure 1035 psia Grid Loss Coefficient 1 Number of Grids 7 Core Entrance and Exit Loss Coefficient 3.5

SOLUTION

The total core pressure drop can be determined from the integrated two-phase momentum equation and is the sum of the acceleration, friction, local (forms) and the elevation terms, i.e.

∆ ∆ ∆ ∆ ∆P P P P Pchannel acceleration fricition local elev= + + + .

where:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=∆ in

exitggconaccelerati

xxgGP υ

ραρα

222 )1(

ll

(1)

⎟⎠⎞⎜

⎝⎛ Φ+≅∆ 2

2

2 oBocfe

ffriction HH

gG

Df

P lρ (2)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+Ψ≅∆ ∑∑

∈∈ oB Hj

jj

Hj

jcf

local KKg

GPρ2

2 (3)

∆P gg

Helevc

core= ρ (4)

Acceleration Pressure Drop To determine the acceleration drop, requires the quality and void fraction at the core exit. The inlet specific volume is given directly in terms of the core inlet temperature, i.e. υ υin F= =( ) .532 0 02123 ft /lbm3 . To determine the fluid conditions at the core exit requires the core exit enthalpy. This is obtained from the energy balance

h h q DHGAexit in

x= +

′′π (5)

The flow area for an arbitrary flow channel is given by

234

A S D

in

x = −

= −

=

2 2

2 2

2

4

0 640 0 493 4

0 219

π

π. ( )( . ) /

.

(6)

For the given data, the core exit enthalpy is

h

Btu lbm

exit = +×

=

526 8144 032 0 493 12 150 12

1 42 10 0 219 144

634 4

6.( , )( )( . )( )

( . )( . / )

. /

π

(7)

giving for the core exit quality

xh h

hexit

exit f

fg

=−

=−

=

634 4 547 85

643 750 1344

. .

..

(8)

The core exit void fraction can be obtained from the Zuber-Findlay correlation

αρρ

ρ=

+−⎡

⎣⎢

⎤

⎦⎥ +

1

1 1C xx

VGxo

g g gj

l

. (9)

For high pressure steam water flows Co = 1.13 (10a)

Vgg

gjc g

=−⎛

⎝⎜

⎞

⎠⎟141 2

14

.( )σ ρ ρ

ρl

l

(10b)

At 1035 psia, the drift velocity is

Vgg

ft

gjc g=

−⎛

⎝⎜

⎞

⎠⎟

=−⎛

⎝⎜

⎞

⎠⎟

=

141

141 0 0013 3217 46 05 2 32946 05

0575

2

2

2

14

14

.( )

. ( . )( . ) ( . . )( . )

. /sec

σ ρ ρ

ρl

l

(11)

giving for the core exit void fraction

235

αρρ

ρ=

+−⎡

⎣⎢

⎤

⎦⎥ +

=+

−⎡

⎣⎢

⎤

⎦⎥ +

××

=

1

1 1

1

113 1 2 32946 05

1 0134401344

2 329 0575 3600142 10 01344

0 656

6

C xx

VGxo

g g gj

l

( . ) ( . )( . )

..

( . )( . )( . )( . )

.

(12)

The acceleration pressure drop is then

∆P Gg

x x

lbf ft lbf in

accelerationc g g exit

in=−

+⎛

⎝⎜⎜

⎞

⎠⎟⎟ −

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

=××

−−

+ −⎧⎨⎩

⎫⎬⎭

= =

2 2 2

6 2

8

2 2

2 2

1

142 10417 10

1 013441 0 656 46 05

013440 656 2 329

0 02123

1832 127

( )

( . )( . )

( . )( . )( . )

( . )( . )( . )

.

. / . /

α ρ α ρυ

l l

(13)

Frictional Pressure Drop The frictional drop is in terms of the friction factor f which is a function of the Reynolds number, the non boiling height and the two-phase friction multiplier. A reasonable approximation for the friction factor in rod bundles is the friction factor in smooth tubes from the Moody Chart. The Reynolds number is

Re =GDe

µ. (14)

The equivalent diameter is

DA

P

S D

De

x

w

= =−4 4 42 2[ / ]π

π (15)

D ine = =4 0 219

0 4930 5656

( . )

( . ).

π (16)

giving for the Reynolds number

Re( . )( . / )

.,=

×=

1 42 10 0 5656 12

0 23290 997

6

. (17)

From the Moody Chart, f ≅ 0 0145. . The non boiling height is given by

( )

( )

Hm h h

q D

ft

of

=−

′′

=× −

=

& ( )

( . )( . / ) . .( , )( )( . / )

.

0

142 10 0 219 144 547 85 5268144 032 0 493 12

2 45

6

π

π (18)

Assuming the Martinelli-Nelson form of the two-phase multiplier, gives Φlo

2 35≅ . , such that the frictional pressure drop is

236

( )

22

8

26

22

/07.5/4.730

5.322.1245.2)1017.4)(05.46)(2(

)1042.1()12/5656.0(

)0145.0(

2

inlbfftlbf

HHg

GDf

P oBocfe

ffriction

==

×+×

×=

⎟⎠⎞⎜

⎝⎛ Φ+≅∆ lρ

(19)

Local Losses The local or forms losses are due to the grid spacers as well as the core inlet and exit losses. To be strictly correct, we should evaluate the two-phase multiplier at the specific locations of the grids. For this example, assume we can use an average two-phase multiplier, similar to the approach taken for the frictional losses.

( )

∆ ΨP Gg

K K

lbf ft lbf in

localf c

j

j H

j

j HB o

≅ +⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=×

×× × + × + +

= =

∈ ∈∑ ∑

2

6 2

8

2 2

2

142 102 4506 417 10

1 6 35 35 35 35 1

2 0355 141

ρ

( . )( )( . )( . )

. . . .

, . / . /

(20)

Elevation Pressure Drop The elevation pressure drop is the sum of the single-phase and two-phase terms. Assuming the void fraction varies linearly over the boiling height, and the subcooling is sufficiently small, the elevation pressure drop can be written as

( )( )

∆Pgg H

gg H

lbf ft lbf in

elev fc

o f g gc

B= + +

= + × + ×

= =

ρ α ρ α ρl

( . )( )( . ) . . . . ( )( . )

. / . /

45 06 1 2 45 0 672 4506 0 328 2 329 1 12 22

489 7 3 42 2

(21)

The total pressure drop is then

∆ ∆ ∆ ∆ ∆P P P P P

lbf in

core acc friction local elev= + + +

= + + +

=

127 5 07 14 1 3 41

2385 2

. . . .

. /

(22)

It should be noted, that even though the core mass flux is approximately half that in a Pressurized Water Reactor, the total core pressure drop is similar. This is due primarily to the enhanced friction and local losses in two-phase systems. In addition, the acceleration drop while not dominant, is still a significant contributor to the total loss. This is in contrast to the acceleration drop in single phase systems.