2 상유동 열전달 공학
Two-phase flow and heat transfer Engineering
2022년 1학기
서울대학교 원자핵공학과
조형규
Two-phase flows in our daily life
Two-phase flow in nuclear system
Two-phase flow
+ phase change heat transfer
Classification of Two-Phase Flow
Diabatic
Heating
Cooling Adiabatic
Flow direction Vertical
Downward Upward Horizontal
Inclined
Geometry
Tube Annulus
Bundle
Fin, twisted fin, etc.
External force
Reduced gravity Zero gravity
Motion condition
Classification of Two-Phase Flow Modelling
https://cfdflowengineering.com/basics-of-multi-phase-flow-and-its-cfd-modeling/
Basic Parameters
❖ Void fraction
✓ Volume fraction of gas
✓ In a local instantaneous analysis
▪ Delta function
✓ Averaged void fraction
▪ Time averaged
▪ Volume averaged
▪ Area averaged
✓ In one-dimensional approaches,
▪ Area averaging is necessary!
1
( )
( ) t v , ' '
t T X t dt
T
=
−x x
( )
1 when vapor is at point at time, 0 otherwise
v
X t t
= x x
1
( ) 1
N
i i
T t
=
=
x
Basic Parameters
❖ Void fraction
✓ Volume average
✓ Area-averaged void fraction
✓ Void fraction range
<< 𝛼(𝑡) >>= 1
𝑉ඵ
𝑉
𝛼(𝑋, 𝑡)𝑑𝑣 = 𝑉𝑔 𝑉𝑙 + 𝑉𝑔
< 𝛼(𝑡) >=
Δ𝑧 𝐴
𝑔𝑑𝐴
Δ𝑧 𝐴𝑑𝐴 = 𝐴𝑔 𝐴𝑙 + 𝐴𝑔
0 1
Basic Parameters
❖ Void fraction measurement
Isupport tube
Resistance Electrical Insulated
Sensing Tip not Insulated
Oscilloscope
200 400 600 800 1000
Voltage
Counts
Basic Parameters
❖ Void fraction measurement
✓ Wire mesh sensor
https://www.hzdr.de/db/Cms?pOid=55368&pNid=393
Basic Parameters
❖ Void fraction measurement
✓ Gamma ray or x-ray densitometer
Basic Parameters
❖ Void fraction measurement
✓ Ultra fast x-ray CT
Basic Parameters
❖ Void fraction measurement
✓ Etc.
https://www.mdpi.com/2311-5521/5/4/216/htm
Basic Parameters
❖ Velocities
✓ Actual velocity (phase velocity)
✓ Superficial velocity
𝑢
𝑙= 𝑄
𝑙𝐴
𝑙𝑢
𝑔= 𝑄
𝑔𝐴
𝑔𝑗
𝐺= 𝑄
𝐺𝐴 = 𝑄
𝐺/𝐴
𝑔(𝐴
𝑙+ 𝐴
𝑔)/𝐴
𝑔= 𝛼
𝑔𝑢
𝐺𝑗 = 𝑄
𝐴 = 𝑄
𝑔+ 𝑄
𝑙𝐴 = 𝑗
𝑔+ 𝑗
𝑙𝑗
𝑙=
𝑄𝑙𝐴
= 𝛼
𝑙𝑢
𝑙= 1 − 𝛼
𝑔𝑢
𝑙For two-phase flows, the phase velocities are larger than the corresponding volumetric fluxes of each phase.
Basic Parameters
❖ Velocities
✓ Relative velocity or slip velocity
✓ Homogeneous flow (?)
▪ No slip between two phases
𝑢
𝑅= 𝑢
𝑔− 𝑢
𝑙𝑢
𝑅= 0 𝑢
𝑅= 𝑢
𝑔− 𝑢
𝑙= 𝑗
𝑔𝛼
𝑔− 𝑗
𝑙(1 − 𝛼
𝑔) = 0 𝛼
𝑔= 1
𝑗
𝑔+ 𝑗
𝑙Basic Parameters
❖ Velocities
✓ Slip ratio
▪ For homogeneous flow, ?
▪ If not,
𝑆 = 𝑢
𝑔𝑢
𝑙𝑆 = 𝑢
𝑔𝑢
𝑙= 𝑚 ሶ
𝑔/𝜌
𝑔𝐴
𝑔ሶ
𝑚
𝑙/𝜌
𝑙𝐴
𝑙= 𝑥 1 − 𝑥
𝜌
𝑙𝜌
𝑔1 − 𝛼
𝑔𝛼
𝑔Basic Parameters
❖ Quality
✓ Thermodynamic equilibrium quality: 𝑥𝑒
✓ Flow quality: 𝑥
✓ Static quality: 𝑥𝑠
𝑥
𝑒= ℎ − ℎ
𝑓ℎ
𝑓𝑔𝑥 = 𝑚 ሶ
𝑔ሶ
𝑚
𝑔+ ሶ 𝑚
𝑙= 𝜌
𝑔𝑢
𝑔𝐴
𝑔𝜌
𝑔𝑢
𝑔𝐴
𝑔+ 𝜌
𝑙𝑢
𝑙𝐴
𝑙𝑥
𝑠= 𝑚
𝑔𝑚
𝑔+ 𝑚
𝑙= 𝜌
𝑔𝐴
𝑔𝜌
𝑔𝐴
𝑔+ 𝜌
𝑙𝐴
𝑙Volumetric quality
𝛽 = 𝑄
𝑔𝑄
𝑔+ 𝑄
𝑙= 𝑗
𝑔𝐴
𝑗𝐴 = 𝑗
𝑔𝑗
Basic Parameters
❖ Quality
✓ Flow quality vs. static quality
▪ For homogeneous flow
𝑥 = 𝑚 ሶ
𝑔ሶ
𝑚
𝑔+ ሶ 𝑚
𝑙= 𝜌
𝑔𝑢
𝑔𝐴
𝑔𝜌
𝑔𝑢
𝑔𝐴
𝑔+ 𝜌
𝑙𝑢
𝑙𝐴
𝑙𝑥
𝑠= 𝑚
𝑔𝑚
𝑔+ 𝑚
𝑙= 𝜌
𝑔𝐴
𝑔𝜌
𝑔𝐴
𝑔+ 𝜌
𝑙𝐴
𝑙𝑥
1 − 𝑥 = 𝜌
𝑔𝑢
𝑔𝐴
𝑔𝜌
𝑙𝑢
𝑙𝐴
𝑙= 𝑢
𝑔𝑢
𝑙𝑥
𝑠1 − 𝑥
𝑠𝑥 = 𝑥
𝑠Basic Parameters
❖ Area-averaging Operator : Zuber notation
✓ Averaged properties
✓ Phasic average
=
AdA
A
1
= =
=
AG A G G G
G G
G
dA
dA A
A
G
1 1
( )
= − − = − −
=
AL A L L L
L L
L
dA
dA A
A
L
1
) 1
) ( 1
1 ( 1 1
A A
G/
=
Basic Parameters
❖ Who is Novak Zuber?
✓ “Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it doesn’t bother you anymore.”
✓ Drift flux model
✓ Uncertainty of the large-break loss-ofcoolant-accident (LOCA) predictions were made in the TRAC code
✓ Developing something called phenomena identification and ranking tables (PIRT) to guide the process
✓ A general scaling approach he called fractional scaling analysis (FSA)
✓ Check the supplement
Basic Parameters
❖ Averaged flow parameters
✓ Mass flow rate
ሶ
𝑚𝑔 = 𝜌𝑔 < 𝑢𝑔 >𝑔 𝐴𝑔 = 𝜌𝑔 < 𝑢𝑔 >𝑔< 𝛼 > 𝐴 = 𝐺 < 𝑥 > 𝐴
ሶ
𝑚𝑙 = 𝜌𝑙 < 𝑢𝑙 >𝑙 𝐴𝑙 = 𝜌𝑙 < 𝑢𝑙 >𝑙 1 −< 𝛼 > 𝐴 = 𝐺 1 −< 𝑥 > 𝐴
ሶ
𝑚 = ሶ𝑚𝑔 + ሶ𝑚𝑙 = 𝐺 < 𝑥 > 𝐴 + 𝐺 1 −< 𝑥 > 𝐴 = 𝐺𝐴
𝐺 = 𝑚ሶ
𝐴 = 𝑚ሶ𝑔
< 𝑥 > 𝐴 = 𝜌𝑔 < 𝑢𝑔 >𝑔< 𝛼 >
< 𝑥 > = 𝜌𝑙 < 𝑢𝑙 >𝑙 1 −< 𝛼 >
1 −< 𝑥 >
𝐺𝑔 = 𝑚ሶ 𝑔
𝐴 𝐺𝑙 = 𝑚ሶ 𝑙
𝐴 𝐺 = 𝐺𝑔 + 𝐺𝑙
< 𝑢𝑔 >𝑔= 𝐺 < 𝑥 >
𝜌𝑔 < 𝛼 > < 𝑢𝑙 >𝑙= 𝐺(1 −< 𝑥 >) 𝜌𝑙 1 −< 𝛼 >
Basic Parameters
❖ Void-quality relation
✓ For homogeneous flow or for the static quality
𝑆 = 𝑢
𝑔𝑢
𝑙= 𝑚 ሶ
𝑔/𝜌
𝑔𝐴
𝑔ሶ
𝑚
𝑙/𝜌
𝑙𝐴
𝑙= 𝑥 1 − 𝑥
𝜌
𝑙𝜌
𝑔1 − 𝛼
𝑔𝛼
𝑔< 𝑥 >
1 −< 𝑥 > = 𝜌𝑙 𝜌𝑔
1 −< 𝛼𝑔 >
< 𝛼𝑔> 𝑆 < 𝛼𝑔> = < 𝑥 >
< 𝑥 > +𝑆 𝜌𝑔
𝜌𝑙 (1 −< 𝑥 >)
< 𝛼𝑔> = < 𝑥 >
< 𝑥 > 𝜌𝑙 + 𝜌𝑔(1 −< 𝑥 >)
Basic Parameters
❖ Two-phase density
✓ Mixture density and specific volume
< ҧ𝜌 >= (1 −< 𝛼 >)𝜌𝑙+< 𝛼 > 𝜌𝑔
< ҧ𝑣 >= 1 −< 𝑥 > 𝑣𝑙+< 𝑥 > 𝑣𝑔
Basic Parameters
❖ Interfacial Area Concentration (IAC)
✓ Importance of IAC
▪ (Interfacial Transfer Terms) ~ (IAC) X (Driving Force)
✓ First Order Importance to Interfacial Transfer Terms
✓ Driving Force: Local Transport Mechanism
» Potential Related to the Momentum & Energy Transport
▪ Significantly Affect the Results of Two-Phase Flow Analysis
❖ IAC models
✓ Correlation bases on static flow regime map
✓ Dynamic Flow Regime Map Based on Transport Equation
𝐼𝐴𝐶 = 𝐼𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑖𝑎𝑙 𝑎𝑟𝑒𝑎
𝑀𝑖𝑥𝑡𝑢𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 [1/𝑚]
