kobe university repository : kernelthe objectives of present study are six-fold. first is to measure...

Kobe University Repository : Kernel

タイトルTit le

Conduct ive and viscous sub-layers on forced convect ion andmechanism of crit ical heat flux during flow boiling of subcooled water ina circular tube at high liquid Reynolds number

著者Author(s) Hata, K. / Liu, Q. S. / Masuzaki, S.

掲載誌・巻号・ページCitat ion Heat and Mass Transfer,55(1):175-195

刊行日Issue date 2019-01

資源タイプResource Type Journal Art icle / 学術雑誌論文

版区分Resource Version author

権利Rights

© Springer-Verlag GmbH Germany, part of Springer Nature 2018. Thisis a post-peer-review, pre-copyedit version of an art icle published inHeat and Mass Transfer. The final authent icated version is availableonline at : ht tps://doi.org/10.1007/s00231-018-2458-4

DOI 10.1007/s00231-018-2458-4

JaLCDOI

URL http://www.lib.kobe-u.ac.jp/handle_kernel/90005760

PDF issue: 2021-08-16

Conductive and viscous sub-layers on forced convection and Mechanism of critical

heat flux during flow boiling of subcooled water in a circular tube at high liquid

Reynolds number

Hata K and

Liu Q S

Graduate School of Maritime Sciences

Kobe University

5-1-1, Fukaeminami, Higashinada, Kobe 658-0022, Japan

[email protected]; [email protected]

Masuzaki S

National Institute for Fusion Science

322-6 Oroshi-cho, Toki, Gifu 509-5292, Japan

[email protected]

ABSTRACT

The turbulent heat transfer, the subcooled boiling heat transfer and the steady state CHF for a Pt-circular test

tube of a 3 mm inner diameter and a 100 mm heated length are measured with a wide range of inlet subcooling

and flow velocity at high liquid Reynolds number, i.e. Red=3.01104 to 1.4310

5. The inner surface temperature

of the Pt-circular test tube calculated by the steady one-dimensional heat conduction equation is compared with

Corresponding author.

mailto:[email protected]

mailto:[email protected]

2

the values derived from authors’ turbulent heat transfer correlation and with the numerical solutions of the

RANS equations (Reynolds Averaged Navier-Stokes Simulation) of k- turbulence model for the flow velocities

ranging from 4 to 21 m/s. The thicknesses of conductive sub-layer from non-boiling regime to CHF are

measured by numerically analyzing the heat transfers with conductive sub-layer on forced convection and with

thinner one dissipated by the evaporation on nucleate boiling. The thicknesses of viscous sub-layer on forced

convection are estimated from the thicknesses of the conductive sub-layer and Prandtl numbers of the surface

temperature on the heated surface. Furthermore, the thicknesses of conductive sub-layer at the CHF point are

extrapolated from the measured values at various flow velocities. The experimental values of the CHF are also

compared with authors’ widely and precisely predictable correlations of critical heat flux during flow boiling of

subcooled water and the corresponding theoretical values of the liquid sub-layer dry-out models suggested by

other researchers, respectively. The authors’ correlations and other researchers’ theoretical values can represent

the subcooled boiling CHFs obtained in this study within the ranges of -13.27 to 6.76% difference and -32.51 to

13.16 % one, respectively. A suggestion based on the experimental data as to what the dominant mechanism is

for critical heat flux during flow boiling of subcooled water on a vertical circular tube is confirmed again at high

liquid Reynolds number. The transitions to film boiling at the subcooled water flow boiling on the Pt test tube of

d=3 mm and L=100 mm would occur due to the liquid sub-layer dry-out model at the steady-state CHF as well

as those on the Pt test tube of d=3 mm and L=66.5 mm, but not due to the heterogeneous spontaneous nucleation

and the hydro-dynamic instability.

KEYWORDS

Conductive and Viscous Sub-layers, Prandtl Number, Mechanism of Critical Heat Flux, Circular Tube, High

Liquid Reynolds Number, Liquid Sub-layer Dry-out Model

3

1. INTRODUCTION

The knowledge of subcooled boiling heat transfer at high liquid Reynolds number is important to discuss the

conductive and viscous sub-layers and the mechanisms of subcooled flow boiling critical heat flux (CHF) in a

vertical circular tube. Many researchers have experimentally studied the steady state CHFs, qcr,sub,st, uniformly

heated on the normal tubes by a steadily increasing current for high liquid Reynolds number and given the

correlations for calculating CHFs on the normal tubes. It has been clarified that the qcr,sub,st against outlet

subcooling, Tsub,out, for Tsub,out30 K are almost proportional to d-0.4

and u0.4

for fixed Tsub,out and L/d, to

(Tsub,out)0.7

for a fixed L/d and to (L/d)-0.1

for a fixed Tsub,out. And, the qcr,sub,st are also proportional to u0.55

for the

experimental data at u>13.3 m/s [1-8]. The authors have given the following steady-state CHF correlations

against outlet and inlet subcoolings based on the effects of test tube inner diameter (d), flow velocity (u), outlet

and inlet subcoolings (Tsub,out and Tsub,in) and ratio of heated length to inner diameter (L/d) on CHF [6].

Outlet subcooling:

70

10

30100820 .

.

..cr Sc

d

LWe*D.Bo

for Tsub,out30 K and u13.3 m/s [1, 4] (1)

70

10

2250175004180 .

.

..cr Sc

d

LWe*D.Bo

for Tsub,out30 K and u>13.3 m/s [6] (2)

Inlet subcooling:

340

2

10

30101

C.

dReC

)d/L(.

..cr *Sce

d

LWe*DCBo

for Tsub,in40 K and u13.3 m/s [2, 3, 4](3)

650

5

10

225017504

C.

dReC

)d/L(.

..cr *Sce

d

LWe*DCBo

for Tsub,in40 K and u>13.3 m/s [6] (4)

where C1=0.082, C2=0.53 and C3=0.7 for L/daround 40 [2] and C1=0.092, C2=0.85 and C3=0.9 for L/daround

40 [3]. C4=0.0418, C5=0.144 and C6=0.7 for L/daround 40 [6] and C4=0.0469, C5=0.231 and C6=0.9 for

L/daround 40 [6]. Bocr, D*, We, Sc and Sc* are boiling number (=qcr,sub/Ghfg), non-dimensional diameter

[D*=d/{/g/(l-g)}0.5

], Weber number (=G2d/l), non-dimensional outlet subcooling (=cplTsub,out/hfg) and non-

dimensional inlet subcooling (Sc*=cplTsub,in/hfg), respectively [1-8]. It has been assumed that flow velocity will

4

affect the incipient boiling superheat and the nucleate boiling heat transfer up to the CHF. Incipient boiling

superheat may shift to a very high value at higher flow velocity and a direct transition to film boiling or a trend

of a decrease in CHF with an increase in the flow velocity may occur due to the heterogeneous spontaneous

nucleation [7-9] but not due to the hydro-dynamic instability [10, 11] or the liquid sublayer dry-out model [12-

14]. The accurate measurement for the subcooled boiling heat transfer up to the CHF is necessary to clarify a

change in the mechanism of CHF.

The objectives of present study are six-fold. First is to measure the turbulent heat transfer, the subcooled boiling

heat transfer and the steady state CHFs for a Pt-circular test tube of a 3 mm inner diameter and a 100 mm heated

length with a wide range of inlet subcoolings (Tsub,in) and flow velocities (u) at high liquid Reynolds number.

Second is to compare the inner surface temperature of the Pt-circular test tube calculated by the steady one-

dimensional heat conduction equation with the values derived from authors’ turbulent heat transfer correlation

and the numerical solutions of the RANS equations of k- turbulence model for the flow velocities ranging from

4 to 21 m/s. Third is to measure the thicknesses of conductive sub-layer from non-boiling regime to CHF by

numerically analyzing the heat transfers with conductive sub-layer on forced convection and with thinner one

dissipated by the evaporation on nucleate boiling. Fourth is to estimate the thicknesses of viscous sub-layer on

forced convection from the thicknesses of the conductive sub-layer and Prandtl numbers of the surface

temperature on the heated surface. Fifth is to extrapolate the thicknesses of conductive sub-layer at the CHF

point from the measured values at various flow velocities. Sixth is to confirm again a suggestion based on the

experimental data as to what the dominant mechanism is for critical heat flux during flow boiling of subcooled

water on a vertical circular tube at high liquid Reynolds number.

2. EXPERIMENTAL APPARATUS AND METHOD

The schematic diagram of the experimental setup comprised of a water loop and a pressurizer is shown in Fig. 1.

The loop is made of SUS304 stainless steel and is capable of working up to 2 MPa. The loop has five test

5

sections whose inner diameters are 2, 3, 6, 9 and 12 mm. Test sections were vertically oriented with water

flowing upward. The test section of the inner diameter of 3 mm was used in this work.

The cross-sectional view of 3-mm inner diameter test section used in this work is shown in Fig. 2. The platinum

(Pt) test tube for the test tube inner diameter, d, of 3 mm, the heated length, L, of 100 mm with the commercial

finish of inner surface (CF) was used in this work. Wall thickness of the test tube, , was 0.5 mm. The silver-

coated 5-mm thickness copper-electrode-plates to supply heating current were soldered to the surfaces of the

both ends of the test tube. The both ends of test tube were electrically isolated from the loop by Bakelite plates of

14-mm thickness. The inner surface condition of the test tube was observed by the scanning electron microscope

(SEM) photograph and inner surface roughness was measured by the surface texture measuring instrument

(SURFCOM 120A) of Tokyo Seimitsu Co., Ltd., (Mitaka, Tokyo, Japan). Figure 3 shows the result of SEM

photograph of the inner surface of platinum (Pt) test tube of d=3 mm with the commercial finish of inner surface.

The inner surface roughness is measured 0.40 m for average roughness, Ra, 2.20 m for maximum roughness

depth, Rmax, and 1. 50 m for mean roughness depth, Rz.

The platinum (Pt) test tube has been heated with an exponentially increasing heat input supplied from a direct

current source (Takasago Ltd., Kawasaki, Kanagawa, Japan, NL035-500R, DC 35 V-3000 A) through the two

copper electrodes shown in Fig. 4. Heat transfer processes caused by exponentially increasing heat inputs, Q0

exp(t/), were measured for the platinum test tube. The exponential periods, , of the heat input ranged from 7.41

to 26.98 s. At the CHF, the test tube average temperature rapidly increases. The current for the heat input to the

test tube was automatically cut off when the measured average temperature increased up to the preset

temperature, which was several degrees (in tens) of Kelvin higher than corresponding CHF surface temperature.

This procedure avoided actual burnout of the test tube.

The average temperature, T , of the Pt test tube shown in Fig. 4 was measured with resistance thermometry

participating as a branch of a double bridge circuit for the temperature measurement. The output voltages from

6

the bridge circuit, VT, together with the voltage drop across the potential taps of the test tube and across a

standard resistance, VI=IRs, were amplified and then were sent via an analog-digital (A/D) converter to a digital

computer. The unbalance voltage, VT, is expressed by means of Ohm’s law as the following form.

32

312

RR

RRRRIV T

T

(5)

These voltages were simultaneously sampled at a constant interval ranging from 60 to 200 ms. The average

temperatures of the Pt test tube between the potential taps was calculated with the aid of previously calibrated

resistance-temperature relation, RT=a(1+bT +cT 2). The heat generation rates of the Pt test tube between the

potential taps, Q=I2RT, was calculated from the measured voltage difference between the potential taps of the Pt

test tube, VR, and that across the standard resistance, VI. The surface heat flux between the potential taps, q, is the

differences between the heat generation rate per unit surface area, Q, and the rate of change of energy storage in

the Pt test tube obtained from the faired average temperature versus time curve as follows:

dt

TdcQ

S

Vq (6)

where , c, V and S are the density, the specific heat, the volume and the inner surface area of the Pt test tube,

respectively.

The heater inner surface temperatures between the potential taps, Ts, was also obtained by solving the steady

one-dimensional heat conduction equation in the test tube under the conditions of measured average temperature,

T , and surface heat flux of the test tube, q. The solutions for the inner and outer surface temperatures of the test

tube, Ts and Tso, are given by the steady one-dimensional heat conduction equation. The basic equation for the

test tube is as follows:

01

2

2

Q

dr

dT

rdr

Td (7)

then integration yields and the mean temperature of the test tube is obtained.

CrlnQrQr

rT o 24

22

(8)

7

or

irio

drrrTrr

T

21

22 (9)

Generating heat in the tube is equal to the heat conduction and the test tube is perfectly insulated.

i

io

irr r

Qrr

dr

dTq

2

22

(10)

0 orrdr

dT (11)

The temperatures of the heater inner and outer surfaces, Ts and Tso, and C in Eq. (8) can be described as follows:

ioi

io

iioiiooo

io

iis rlnrr

)rr(

qrrrrlnrrlnrr

)rr(

qrT)r(TT 22

22

44222

2222

22

1

2

14

4

(12)

o

io

oiioiiooo

io

ioso rln

)rr(

rqrrrrlnrrlnrr

)rr(

qrT)r(TT 21

22

1

2

14

4 22

244222

222

(13)

44222

222 2

1

2

14

4ioiiooo

io

i rrrlnrrlnrr)rr(

qrTC

(14)

where T , q, , ri and ro are average temperature of the test tube, heat flux, thermal conductivity, test tube inner

radius and test tube outer radius, respectively.

In case of the 3-mm inner diameter test sections, before entering the test tube, the test water flows through the

tube with the same inner diameter of the Pt test tube to form the fully developed velocity profile. The entrance

tube lengths, Le, are given 240 mm (Le/d=80). The values of Le/d for the test tube of d=3 mm at which the center

line velocity reaches 99 % of the maximum value for turbulence flow were obtained ranging from 9.8 to 21.9

(3.01104Red1.4310

5) by the correlation of Brodkey and Hershey [15] as follows:

416930 /d

e Re.d

L (15)

The inlet and outlet liquid temperatures, Tin and Tout, were measured by 1-mm o.d., sheathed, K-type

thermocouples (Nimblox, sheath material: SUS316, hot junction: ground, response time (63.2 %): 46.5 ms)

which are located at the centerline of the tube at the upper and lower stream points of 262 and 53 mm from the

tube inlet and outlet points for the 3-mm inner diameter test section. The inlet and outlet pressures, Pipt and Popt,

were measured by the strain gauge transducers (Kyowa Electronic Instruments Co., Ltd., PHS-20A, Natural

8

frequency: approximately 30 kHz), which were located near the entrance of conduit at upper and lower stream

points of 53 mm from the tube inlet and outlet points for d=3 mm inner diameter test section. The thermocouples

and the transducers were installed in the conduits as shown in Fig. 2.

The inlet and outlet pressures, Pin and Pout, for the 3-mm inner diameter test section were calculated from the

pressures measured by inlet and outlet pressure transducers, Pipt and Popt, as follows:

optipt

ipt

wnhoptwnhiptiptinLLL

LPPPP

(16)

opt

optininoutLL

LPPPP

(17)

where Lipt=0.053 m and Lopt=0.053 m for the 3-mm inner diameter one. Experimental errors are estimated to be

1 K in inner tube surface temperature and 2 % in heat flux. Mass velocity, inlet and outlet subcoolings, inlet

and outlet pressures and exponential period were measured within the accuracy 2 , 1 K, 4 kPa and 2 %,

respectively.

3. NUMERICAL SOLUTION OF TURBULENT HEAT TRANSFER

3.1. RANS Equations for k- Turbulence Model with High Reynolds Number Form

The RANS equations for k- turbulence model [16] in a circular tube of 3 mm in diameter and 559 mm long

were numerically solved for heating of water with heated section of 3 mm in diameter and 100 mm long by using

PHOENICS code under the same conditions as the experimental ones and with temperature dependent thermo-

physical fluid properties [17]. The unsteady RANS equations for turbulent heat transfer are solved in the three

dimensional coordinate shown in Fig. 5 as follows [18].

(Continuity Equation) Cylindrical coordinates (r, , z):

9

011

)u(

z)u(

r)ru(

rrtzr

(18)

(Momentum Equation) Cylindrical coordinates (r, , z)

r-component:

rrzrrrr grzzrrrrr

rrrr

Puu

zr

uuu

ruu

ru

t

1112

(19)

-component:

g

zzrrr

rr

P

ruu

zr

uuuu

ruu

ru

tz

rr

1111 2

2 (20)

z-component:

zzzzzrz gzzzzrrz

rrrz

Puu

zuu

ruu

ru

t

111 (21)

(Energy Equation) Cylindrical coordinates (r, , z):

Qz

Tc

z

Tc

rr

Trc

rr

)hu(z

)hu(r

)hu(rt

h

t

tp

t

tp

t

tp

zr

2

11

1

(22)

(Transport Equation for k) Cylindrical coordinates (r, , z):

)P(z

k

z

k

rr

kr

rr

)ku(z

)ku(r

)ku(rt

k

bk

k

t

k

t

k

t

zr

2

11

1

(23)

(Transport Equation for ) Cylindrical coordinates (r, , z):

)CCPC(kzzrr

rrr

)u(z

)u(r

)u(rt

ebekettt

zr

2312

11

1

(24)

where r

u

rrr

2 (25)

r

uu

r

r

1

2 (26)

z

u

zzz

2 (27)

10

ru

rr

u

rr

rr

1 (28)

zu

rz

u

zz

1 (29)

z

u

r

u

rzzrrz (30)

0rg , 0g , ggz (31)

Tch p (32)

2kCt (33)

222222111

2z

u

r

uu

rz

u

r

u

r

uu

rz

u

r

uu

rr

uP rzzrzrr

tk

(34)

zg

rg

rg zr

t

tb

1 (35)

te (36)

t

tp

t

c

(37)

t

(38)

ur, u and uz are the r, and z components of a velocity vector, respectively. The constants, k, , t, C1e, C2e,

C3e and C, appearing in the unsteady RANS equations for turbulent heat transfer, Eqs. (22) to (24), (33), (35)

and (37), take the values given in Table 1 [18].

3.2. Boundary Conditions

The fundamental equations are numerically analyzed together with the following boundary conditions. On the

outer boundary of heated section: constant heat flux, and non-slip condition.

ttanconsr

Tq

(39)

11

At the outer boundary of non-heated section:

0

r

T (40)

At the lower boundary:

uuandu,u,TT zrin 00 for in-flow (41)

where Tin and u are a inlet liquid temperature and a flow velocity at the entrance of the test section.

3.3. Method of solution

The control volume discretization equations were derived from these fundamental equations by using the hybrid

scheme [19]. The thermo-physical properties for each control volume are given by each control volume

temperature numerically analyzed. The procedure for the calculation of the flow field is the SIMPLE algorithm

which stands for Semi-Implicit Method for Pressure-Linked Equations [20]. A uniform heat flux, q, was

prescribed at the heated pipe wall for the range of 5.71105 to 4.4710

7 W/m

2 as a boundary condition, and

numerical calculation was continued until the steady-state was obtained. The surface temperature on the test

tube, Ts, was analyzed from the calculated temperature of the first control volume on the test tube surface, TEM,

which is located on the center of the control volume, by solving the heat conduction equation in liquid as follows

[21-26].

TEM)r(q

T out

l

s

2

(42)

where, (r)out is the first control volume width on the r-component. In Fig. 6, the test tube surface is located at

r=-1.5 mm and the conductive sub-layer [21-26] exists on the test tube surface. The liquid temperatures in the

conductive sub-layer on the test tube surface will become linearly lower with a decrease in the radius by the heat

conduction from the surface temperature on the test tube, Tf=Ts-rq/l. And let those, Tf, equal the calculated

liquid temperature of the outer control volume on the test tube surface, TEM, in the turbulent flow region, which

is located on the center of the control volume as given in Eq. (42). Half the outer control volume width for the r-

component, (r)out/2, would become the thickness of the conductive sub-layer, CSL, for the turbulent heat

12

transfer in a circular tube under two-phase model classified into conductive sub-layer and inner region of the

turbulent flow [21-26]. Average heat transfer coefficient on the test tube surface was obtained by Ts,av averaging

the calculated local surface temperatures, (Ts)z, at every 0.5 mm in the heated length, L. The flow and

temperature field predictions were obtained using the PHOENICS CFD code [17].

4. RESULTS AND DISCUSSION

4.1. Experimental Conditions and Parameters used for Calculation

Steady-state heat transfer processes on the Pt test tube of 3 mm inner diameter that caused by the exponentially

increasing heat inputs, Q0exp(t/), were measured. The exponential periods, , of the heat input ranged from 7.41

to 26.98 s. The initial experimental conditions such as inlet flow velocity, inlet liquid temperature, inlet pressure

and exponential period for the CHF experiment were determined independent of each other before each

experimental run.

The experimental conditions were as follows:

Test Tube Number THD-F174

Heater material platinum

Surface condition Commercial finish of inner surface (CF)

Surface roughness 0.40 m for Ra, 2.20 m for Rmax and 1.50 m for Rz

Inner diameter (d) 3 mm

Heated length (L) 100 mm

L/d 33.33

Wall thickness () 0.5 mm

Inlet flow velocity (u) 4.34, 7.71, 10.50, 14.12, 17.83 and 21.79 m/s on forced convection

3.70, 7.07, 9.84, 13.52, 17.30 and 21.30 m/s at CHF

13

Liquid Reynolds numbers (Red) 3.01104 to 1.4110

5

Inlet pressure (Pin) 913.00 to 1025.46 kPa

Outlet pressure (Pout) 830.67 to 865.87 kPa

Inlet subcooling (Tsub,in) 140.04 to 142.26 K

Outlet subcooling (Tsub,out) 53.40 to 85.83 K

Inlet liquid temperature (Tin) 307.09 to 312.08 K

Exponentially increasing heat input (Q) Q0exp(t/), =7.41 to 26.98 s

The parameters used for calculation were as follows:

Inner diameter (d) 3 mm

Heated length (L) 100 mm

Entrance length (Le) 282 mm

Exit length (Lex) 177 mm

Test section length (Lts) 559 mm

Heat flux (q) 5.71105 to 4.4710

7 W/m

2 (q0

exp(t/), =24.29 to 26.98 s)

Inlet flow velocity (u) 4.34, 7.71, 10.50, 14.12, 17.83 and 21.79 m/s on forced convection

3.70, 7.07, 9.84, 13.52, 17.30 and 21.30 m/s at CHF

Liquid Reynolds numbers (Red) 3.01104 to 1.4310

5

Inlet liquid temperature (Tin) 307.09 to 312.08 K

Coordinate system cylindrical coordinate (r, , z)

Control volume number (17 to 44, 60, 962)

Physical model k- turbulence model with high Reynolds number form

Wall functions logarithmic law (9.132y+16.05)

4.2. Steady State Heat Transfer Characteristics

14

4.2.1. Platinum test tube of 3 mm inner diameter and 100 mm heated length

Figure 7 has been shown typical examples of the heat transfer curves for the exponential period, , of around 26

s on the Platinum test tube of d=3 mm and L=100 mm at the inlet liquid temperature, Tin, of around 309.26 K and

the flow velocities, u, of 4 to 21 m/s. At a fixed flow velocity, the heat flux gradually becomes higher with an

increase in Tsat (=T s-Tsat) on the non-boiling forced convection curve derived from our correlation, Eq. (43),

[27] up to the point where the slope begins to increase with heat flux following the onset of nucleate boiling.

140080

40850020

.

w

.

..dd

d

LPrRe.Nu

(43)

All properties in the equation are evaluated at the average bulk liquid temperature, TL, [=(Tin+(Tout)cal)/2], except

w, which is evaluated at the heater inner surface temperature. And the heat flux increases up to the CHF where

the heater surface temperature rapidly jumps from the nucleate boiling heat transfer regime to the film boiling

one. The CHF and its superheat become higher with an increase in flow velocity. The nucleate boiling curves in

higher heat flux range for each flow velocity agree with each other forming a single straight line on the log q

versus log Tsat graph. The analytical solution of incipient boiling superheat given by Sato and Matsumura [28]

is shown in the figure for comparison. The solution was derived based on the initiation model of bubble growth.

lgsat

satsfgl

vvT

TThq

8

2

(44)

The thermo-physical properties are defined at the saturated temperature, Tsat. The experimental data of the

incipient boiling superheat for each flow velocity almost agree with the values predicted by Eq. (44). The

equation of incipient boiling superheat given by Bergles and Rohsenow [29] is also shown in the figure for

comparison.

023404630

156110825560

.P.

.ONBsatP

q.)T(

(45)

where q is the surface heat flux in W/m2, P is the system pressure in bar and Tsat is in K. The values of

(Tsat)ONB calculated from Eq. (45) are in good agreement with the experimental data of the incipient boiling

superheat for each flow velocity. The fully developed subcooled boiling curve for Platinum test tube with a

15

commercial finish of inner surface can be expressed by the following empirical correlation.

3760 satn

sat TTCq (46)

where C and n are coefficient and exponent, and equivalent to 760 and 3 respectively. The correlation can almost

describe the fully developed subcooled boiling curves for the Platinum test tube of d=3 mm and L=100 mm with

the commercial finish of inner surface at the outlet pressure of around 800 kPa obtained in this work within 15 %

difference under the wide range of flow velocities. The corresponding curve derived from the correlation, Eq.

(47) with Csf=0.012, Eqs. (48) to (51), for fully developed subcooled boiling given by Rohsenow [30], McAdams

et al. [31], Jens and Lottes [32] and Thom et al. [33] are also shown in Fig. 7 for comparison.

Rohsenow [30]

71330 .

l

lpl

.

glfgl

sf

fg

satpl c

)(gh

qC

h

Tc

(47)

where the various fluid properties are evaluated at the saturation temperature corresponding to the local pressure

and Csf is a function of the particular heating surface-fluid combination.

McAdams et al. [31]:

25906222 .sat q.T for dissolved gas concentration of 0.3 ml of air per liter of water (48)

25909228 .sat q.T for dissolved gas concentration of 0.06 ml of air per liter of water (49)

where Tsat is in K and q is in MW/m2.

Jens and Lottes [32]:

26250790 ./P.sat eq.T (50)

where P is the absolute pressure in MPa, Tsat is in K and q is in W/m2.

Thom et al. [33]:

68500220 ./P.sat eq.T (51)

where P is the absolute pressure in MPa, Tsat is in K and q is in W/m2. The values of n given by McAdams et al.

correlation and Jens and Lottes one are about 30 % larger than that of our correlation, although that of Thom et

al. correlation is about 30 % smaller. The values calculated from our correlation are in good agreement with the

16

corresponding values for Rohsenow correlation, Eq. (47), with Csf=0.012 on the log q versus log Tsat graph. The

values of the lower limit of the heterogeneous spontaneous nucleation temperature, THET, [9] and the

homogeneous spontaneous nucleation temperature, TH, [34] at the pressure of 800 kPa are shown in the figure

for comparison. The inner surface temperature of the test tube at CHF with the flow velocity, u, of 21 m/s is

46.92 and 101.64 K lower than the THET and the TH, respectively.

4.2.2. Thickness of conductive sub-layer on forced convection, CSL

The numerical solutions of the theoretical equations for k- turbulence model with high Reynolds number form

are in good agreement with the experimental data and the values derived from Eq. (43) within -15 % differences

as shown in Fig. 7. The thicknesses of the conductive sub-layer, CSL, for the turbulent heat transfer on the Pt test

tube of d=3 mm and L=100 mm in Fig 7 are 16.38, 9.32, 7.56, 5.71, 4.25 and 3.48 m at flow velocities, u, of

4.0, 6.9, 9.9, 13.3, 17 and 21 m/s, respectively. These thicknesses of the conduction sub-layer do not depend on

the heat flux and is a constant value 16.38, 9.32, 7.56, 5.71, 4.25 and 3.48 m in the wide Tsat region.

Relationships between CSL numerically solved for steady-state turbulent heat transfer, and u for d=3 mm, L=100

mm, and L/d= 33.33 with Tin of approximately 309.26 K are listed in Table 2 and shown in Fig. 8. These

numerical solutions of CSL for steady-state turbulent heat transfer can be expressed by the following equation

determined by the least squares method of power-law:

0 947966 752 .CSL . u (52)

4.2.3. Thickness of viscous sub-layer on forced convection, VSL

4.2.3.1. In case of u=4.34 m/s

The typical example of the steady-state turbulent heat transfer curve for Platinum circular tube, d, of 3 mm,

heated length, L, of 100 mm and heated length-to-inner diameter ratios, L/d, of 33.3 with the exponential period,

17

, of around 24.29 s at the flow velocity, u, of 4.34 m/s are re-plotted versus the temperature difference between

average inner surface temperature and liquid bulk mean temperature, TL (=Ts,av-TL), in Fig. 9. The numerical

solutions for the relation between the heat flux, q, and the temperature difference between average inner surface

temperature and liquid bulk mean temperature, TL, are shown at the flow velocity of 4.34 m/s as green solid

circles. The thicknesses of the conductive sub-layer, CSL, for the turbulent heat transfer on the Pt test tube of d=3

mm and L=100 mm in Fig. 9 is almost constant at 16.38 m in the wide range of TL at flow velocity, u, of 4.34

m/s. The Prandtl numbers of the surface temperature on the heated surface of the Platinum circular tube on

forced convection, (Pr)Ts, are calculated with respect to the heater inner surface temperature of the heating

element, Ts, and are indicated by a blue solid line in Fig. 9. As shown in Fig. 10, the Prandtl numbers of water

are given by the following equation with respect to the water temperature.

6 5 4 3

2

Pr=3.731204E-13 T -4.352312E-10 T +2.010978E-07 T -4.696383E-05 T +

5.897680E-03 T -0.3942907 T+13.11831 (53)

The Prandtl number is 3.57 at TL=11 K in Fig. 9. It decreases gradually with an increase in TL and becomes

the minimum value 0.84 at TL =138.5 K. Furthermore, when TL is 138.5 K or more, it becomes almost

constant value 0.88. The Prandtl number is approximately 1 at TL of 117.30 K. The thicknesses of conductive

sub-layer, CSL, are also shown in Fig. 9. The thickness of the conduction sub-layer does not depend on the heat

flux and is a constant value 16.38 m in the wide TL region. Results of scale analysis in laminar thermal

boundary layers give the order of magnitude of the main values shown in following equations [25, 26]:

1 2 /VSL

CSLPr

for Pr1 (54)

1VSL

CSL

for Pr=1 (55)

1 3 /VSL

CSLPr

for Pr>1 (56)

The thicknesses of the viscous sub-layer, VSL, for the turbulent heat transfer on the Pt test tube of d=3 mm and

L=100 mm at flow velocity, u, of 4.34 m/s are estimated from the thicknesses of the conductive sub-layer and

Prandtl numbers of the surface temperature on the heated surface and those are shown with a red broken line in

Fig. 9, and are also listed in Table 3. The thickness of the viscous sub-layer, VSL, is 25.04 m at TL=11 K. It

18

decreases gradually with decreasing trend similar to Prandtl number, (Pr)Ts, as TL increases and becomes the

minimum value 15.03 m at TL =138.5 K. Furthermore, when TL is 138.5 K or more, it becomes almost

constant value 15.37 m. The Prandtl number is approximately 1 at TL of 117.30 K and the thickness of the

viscous sub-layer is almost the same thickness as the thickness of conductive sub-layer 16.38 m.

4.2.3.2. In cases of u=7.71, 10.50, 14.12, 17.83 and 21.79 m/s

Heat transfer processes on the Pt test tube of d=3 mm and L=100 mm compared with heat transfer curves

numerically analyzed, Prandtl numbers of the surface temperature on the heated surface under forced convection

and thicknesses of conductive and viscous sub-layers at various TL are shown in Fig. 11 and Table 4 with

u=7.71 m/s, Fig. 12 and Table 5 with u=10.50 m/s, Fig. 13 and Table 6 with u=14.12 m/s, Fig. 14 and Table 7

with u=17.83 m/s, and Fig. 15 and Table 8 with u=21.79 m/s, respectively. The Prandtl numbers of the surface

temperature on the heated surface of the Platinum circular tube on forced convection, (Pr)Ts, decrease gradually

with an increase in TL and becomes the minimum value. Furthermore, those become almost constant value. The

Prandtl number is approximately 1 at TL of 119.10, 119.55, 119.69, 116.89 and 113.74 K at u=7.71, 10.50,

14.12, 17.83 and 21.79 m/s, respectively. The thicknesses of the conduction sub-layer, CSL, do not depend on the

heat flux and are constant values 9.32, 7.56, 5.71, 4.25 and 3.48 m in the wide TL region. The thicknesses of

the viscous sub-layer, VSL, for the turbulent heat transfer on the Pt test tube of d=3 mm and L=100 mm are

estimated from the thicknesses of the conductive sub-layer and Prandtl numbers of the surface temperature on

the heated surface and those are shown with a red broken line in Figs. 11, 12, 13, 14 and 15, and are also listed in

Tables 4 to 8. These thicknesses of the viscous sub-layer, VSL, are 14.44 to 8.43 m, 11.73 to 6.85 m, 9.15 to

5.24 m, 6.76 to 3.89 m, and 5.47 to 3.18 m in the wide range of TL at u=7.71, 10.50, 14.12, 17.83 and

21.79 m/s, respectively. Those decrease gradually with decreasing trend similar to Prandtl number, (Pr)Ts, as TL

increases and becomes the minimum value 8.43, 6.85, 5.24, 3.89 and 3.18 m at TL =133.00, 140.15, 140.12,

140.68 and 140.19 K. Furthermore, when TL is 133.00, 140.15, 140.12, 140.68 and 140.19 or more, it becomes

almost constant value 8.91, 7.43, 5.53, 4.11 and 3.43 m, respectively. At TL of 119.10 119.55, 119.69, 116.89

19

and 113.74 K (Pr=1) the thicknesses of the viscous sub-layer are almost the same thickness as the thicknesses of

conductive sub-layer 9.32, 7.56, 5.71, 4.25 and 3.48 m.

4.2.4. Thickness of conductive sub-layer on nucleate boiling heat transfer,

The heat transfer with thinner conductive sub-layer dissipated by the evaporation on nucleate boiling is

numerically analyzed by the theoretical equations for steady-state turbulent heat transfer. The numerical

solutions for the relation between the heat flux, q, and the temperature difference between average inner surface

temperature and liquid bulk mean temperature, TL, are shown for the heat flux, q, ranging from 5.81105 to

1.22107 W/m

2, from 5.7110

5 to 1.7710

7 W/m

2, from 6.4210

5 to 2.5710

7 W/m

2, from 6.4210

5 to 2.9810

7

W/m2, from 6.4110

5 to 3.1510

7 W/m

2 and from 6.4310

5 to 4.4710

7 W/m

2 at the flow velocities of 4.34, 7.71,

10.50, 14.12, 17.83 and 21.79 m/s as colored solid circles in Figs. 16 to 21, respectively. The 11-12, 11-14, 12-

14, 14, 14-17 and 14 different values for the numerical solutions are plotted for the TL ranging from 6.00 to

396.11 K, from 5.69 to 350.65 K, from 7.01 to 395.24 K, from 5.71 to 384.13 K, from 4.74 to 296.68 K and

from 5.33 to 250.91 K on the log q versus log TL graph, respectively. These solutions for =8.24, 2.86, 0.81 and

0.15 m, =5.24, 1.97, 0.57 and 0.11 m, =4.25, 1.64 and 0.48 m, =3.48, 1.37, 0.57 and 0.41 m, =2.86,

1.15 and 0.35 m and =1.97, 0.81 and 0.25 m become also higher with an increase in the TL on the higher

curves parallel to the steady-state turbulent heat transfer one derived from Eq. (43) as shown in Figs. 16 to 21,

respectively. The outer control volume widths for the r-component, (r)out, which are twice as large as the

thicknesses of the conductive sub-layer, , are ranging from 16.48 to 0.3 m, ranging from 10.48 to 0.22 m,

ranging from 8.5 to 0.96 m, ranging from 6.96 to 0.82 m, ranging from 5.72 to 0.7 m and ranging from 3.94

to 0.5 m at flow velocities, u, of 4.34, 7.71, 10.50, 14.12, 17.83 and 21.79 m/s, respectively. These curve-fitted

numerical solutions based on the least squares method of power-law are shown in Figs. 16 to 21. The thicknesses

of the conductive sub-layer, , on the nucleate boiling heat transfer experimentally obtained would become half

the outer control volume width for the r-component, (r)out/2, numerically solved at an intersection point

between the nucleate boiling heat transfer curves and the smoothing curves of conductive sub-layer.

20

It has been assumed that the measurement of thickness of the conductive sub-layer at CHF point would be very

useful to discuss the mechanism of the critical heat flux during flow boiling of subcooled water, which would

occur due to the heterogeneous spontaneous nucleation at the lower limit of the heterogeneous spontaneous

nucleation temperature [9], due to the hydro-dynamic instability suggested by Kutateladze [10] and Zuber [11]

or due to the liquid sub-layer dry-out models suggested by Lee and Mudawar [12], Katto [13] and Celata et. al.

[14]. The thicknesses of the conductive sub-layer on forced convection and nucleate boiling heat transfer, CSL

and , are shown versus heat flux, q, for the u ranging from 4 to 21 m/s in Fig. 22. These thicknesses on the

semi-log graph (CSL and versus log q) become linearly lower with an increase in the q. Those look like being

almost 0 m at the CHF points, although that at u=4 m/s becomes almost 0 m at the heat flux 13.17 % lower

than the CHF one. The conductive sub-layer at CHF point would have almost disappeared, although the violent

boiling noise was made for a period of time before the CHF point. As shown in Fig. 23, the ratios of conductive

sub-layer on nucleate boiling heat transfer to conductive sub-layer on forced convection, CSL/CSL and /CSL, for

the Pt test tube of d=3 mm and L=100 mm with the commercial finish of inner surface can be expressed for the

ratios of boiling number on nucleate boiling heat transfer to boiling number at CHF point, Bo/Bocr, for the u

ranging from 4 to 21 m/s by the following correlations:

crCSLBo

Bo

.ln.

81950

194010

for u=4 m/s (57)

crCSLBo

Bo

.ln.

91740

112051

for u=6.9 m/s (58)

crCSLBo

Bo

.ln.

92610

112141

for u=9.9 m/s (59)

crCSLBo

Bo

.ln.

03421

108751

for u=13.3 m/s (60)

crCSLBo

Bo

.ln.

01991

145201

for u=17 m/s (61)

crCSL Bo

Bo

.ln.

03841

131281

for u=21 m/s (62)

21

It is assumed that the transition to film boiling at the subcooled water flow boiling on the Pt test tube of d=3 mm

and L=100 mm would occur due to the liquid sub-layer dry-out model at the steady-state CHF but not due to

the heterogeneous spontaneous nucleation and the hydro-dynamic instability. The conductive sub-layer at CHF

point has almost disappeared under the u ranging from 4 to 21 m/s, and it would not be seen that the CHF

phenomenon occurs at some critical velocity in the vapor phase when the vapor jets and the liquid ones start

interfering with each other. It is the reason not to occur the hydrodynamic instability on the vapor-liquid

interface at the CHF.

The values of CHF numerically analyzed from Celata et. al.’s liquid sub-layer dry-out model [14] are shown in

Figs. 16 to 23 for comparison. The values derived from liquid sub-layer dry-out model are in good agreement

with the experimental values of CHF for the Pt test tube of d=3 mm and L=100 mm within -5.99 to 10.91 %

differences at whole u range of 4.34, 7.71, 10.50, 14.12, 17.83 and 21.79 m/s tested here. The relationship

between CSL/CSL and /CSL, and Bo/Bocr at a flow velocity from 4 to 21 m/s is almost represented by one

display formula shown below instead of Eqs. (57) to (62) in Fig. 23.

42115148630

04920267790

.

Reln.

Bo

Bo

ln.

Reln.

d

crd

CSL

(63)

4.3. Comparison of the Measured CHFs with Author’s Correlations, and Other Researchers’ CHF Model

and Correlations.

Figure 24 shows the steady-state CHFs, qcr,sub,st, versus the outlet subcoolings, Tsub,out, for the vertical Pt circular

test tube of the inner diameter (d=3 mm), the heated length (L=100 mm), L/d (=33.33) and the wall thickness

(=0.5 mm) obtained for the flow velocities, u, ranging from 4 to 21 m/s at the outlet pressure, Pout, of around

800 kPa. As shown in the figure, the qcr,sub,st for each flow velocity become higher with an increase in Tsub,out

and the increasing rate becomes lower for higher Tsub,out. The CHFs in the whole experimental range become

higher with an increase in the flow velocity at a fixed Tsub,out. The curves given by Eqs. (1) and (2) for the

22

vertical SUS304 circular test tube are shown in Fig. 24 at each flow velocity for comparison. The CHF data for

Tsub,out30 K are in good agreement with the values given by the correlations. Equations (1) and (2) were

derived based on the experimental data for the vertical SUS304 test tube with the flow velocities ranging from 4

to 13.3 m/s [1-4] and ranging from 17 to 40 m/s [6] respectively. To confirm the applicability of Eqs. (1) and (2)

to the data for the flow velocity of 4 to 21 m/s, the ratios of these CHF data to the corresponding values

calculated by Eqs. (1) and (2) are shown versus Tsub,out in Fig. 25. Most of the data for the vertical circular test

tube (13 points) are within -13.27 to 2.45 % differences for 3.70 m/su21.30 m/s and 53.40 KTsub,out87.78

K. It can be considered that the CHFs are determined not by the outlet conditions but by the inlet ones. The

steady-state CHFs, qcr,sub,st, for the vertical Pt circular test tube of the inner diameter of 3 mm, L=100 mm,

L/d=33.33 and =0.5 mm were shown versus the inlet subcooling, Tsub,in, with the flow velocities of 4 to 21 m/s

in Fig. 26. The qcr,sub,st for each flow velocity become higher with an increase in Tsub,in. The increasing rate

becomes also lower for higher Tsub,in. The qcr,sub,st increase with an increase in the flow velocity at a fixed

Tsub,in. The qcr,sub,st for the wide range of flow velocities are proportional to Tsub,in0.7

for Tsub,in40 K. The

curves derived from Eqs. (3) and (4) for the vertical SUS304 circular test tube are shown in Fig. 26 for

comparison. The CHF data for Tsub,in40 K are in good agreement with the values given by authors’ correlation.

To confirm the applicability of Eqs. (3) and (4), the ratios of these CHF data for the d=3 mm vertical circular test

tube (13 points) to the corresponding values calculated by Eqs. (3) and (4) are shown versus Tsub,in in Fig. 27.

Most of the data for Tsub,in40 K are within -3.05 to 6.76 % differences of Eqs. (3) and (4) for 3.70

m/su21.30 m/s and 136.16 KTsub,in142.33 K.

Celata et al. [14] presented a mechanistic model for prediction of CHF in flow boiling of subcooled water. Hall

and Mudawar [35] developed the inlet and outlet condition correlations for subcooled high-CHF base on the

experimental data. Hall and Mudawar correlation [35] for the outlet conditions is as follows:

out

.

g

l

.

g

l.cr .We.Bo

83206810

2350 68370103320 for outlet (64)

23

Shah [36, 37] presented the upstream-conditions correlation (UCC) and the local-conditions correlation (LCC)

for CHF in vertical tubes. Shah correlation for the LCC version [37] is as follows:

0BoFFBo xEcr for the LCC version (65)

Parameter Bo0 has the highest value provided by the following three expressions:

61200 15 .YBo (66)

034300 45110820 .. Pr.Y.Bo (67)

39310500 151100240 .. Pr.Y.Bo (68)

where Pr=Pout/Pcr is the reduced pressure and Pcr (=22064 kPa) is the critical pressure. The Shah correlating

parameter Y [37] is defined as

60

402

.

g

l.

l

plFr

GdcY

(69)

where Fr=u/(gd) is the Froude number.

If c < 0, b

rx

.

).P)(F(FF

350

6011 2

1 (70)

4108801 005201 ..

c Y)(.F (71)

If Y ≥ 1.4107, then Y=1.410

7 must be used in Eq. (70). Also

F2=F1-0.42

when F1 4 (72)

F2=0.55 when F1> 4 (73)

b=0 for Pr 0.6 (74)

b=1 for Pr> 0.6 (75)

The experimental data for u=4 to 21 m/s at Pout=800 kPa are compare with authors’ correlations, Eqs. (1) and (2),

solutions of the model by Celata et al. [14], Hall and Mudawar correlation, Eq. (64), and Shah correlation for the

LCC version, Eq. (65), in Fig. 28 for the THD-F174 Pt test tube of d=3 mm and L=100 mm with the commercial

finish of inner surface. The authors’ correlations, Eqs. (1) and (2), solutions of the model by Celata et al., Hall

and Mudawar correlation, Eq. (64), and Shah correlation for the LCC version, Eq. (65), are in good agreement

24

with the experimental data for u=4 to 21 m/s within -13.27 to 2.45 % differences, -15.51 to 10.91 % differences,

-20.52 to 9.61 % differences and -32.51 to 13.16 % differences, respectively.

5. CONCLUSIONS

The subcooled boiling heat transfer and the steady state critical heat flux (CHF) in a vertical circular tube for the

liquid Reynolds numbers (Red=3.01104 to 1.4310

5), the flow velocities (u=3.70 to 21.79 m/s), the inlet liquid

temperatures (Tin=307.09 to 312.08 K), the inlet pressures (Pin=913.00 to 1025.46 kPa) and the exponentially

increasing heat input (Q=Q0exp(t/), =7.41 to 26.98 s) are systematically measured. The Pt test tube of inner

diameter (d=3 mm), heated length (L=100 mm), L/d (=33.33) and wall thickness (=0.5 mm) with average

surface roughness (Ra=0.40 m) is used. On the other hand, the RANS equations with k- turbulence model in a

circular tube of a 3 mm in diameter and a 559 mm long are numerically solved for heating of water on heated

section of a 3 mm in diameter and a 100 mm long with various thicknesses of conductive sub-layer by using

PHOENICS code under the same conditions as the experimental ones previously obtained and with temperature

dependent thermo-physical fluid properties. Experimental and computational study results lead as follows:

1) The thicknesses of the conductive sub-layer, CSL, for the turbulent heat transfer on the Pt test tube of d=3

mm and L=100 mm are 16.38, 9.32, 7.56, 5.71, 4.25 and 3.48 m at flow velocities, u, of 4.34, 7.71, 10.50,

14.12, 17.83 and 21.79 m/s, respectively. These thicknesses of the conduction sub-layer do not depend on

the heat flux and is a constant value 16.38, 9.32, 7.56, 5.71, 4.25 and 3.48 m in the wide Tsat region.

2) The thicknesses of the viscous sub-layer on forced convection are estimated from the thicknesses of the

conductive sub-layer and Prandtl numbers of the surface temperature on the heated surface. These

thicknesses of the viscous sub-layer, VSL, are 25.04 to 15.03 m, 14.44 to 8.43 m, 11.73 to 6.85 m, 9.15

to 5.24 m, 6.76 to 3.89 m and 5.47 to 3.18 m in the wide range of TL at u=4.34, 7.71, 10.50, 14.12,

17.83 and 21.79 m/s, respectively.

3) The relationship between CSL/CSL and /CSL, and Bo/Bocr at a flow velocity from 4 to 21 m/s is almost

25

represented by one display formula shown below:

42115148630

04920267790

.

Reln.

Bo

Bo

ln.

Reln.

d

crd

CSL

(63)

4) The transition to film boiling at the subcooled water flow boiling on the Pt test tube of d=3 mm and L=100

mm would occur due to the liquid sub-layer dry-out model at the steady-state CHF but not due to the

heterogeneous spontaneous nucleation and the hydro-dynamic instability.

5) Most of the steady-state CHF data for Pt circular tube of d=3 mm and L=100 mm at high liquid Reynolds

number with 3.70 m/su21.30 m/s (13 points) are within -13.27 to 2.45 % differences of Eqs. (1) and (2) for

53.40 KTsub,out87.78 K and within -3.05 to 6.76 % differences of Eqs. (3) and (4) for 3.70 m/su21.30

m/s and 136.16 KTsub,in142.33 K.

6) The solutions of the model by Celata et al., Hall and Mudawar correlation, Eq. (64), and Shah correlation for

the LCC version, Eq. (65), are in good agreement with the experimental data for u=4 to 21 m/s within -15.51

to 10.91 % differences, -20.52 to 9.61 % differences and -32.51 to 13.16 % differences, respectively.

NOMENCLATURE

Bo =q/Ghfg, boiling number

Bocr =qcr,sub,st/Ghfg, boiling number at CHF point

C1, C2, C3, C4, C5, C6 constants in Eqs. (3) and (4)

cpl specific heat at constant pressure, J/kgK

d test tube inner diameter, m

fF Fanning friction factor

G =lu, mass velocity, kg/m2s

hfg latent heat of vaporization, J/kg

L heated length, m

Leff effective length, m

26

Nud =hd/l, nusselt number

Pcr =22064 kPa, critical pressure, kPa

Pin pressure at inlet of heated section, kPa

Pipt pressure measured by inlet pressure transducer, kPa

Pout pressure at outlet of heated section, kPa

Popt pressure measured by outlet pressure

transducer, kPa

Pr =cp/, Prandtl number

(Pr)Ts Prandtl number of the surface temperature on the heated surface under forced convection

Q heat input per unit volume, W/m3

Q0 initial exponential heat input, W/m3

q heat flux, W/m2

qcr,sub,st steady-state CHF for subcooled condition, W/m2

Ra average roughness, m

Red =Gd/l, Reynolds number

Rmax maximum roughness depth, m

Rz mean roughness depth, m

ri test tube inner radius, m

ro test tube outer radius, m

(r)out outer control volume width for r-component, m

TEM calculated temperature of the outer control volume, K

T water temperature, C

average temperature of test tube, K

Tf,av average liquid temperature, K

Tin inlet liquid temperature, K

TL =(Tin+Tout)/2, liquid bulk mean temperature, K

T

27

Tout outlet liquid temperature, K

Ts heater inner surface temperature, K

Tsat saturation temperature, K

Tso heater outer surface temperatures, K

Ts,av average inner surface temperature, K

TL =(Ts,av-TL), temperature difference between average inner surface temperature and liquid bulk mean

temperature, K

Tsat =Ts-Tsat, inner surface superheat, K

u flow velocity, m/s

y+ =y(wl)

0.5/l, dimensionless normal-distance

coordinate

y+

CSL =(fF/2)0.5luCSL/l, non-dimensional thickness of conductive sub-layer

conductive sub-layer on nucleate boiling heat transfer

CSL =(r)out/2, thickness of conductive sub-layer and conductive sub-layer on forced convection

VSL thickness of viscous sub-layer on forced convection

rate of dissipation of turbulent energy, m2/s

3

l viscosity, Ns/m2

w viscosity at tube wall temperature, Ns/m2

l =l/l, kinematic viscosity of fluid, Ns m/kg

l density of fluid, kg/m3

w shear stress at the wall, N/m2

vaper quality

ACKNOWLEDGMENTS

28

This research was performed as a LHD joint research project of NIFS (National Institute for Fusion Science),

Japan, NIFS15KEMF066, 2015, 2016 and 2017.

REFERENCES

1. K. Hata, M. Shiotsu and N. Noda, "Critical Heat Fluxes of Subcooled Water Flow Boiling against Outlet

Subcooling in Short Vertical Tube," Journal of Heat Transfer, Trans. ASME, Series C, 126, pp. 312-320

(2004).

2. K. Hata, H. Komori, M. Shiotsu and N. Noda, "Critical Heat Fluxes of Subcooled Water Flow Boiling

against Inlet Subcooling in Short Vertical Tube," JSME International Journal, Series B, 47(2), pp. 306-315

(2004).

3. K. Hata, M. Shiotsu and N. Noda, "Critical Heat Flux of Subcooled Water Flow Boiling for High L/d

Region," Nuclear Science and Engineering, 154(1), pp. 94-109 (2006).

4. K. Hata and S. Masuzaki, “Subcooled Boiling Heat Transfer in a Short Vertical SUS304-Tube at Liquid

Reynolds Number Range 5.19×104 to 7.43×10

5,” Nuclear Engineering and Design, 239, pp. 2885-2907

(2009).

5. K. Hata and S. Masuzaki, “Subcooled Boiling Heat Transfer for Turbulent Flow of Water in a Short Vertical

Tube,” Journal of Heat Transfer, Trans. ASME, Series C, 132, pp. 011501-1-11 (2010).

6. K. Hata and S. Masuzaki, "Critical Heat Fluxes of Subcooled Water Flow Boiling in a Short Vertical Tube at

High Liquid Reynolds Number," Nuclear Engineering and Design, 240, pp. 3145-3157 (2010).

7. K. Hata, K. Fukuda and S. Masuzaki, “Mechanism of Critical Heat Flux during Flow Boiling of Subcooled

Water in a Circular Tube at High Liquid Reynolds Number,” Experimental Thermal and Fluid Science,

http://dx.doi.org/10.1016/j.expthermflusci.2015.09.015, 70, pp. 255-269 (2016).

8. K. Hata, K. Fukuda and S., Masuzaki, “Influence of Boiling Initiation Surface Superheat on Subcooled

Water Flow Boiling Critical Heat Flux in a SUS304 Circular Tube at High Liquid Reynolds Number,”

International Journal of Heat and Mass Transfer, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.03.017,

98, pp. 299-312 (2016).

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.03.017

29

9. C. Cole, Homogeneous and heterogeneous nucleation in Boiling Phenomena, 1, Stralen, S. van, and Cole, R.

eds., Hemisphere Pub. Corp., p. 71, (1979).

10. S.S. Kutateladze, Heat Transfer in Condensation and Boiling, AEC-tr-3770, USAEC (1959).

11. N. Zuber, Hydrodynamic Aspects of Boiling Heat Transfer, AECU-4439, USAEC (1959).

12. C.H. Lee and I. Mudawar, “A mechanistic critical heat flux model for subcooled flow boiling based on local

bulk flow conditions,” International Journal of Multiphase Flow, 14, pp. 711-728 (1988).

13. Y., Katto, "A physical approach to critical heat flux of subcooled flow boiling in round tubes," International

Journal of Heat and Mass Transfer, 33(4), pp. 611-620 (1990).

14. G. P. Celata, M. Cumo, A. Mariani, M. Simoncini and G. Zummo, "Rationalization of existing mechanistic

models for the prediction of water subcooled flow boiling critical heat flux," International Journal of Heat

and Mass Transfer , 37, suppl. 1, pp. 347-360 (1994).

15. R. S. Brodkey and H. C. Hershey, Transport Phenomena, McGraw-Hill, New York, p. 568 (1988).

16. W.P. Jones and B.E., Launder “The Prediction of Laminarization with a Two- equation Model of

Turbulence,” Int. J. Heat Mass Transfer, 15, pp.301–314 (1972).

17. D.B. Spalding, The PHOENICS Beginner's Guide, CHAM Ltd., London, United Kingdom (1991).

18. PHOENICS Nihongo Manyuaru (PHOENICS Japanese Manual), CHAM-Japan, Tokyo (in Japanese)

(2010).

19. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Pub. Corp., New York (1980).

20. S.V. Patankar, and D.B. Spalding, “A Calculation Procedure for Heat, Mass and Momentum Transfer in

Three-Dimensional Parabolic Flows,” Int. J. Heat Mass Transfer, 15, pp.1787–1806 (1972).

21. K. Hata, Y. Shirai, S. Masuzaki and A. Hamura, “Computational Study of Turbulent Heat Transfer for

Heating of Water in a Short Vertical Tube under Velocities Controlled,” Nuclear Engineering and Design,

249, pp. 304-317 (2012).

22. K. Hata, Y. Shirai, S. Masuzaki and A. Hamura, “Computational Study of Turbulent Heat Transfer for

Heating of Water in a Vertical Circular Tube –Influence of Tube Inner Diameter on Thickness of

Conductive Sub-layer–,” Journal of Power and Energy Systems, 6(3), pp. 446-461 (2012).

23. K. Hata, K. Fukuda, S. Masuzaki and A. Hamura, “Conductive Sub-layer of Turbulent Heat Transfer for

30

Heating of Water in a Circular Tube,” Proceedings of the 25th International Symposium on Transport

Phenomena, November 5-7, 2014, Krabi Thailand, ISTP-25 Paper 6, pp. 1-10 (2014).

24. K. Hata, K. Fukuda and S. Masuzaki , “Conductive Sub-layer of Turbulent Heat Transfer for Heating of

Water in a Circular Tube,” Heat and Mass Transfer, DOI: 10.1007/s00231-017-1996-5, pp. 1-18 (2017).

25. Favre-Marinet, Michel, and Tardu, Sedat, 2009, "Convective Heat Transfer," ISTE Ltd and John Wiley &

Sons, Inc., Great Britain and United States.

26. Hanjalic, K., Kenjeres, S., Tummers, M.J., and Jonker, H.J.J., 2009, "Analysis and Modeling of Physical

Transport Phenomena," Published by VSSD.

27. K. Hata and N. Noda, “Turbulent Heat Transfer for Heating of Water in a Short Vertical Tube,” Journal of

Power and Energy Systems, 2(1), pp. 318-329 (2008).

28. T. Sato and H. Matsumura, "On the Conditions of Incipient Subcooled-Boiling with Forced Convection,"

Bulletin of JSME, 7, pp. 392-398 (1963).

29. A. E. Bergles and W.M. Rohsenow, "The Determination of Forced-Convection Surface-Boiling Heat

Transfer," Journal of Heat Transfer, Trans. ASME, Series C, 86, pp. 365-372 (1964).

30. W. M. Rohsenow, "A Method of Correlating Heat-Transfer Data for Surface Boiling of Liquids,"

Transactions of ASME, 74, pp. 969-976 (1952).

31. W. H. McAdams, W. E. Kennel, C. S. L. Minden, R. Carl, P. M. Picornell and J. E. Dew, "Heat Transfer at

High Rates to Water with Surface Boiling," Ind. Engng. Chem., 41(9), pp. 1945-1953 (1949).

32. W. H. Jens and P. A. Lottes, "Analysis of Heat Transfer Burnout, Pressure Drop and Density Data for High

Pressure Water," ANL-4627, May (1951).

33. J. R. S. Thom, W. M. Walker, T. A. Fallon and G. F. S. Reising, "Boiling in Subcooled Water during Flow

up Heated Tubes or Annuli," Proc. Inst. Mech. Engrs, 180, Pt 3C, pp. 226-246 (1966).

34. J. H. Lienhard, "Correlation of Limiting Liquid Superheat," Chem. Eng. Science, 31, pp. 847-849 (1976).

35. D. D. Hall and I. Mudawar, “Ultra-high Critical Heat Flux (CHF) for Subcooled Water Flow Boiling-II:

high-CHF Database and Design Equation,” International Journal of Heat and Mass Transfer, 42, pp. 1429-

1456 (1999).

36. M. M. Shah, “Improved General Correlation for Critical Heat Flux during Upflow in Uniformly Heated

31

Vertical Tubes,” International Journal of Heat and Fluid Flow, 8, pp. 326-335 (1987).

37. S. M. Ghiaasiaan, “Two-Phase Flow, Boiling, and Condensation in Conventional and Miniature Systems,”

Cambridge University Press, p. 381 (2008).

32

Figure and Table Captions

Fig. 1 Schematic diagram of experimental water loop.

Fig. 2 Vertical cross-sectional view of 3-mm inner diameter test section.

Fig. 3 Result of SEM photograph of platinum test tube of d=3 mm with the commercial finish of inner surface.

Fig. 4 Measurement and data processing system.

Fig. 5 Physical model for numerical analysis.

Table 1 The values of the constants in the Chen-Kim k- turbulence model [18].

Fig. 6 Liquid temperatures in the conductive sub-layer, CSL, based on numerically predicted data points

(solution of RANS equations) for d=3 mm test tube.

Fig. 7 Typical heat transfer processes on the Pt test tube of d=3 mm and L=100 mm for =around 26 s with

u=4.0, 6.9, 9.9, 13.3, 17 and 21 m/s compared with numerical solutions of inner surface temperature.

Table 2 Relationship between CSL for turbulent heat transfer numerically solved for circular tube of d=3 mm

and L=100 mm, and u.

Fig. 8 Relationship between CSL for turbulent heat transfer numerically solved for circular tube of d=3 mm and

L=100 mm, and u.

Fig. 9 Heat transfer process on the Pt test tube of d=3 mm and L=100 mm for =24.29 s with u=4.34 m/s

compared with heat transfer curves numerically analyzed by CSL=16.38 m, Thicknesses of conductive and

viscous sub-layers, CSL and VSL, and Prandtl numbers of the surface temperature on the heated surface under

forced convection, (Pr)Ts.

Fig. 10 The relation between Prandtl number, Pr, and water temperature, T.

Table 3 Thicknesses of conductive and viscous sub-layers, CSL and VSL, and Prandtl numbers of the surface

temperature on the heated surface under forced convection, (Pr)Ts, at various TL with u=4.34 m/s.





33

























34




compared with heat transfer curves numerically analyzed by CSL, =16.38 to 0.15 m.











Fig. 22 Experimentally measured thicknesses of the conductive sub-layer on forced convection and nucleate

boiling heat transfer, CSL and , vs. heat flux for Pt test tube of d=3 mm and L=100 mm with u=4 to 21 m/s.

Fig. 23 Relationship between CSL/CSL, /CSL and Bo/Bocr for Pt test tube of d=3 mm and L=100 mm with u=4

to 21 m/s.

Fig. 24 qcr,sub,st vs. Tsub,out for an inner diameter of 3 mm with the heated length of 100 mm at an outlet pressure

of around 800 kPa.

Fig. 25 Ratios of CHF data for the inner diameter of 3 mm to the values derived from the outlet CHF correlation

versus Tsub,out at outlet pressure of around 800kPa.

Fig. 26 qcr,sub,st vs. Tsub,in for an inner diameter of 3 mm with the heated length of 100 mm at an inlet pressures

of 835.65 to 1043.03 kPa.

35

Fig. 27 Ratios of CHF data for the inner diameter of 3 mm to the values derived from the inlet CHF correlation

versus Tsub,in at the inlet pressures of 835.65 to 1043.03 kPa.

Fig. 28 Comparison of CHF data for the THD-F174 Pt test tube of d=3 mm and L=100 mm with the commercial

finish of inner surface with authors’ correlations, Eqs. (1) and (2), solutions of Celata et. al.’s liquid sub-layer

dry-out model [14], Hall and Mudawar correlation, Eq. (64), [35] and Shah correlation for LCC version, Eq.

(65), [36, 37].

36

Fig. 1 Schematic diagram of experimental water loop.

37

Fig. 2 Vertical cross-sectional view of 3-mm inner diameter test section.

38

Fig. 3 Result of SEM photograph of platinum test tube of d=3 mm with the commercial finish of inner surface.

Flow direction

39

Fig. 4 Measurement and data processing system.

40

Fig. 5 Physical model for numerical analysis.

41

Table 1 The values of the constants in the Chen-Kim k- turbulence model [18]. k t C1e C2e C3e C 0.75 1.3 1.0 1.15 1.9 0.25 0.5478

42

Fig. 6 Liquid temperatures in the conductive sub-layer, CSL, based on numerically predicted data points (solution of RANS equations) for d=3 mm test tube.

-1.5 -1 -0.5 0 0.5 1 1.5250

300

350

400

450

500

0

5

10

15

20

25

30

r (mm)

Tf (K

)

Numerical Solution m120-TEM-72 s

d=3 mm L=100 mm Tin=312.350 K q=20.2793 MW/m

2

u=21.787 m/s

vz

(m/s

)Ts

Test tube

Tf at L=0.04425 mvz at L=0.04425 m

Ts

Test tube

-1.5 -1.49-1.48-1.47-1.46-1.45-1.44-1.43-1.42-1.41 -1.4300

350

400

450

500

0

5

10

15

20

25

30

r (mm)

Tf

(K)

Numerical Solution m120-TEM-72 s Tin=312.350 K q=20.2793 MW/m

2

u=21.787 m/s

vz

(m/s

)

CSL=(r)out/2Thickness of Conductive Sub-Layer

Ts

(r)2

(r)3

(r)4

(r)5

TEM

q=2l(Ts-TEM)/( r)out

(r)out

Ts

Test tubeConductive Sub-layer

Tf at L=0.04425 mvz at L=0.04425 m

Molecular transport

dominates.

(r)6 (r)7 (r)8 (r)9

43

Fig. 7 Typical heat transfer processes on the Pt test tube of d=3 mm and L=100 mm for =around 26 s with u=4.0, 6.9, 9.9, 13.3, 17 and 21 m/s compared with numerical solutions of inner surface temperature.

1 10 102106

107

108

Tsat=Ts-Tsat (K)

q

(

W/m

2)

Experimental Data Pt Tube d=3 mm L=100 mm Pout=800 kPa Tin=309.26 K =26 s

u 4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s

THET TH

Eq. (45)

Eqs. (48) and (49)Eq. (50)Eq. (51)

Eq. (43) Eq. (46)

+15%-15%

Eq. (44)

Eq. (47) with C sf=0.012

Hata

Numerical Solution CSL u

16.38 m 4.0 m/s9.32 m 6.9 m/s7.56 m 9.9 m/s5.71 m 13.3 m/s4.25 m 17 m/s3.48 m 21 m/s

44

Table 2 Relationship between CSL for turbulent heat transfer numerically solved for circular tube of d=3 mm

and L=100 mm, and u.

d = 3 mm and L = 100 mm

u (m/s) CSL (m)

4.343 16.38

7.713 9.32

10.497 7.56

14.123 5.71

17.833 4.25

21.787 3.48

45

Fig. 8 Relationship between CSL for turbulent heat transfer numerically solved for circular tube of d=3 mm and L=100 mm, and u.

1 5 10 50 1001

5

10

50

100

CS

L (

m)

u (m/s)

d=3 mm L=100 mm L/d= 33.33 Pin=871.85-1027.56 kPa Tin=307.09-312.08 K =26.45 s

Numerical Solution CSL L 100 mm

Eq. (52)

46

u=4.0 m/s

Fig. 9 Heat transfer process on the Pt test tube of d=3 mm and L=100 mm for =24.29 s with u=4.34 m/s compared with heat transfer curves numerically analyzed by CSL=16.38 m, Thicknesses of conductive and viscous sub-layers, CSL and VSL and Prandtl numbers of the surface temperature on the heated surface under forced convection, (Pr)Ts.

1 10 102 103

106

107

108

0

10

20

30

40

50

TL (K)

q

(W/m

2)

Experimental Data SF6562 Pt Tube d=3 mm L=100 mm L/d=33.3 Pin=847.4 kPa Tin=309.23 K u=4.34 m/s =24.29 s

Eq. (43)

Numerical SolutionCSL=16.38 m

C

SL,

VS

L

(m

)

+20%

-20%

(Pr)Ts

VSL

CSL

VSL=CSL/Pr-1/3

for (Pr>1)

VSL= CSL/Pr

-1/2

for (Pr<1)

VSL=CSL

for (Pr=1)

1

10

102

(P

r) T

s

47

Fig. 10 The relation between Prandtl number, Pr, and water temperature, T.

50 100 150 200 250 300 350 400

5

10

15

0

T (C)

Pr

Saturated Water

Eq. (53)

48



TL (K) 10.994 20.014 40.461 60.076 79.52 100.704 117.301 138.492 237.909

(Pr)Ts 3.5706 2.9600 2.1832 1.8034 1.5224 1.2134 1.0027 0.8423 0.8806

CSL (m) 16.38 16.38 16.38 16.38 16.38 16.38 16.38 16.38 16.38

VSL (m) 25.0358 23.5187 21. 2494 19.9378 18.8433 17.4710 16.3945 15.0333 15.3714

49

u=6.9 m/s

Fig. 11 Heat transfer process on the Pt test tube of d=3 mm and L=100 mm for =25.92 s with u=7.71 m/s compared with heat transfer curves numerically analyzed by CSL=9.32 m, Thicknesses of conductive and viscous sub-layers, CSL and VSL, and Prandtl numbers of the surface temperature on the heated surface under forced convection, (Pr)Ts.

1 10 102 103

106

107

108

0

10

20

30

TL (K)

q

(W/m

2)


Eq. (43)


C

SL,

VS

L

(m

)

+20%

-20%

(Pr)Ts

CSL

VSL

VSL=CSL/Pr-1/3

for (Pr>1)VSL=CSL

for (Pr=1)

VSL= CSL/Pr

-1/2

for (Pr<1)

1

10

102

(P

r) T

s

50



TL (K) 10.208 19.933 40.23 59.611 79.963 100.611 119.1 133.008 234.060

(Pr)Ts 3.722 3. 0126 2.2365 1.8365 1.5305 1.2479 1.0108 0.8181 0.9144

CSL (m) 9.32 9.32 9.32 9.32 9.32 9.32 9.32 9.32 9.32

VSL (m) 14.4437 13.4605 12.1882 11.4134 10.7405 10.0341 9.3534 8.4296 8.9120

51

u=9.9 m/s


1 10 102 103

106

107

108

0

10

20

30

TL (K)

q

(W/m

2)


Eq. (43)


C

SL,

VS

L

(m

)

(Pr)Ts+20%

-20%

CSL

VSL=CSL/Pr-1/3

for (Pr>1)VSL=CSL

for (Pr=1)

VSL= CSL/Pr

-1/2

for (Pr<1)

VSL

1

10

102

(P

r) T

s

52



TL (K) 11.732 20.205 40.276 59.818 79.35 100.04 119.552 140.154 205.559

(Pr)Ts 3.7369 3. 0323 2.2381 1.8393 1.5388 1.2450 1.0151 0.8204 0.9663

CSL (m) 7.56 7.56 7.56 7.56 7.56 7.56 7.56 7.56 7.56

VSL (m) 11.7316 10.9425 9.8890 9.2628 8.7281 8.1330 7.5977 6.8477 7.4316

53

u=13.3 m/s


1 10 102 103

106

107

108

0

5

10

15

20

25

TL (K)

q

(W/m

2)


Eq. (43)


C

SL,

VS

L

(m

)

+20%

-20%

(Pr)TsCSL

VSL

VSL=CSL/Pr-1/3

for (Pr>1)VSL=CSL

for (Pr=1)

VSL= CSL/Pr

-1/2

for (Pr<1)

1

10

102

(P

r) T

s

54



TL (K) 5.076 20.346 39.921 59.642 78.488 99.614 119.693 140.116 195.206

(Pr)Ts 4.1166 2.9384 2.2248 1.8344 1.5345 1.2486 1.0069 0.8411 0.9383

CSL (m) 5.71 5.71 5.71 5.71 5.71 5.71 5.71 5.71 5.71

VSL (m) 9.1513 8.1785 7.4542 6.9898 6.5860 6.1487 5.7232 5.2368 5.5309

55

u=17 m/s


1 10 102 103

106

107

108

0

5

10

15

TL (K)

q

(W/m

2)


Eq. (43)


C

SL,

VS

L

(m

)

+20%

-20%

(Pr)Ts

CSL

VSL

VSL=CSL/Pr-1/3

for (Pr>1)VSL=CSL

for (Pr=1)

VSL= CSL/Pr

-1/2

for (Pr<1)

1

10

102

(P

r) T

s

56



TL (K) 5.07 19.738 40.139 60.115 80.807 100.578 116.894 140.676 192.612

(Pr)Ts 4.0258 2.9357 2.1777 1.8051 1.4873 1.2155 1.0112 0.8365 0.9345

CSL (m) 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25

VSL (m) 6.7609 6.0855 5.5088 5.1748 4.8512 4.5357 4.2658 3.8870 4.1084

57

u=21 m/s


1 10 102 103

106

107

108

0

5

10

15

TL (K)

q

(W/m

2)


Eq. (43)


C

SL,

VS

L

(m

)

+20%

-20%

(Pr)Ts

CSL

VSL

VSL=CSL/Pr-1/3

for (Pr>1)VSL=CSL

for (Pr=1)

VSL= CSL/Pr

-1/2

for (Pr<1)

1

10

102

(P

r) T

s

58



TL (K) 3.744 19.96 40.002 59.907 80.355 100.469 113.743 140.187 207.702

(Pr)Ts 3.8792 2.8086 2.1424 1.7698 1.4796 1.1948 1.0279 0.8328 0.9726

CSL (m) 3.48 3.48 3.48 3.48 3.48 3.48 3.48 3.48 3.48

VSL (m) 5.4680 4.9099 4.4862 4.2094 3.9655 3.6927 3.5121 3.1758 3.4320

59

Fig. 16 Heat transfer process on the Pt test tube of d=3 mm and L=100 mm for =24.29 s with u=4.34 m/s compared with heat transfer curves numerically analyzed by CSL, =16.38 to 0.15 m.

1 10 102 103

106

107

108

TL (K)

q

(W/m

2)


Eq. (43)

Numerical SolutionCSL=16.38 m=8.24 m=2.86 m=0.81 m=0.15 m

(qcr,sub,st)Celata Model

60


1 10 102 103

106

107

108

TL (K)

q

(W/m

2)


Eq. (43)


Numerical SolutionCSL=9.32 m=5.24 m=1.97 m=0.57 m=0.11 m

61


1 10 102 103

106

107

108

TL (K)

q

(W/m

2)


Eq. (43)

Numerical SolutionCSL=7.56 m=4.25 m=1.64 m=0.48 m


62


1 10 102 103

106

107

108

TL (K)

q

(W/m

2)


Eq. (43)



63


1 10 102 103

106

107

108

TL (K)

q

(W/m

2)


Eq. (43)



64


1 10 102 103

106

107

108

TL (K)

q

(W/m

2)


Eq. (43)



65

107-5

0

5

10

15

20

25

exp Celata u 4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s

C

SL, (

m)

q (W/m2)

(qcr,sub,st)exp


Experimental Data Pt Tube d=3 mm L=100 mm L/d=33.33 Pout=830.67-865.87 kPa Tin=307.09-312.08 K

CSL u4.0 m/s6.9 m/s9.9 m/s13.3 m/s17 m/s21 m/s

Fig. 22 Experimentally measured thicknesses of the conductive sub-layer on forced convection and nucleate boiling heat transfer, CSL and , vs. heat flux for Pt test tube of d=3 mm and L=100 mm with u=4 to 21 m/s.

66

Fig. 23 Relationship between CSL/CSL, /CSL and Bo/Bocr for Pt test tube of d=3 mm and L=100 mm with u=4 to 21 m/s.

0.5 1

0

0.5

1

1.5

2

CS

L/

CS

L, /

CS

L

Bo/Bocr

Experimental Data Pt Tube d=3 mm L=100 mm L/d=33.33 Pout=830.67-865.87 kPa Tin=307.09-312.08 K

CSL u4.0 m/s6.9 m/s9.9 m/s13.3 m/s17 m/s21 m/sEq. (57)

Eq. (58)Eq. (59)Eq. (60)Eq. (61)Eq. (62)

(qcr,sub,st)exp

+15 %-15 %

exp Celata u 4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s


Eq. (63)

67

Fig. 24 qcr,sub,st vs. Tsub,out for an inner diameter of 3 mm with the heated length of 100 mm at an outlet pressure of around 800 kPa.

50 100 150

10

20

30

40

50

0

qcr,s

ub

,st (M

W/m

2)

Pt TubeCFd=3 mmL=100 mmL/d=33.33Pout=800 kPa

u 4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s

Tsub,out (K)

Eq. (1) Eq. (2)

Outlet subcooling

68

Fig. 25 Ratios of CHF data for the inner diameter of 3 mm to the values derived from the outlet CHF correlation versus Tsub,out at outlet pressure of around 800kPa.

50 100 150

0.5

1

1.5

2

0

(qcr,s

ub

,st)

exp/(

qcr,s

ub

,st)

cal

Tsub,out (K)

+15%

-15%


u(m/s) 4.0 6.9 9.9 13.3

17 21

Eq. (1)

Eq. (2)

69

Fig. 26 qcr,sub,st vs. Tsub,in for an inner diameter of 3 mm with the heated length of 100 mm at an inlet pressures of 835.65 to 1043.03 kPa.

50 100 150

5

10

15

20

25

30

35

40

0

qcr,s

ub

,st(

MW

/m2)

u 4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s

Tsub,in (K)

Eq. (3) Eq. (4)

Pt TubeCFd=3 mmL=100 mmL/d=33.33Pin=835.65-1043.03 kPa

Inlet subcooling

70

Fig. 27 Ratios of CHF data for the inner diameter of 3 mm to the values derived from the inlet CHF correlation versus Tsub,in at the inlet pressures of 835.65 to 1043.03 kPa.

50 100 150

0.5

1

1.5

2

0

(qcr,s

ub

,st)

exp/(

qcr,s

ub

,st)

cal

Tsub,in (K)

Pt TubeCFd=3 mmL=100 mmL/d=33.33Pin=835.65-1043.03 kPa

+15%

-15% u

4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s

17 m/s 21 m/s

Eq. (3)

Eq. (4)

71

Fig. 28 Comparison of CHF data for the THD-F174 Pt test tube of d=3 mm and L=100 mm with the commercial finish of inner surface with authors’ correlations, Eqs. (1) and (2), solutions of Celata et. al.’s liquid sub-layer dry-out model [14], Hall and Mudawar correlation, Eq. (64), [35] and Shah correlation for LCC version, Eq. (65), [36, 37].

50 100 150

0.5

1

1.5

2

0

(qcr,s

ub

,st)

exp/(

qcr,s

ub

,st)

cal

Tsub,out (K)

+15%

-15%


Current Study d=3 mm L=100 mm L/d=33.33 Pout=800 kPa u

4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s

Current Study

Outlet subcooling

Hall and Mudawar [35] u

4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s

Celata et al.'s model [14] u

4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s

S hah (LCC) [36, 37] u

4.0 m/s 6.9 m/s 9.9 m/s 13.3 m/s 17 m/s 21 m/s

kobe university repository : kernelthe objectives of present study are six-fold. first is to measure...

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