MODELING THE NUKIYAMA CURVE FOR WATER-COOLED FUSION DIVERTOR CHANNELS

Theron D. Marshall CEA Cadarache - SERSI/LECC 13108 St. Paul-lez-Durance, FRANCE [email protected]

Dennis L. Youchison Sandia National Laboratories P.O. Box 5800 Albuquerque, New Mexico 87185 USA (505) 845-3138 [email protected]

ABSTRACT A conclusive safety assessment of a fusion reactor requires that the thermal response of the divertor assembly is known with a high degree of accuracy. Such accuracy is mandated because the divertor assembly is subjected to the highest levels of incident heat flux within the reactor. In order to accurately predict the thermal response of the divertor’s cooling channels, it is necessary to have a complete model of the Nukiyama boiling curve for the water conditions of interest. Currently published models of the Nukiyama curve for fusion divertor channels have only included the regimes of forced convection, partially and fully developed nucleate boiling, and the local CHF. This paper presents a model that includes these pre-CHF regimes and the post-CHF regime of transition boiling. The model is unique because (1) it tightly integrates the respective heat transfer correlations and makes heat transfer predictions for the water conditions and incident heat fluxes that are fusion-specific, (2) predicts post-CHF heat transfer properties for a swirl tape divertor channel, and (3) validates its predictions via comparison with experimental data. Based on these three points, this model is considered as one of the best available methods for predicting the Nukiyama curve for a water-cooled fusion device.

Lee C. Cadwallader Idaho National Engineering And Environmental Laboratory P.O. Box 1625 Idaho Falls, Idaho 83415 USA (208) 526-1232 [email protected]

model of the Nukiyama curve. For the high-heat flux levels and one-sided heating conditions of fusion devices, such a model would have to additionally demonstrate its applicability at fusion-relevant conditions and with prototypical fusion divertor channels.

I. INTRODUCTION To receive regulatory licensing, commercial fusion power plants will be required to demonstrate the accuracy of their thermal predictions for the internal components of the reactor. It is reasonable to expect that the regulatory agency will be especially attentive to the thermal calculations of the divertor coolant channels since by the nature of the divertor’s design, these channels will receive the highest levels of incident heat flux within the reactor. Currently, there are many available correlations for predicting heat transfer in the individual regimes of the Nukiyama boiling curve [1] (Figure 1) when water is used as the coolant. Unfortunately, there is no physical model that intimately integrates the correlations and forms a cohesive

Figure 1: Nukiyama’s Boiling Curve.

This paper discusses a physical heat transfer model that addresses the above requirements. The model includes all regimes of the Nukiyama boiling curve that can be anticipated for a water-cooled divertor channel. Furthermore, the model has shown excellent agreement with experimental data from fusion-relevant experiments. The remainder of this paper discusses the correlations of the model, the comparison of the model’s prediction with experimental data, and the engineering software that was developed from the model.

II. HEAT TRANSFER CORRELATIONS A. Bare Channel Mockup 1. Forced Convection. For the forced convection regime, Sieder-Tate [2] was the selected correlation. This selection was based upon: (1) a thorough review of the forced convection literature for fusion-relevant conditions [3],[4],[5] and (2) the correlation’s excellent agreement with previous heat transfer experiments at Sandia National Laboratories [6]

3. Partially Developed Nucleate Boiling. The Bergles-Rohsenow [7] partial nucleate boiling correlation was selected based on three factors: (1) the correlation’s good agreement with data from non-uniform heating experiments at fusion-relevant water conditions [4], (2) the logic of the correlation’s graphical approach, and (3) the continuity of the correlation with Bergles-Rohsenow’s onset to nucleate boiling correlation. The correlation is written as:

The correlation demonstrates the correct trends for the heat transfer coefficient response and generally has very good agreement with experimental data. It is written as: 0.14 ù æ k ö é æµ ö ú ÷ ⋅ ê 0.027 Re 0.8 Pr1/ 3 ç b ÷ h fc = ç çD ÷ ê çµ ÷ ú (1) è hø ê è wø úû ë

æv ρ ö = Dh ⋅ ç b b ÷ è µb ø

Re Pr

=

C p µb

(2)

k

Cp =

specific heat at constant pressure (J/kg-K)

Dh =

hydraulic diameter (m)

hfc =

forced convection heat transfer coefficient (W/m2-K)

k

=

bulk liquid thermal conductivity (W/m-K)

Db

=

bulk liquid density (kg/m3)

:b

=

bulk liquid viscosity (kg/m-s)

:w =

wall liquid viscosity (kg/m-s)

vb

bulk liquid velocity (m/s)

=

[

P

=

2.1598 P 0.0234

(3)

pressure (bar)

Mbi =

incipient boiling heat flux (MW/m2)

Tw =

wall temperature (°C)

Tsat =

saturation temperature (°C)

æ Φ öù ç1 − bi ÷ ú ç ÷ Φ è fdb ø ú û

2

(4)

Mbi =

incipient boiling flux (W/m2)

Mfc =

forced convection flux (W/m2)

Mfdb =

fully developed nucleate boiling flux (W/m2)

Mpb =

partially developed nucleate boiling flux (W/m2)

4. Fully Developed Nucleate Boiling. The Araki [9] correlation was selected based on three factors: (1) the correlation’s good agreement with data from non-uniform heating experiments at fusion-relevant water conditions [6], (2) the correlation’s range of experimental data envelopes the range for fusion devices, and (3) the mockup was highly instrumented and the experiment meticulously performed. The correlation is:

P

]

= Φ fc

Φ fdb =

2. Incipience of Boiling. Experimenters [7][4] have shown the Bergles-Rohsenow [8] incipient boiling correlation to have good agreement with thermal data produced during one-sided heating. The correlation is:

. Φ bi = 1082 P1156 1799 . ( Tw − Tsat )

Φ pb

é Φ fdb 1+ ê êë Φ fc

=

é ê ∆Tsat P ê − êë 25.72 ⋅ e 8.6

ù ú ú úû

3

(5)

pressure (MPa)

Mfdb =

fully developed nucleate boiling flux, (MW/m2)

)Tsat=

wall superheat, Tw - Tsat (°C)

5. Critical Heat Flux. The Tong-75 [10] critical heat flux (CHF) correlation was selected because it (1) satisfactorily incorporates the thermal and hydrodynamic effects associated with the onset and progression of CHF, (2) predicts the local CHF, (3) has been recommended for fusion CHF predictions [11], and (4) compares well with data [12][13]. The correlation is written as:

[

1.8 Φ CHF = 0.23 f o GH fg 1 + 0.00216 Pratio Re 0.5 Ja

] (6)

0.32 f o = 8.0 Re − 0.6 Dratio D Dratio = h Do P Pratio = Pcrit (7)

Ja = − χ sub

χ sub =

− Cp

ρl ρv ∆Tsub

H fg

channel tubes. The following subsections discuss the swirl tape factors that were experimentally derived for each of the correlations. It is important to note that the factors are without units. Thus, the units that were presented in Section I.A remain unchanged and are not repeated here. Prior to discussing the swirl tape factors, it is important to explain the term “swirl tape twist ratio”. This defining characteristic of a swirl tape insert is the tape’s twist ratio. The twist ratio is defined as the number of tube inner diameters per the pitch length for 180° rotation of the tape. 1. Forced Convection. Sieder-Tate is modified as:

Cp =

isobaric specific heat (J/kg-°C)

D0 =

reference inner diameter (0.0127 m)

Dh =

hydraulic diameter of cooling channel (m)

fo

=

Fanning friction factor

G

=

the mass flux (kg/m2-s)

0.14 éæ k ö æµ ö ù hsw = êç . 2.26 ⋅ Y − 0.248 ÷ 0.027 Re 0.8 Pr 1/ 3 ç b ÷ ú ⋅ 142 è µw ø ú êè D h ø ë û

[ (

)] (9)

Hf g =

latent heat of vaporization (J/kg)

hSW =

swirl tape heat transfer coefficient (W/cm2)

Ja =

Jakob number

hbt =

bare tube heat transfer coefficient (W/cm2)

P

water pressure (MPa)

Y

swirl tape twist ratio

=

=

Pcrit =

critical pressure of water (22.089 MPa)

2. Boiling Incipience. Same.

Dl

=

density of liquid bulk (kg/m3)

3. Partially Developed Nucleate Boiling. Same.

Dv

=

density of vapor at the liquid bulk temperature (kg/m3)

4. Fully Developed Nucleate Boiling. Same.. 5. Critical Heat Flux. The modified Tong-75 is:

)Tsub = degree of subcooling, Tsat - Tb (°C)

Xsub = quality of subcooled liquid bulk

[

1.8 Φ CHF = 0.23 f sw GH fg 1 + 0.00216 Pratio Re 0sw.5 Ja

MCHF = critical heat flux (W/m ) 2

6. Transition Boiling. The Marshall-98 [14] correlation was selected because it was the only published correlation[15][16][17][18][19] that demonstrated agreement with data from fusion-relevant experiments. It is: é T − Tsat ù Φ TB = Φ CHF ê w ú ë TCHF − Tsat û

[

f sw = f o 0.95 2.75Y − 0.406 fo = Dratio,sw =

− 0.23

(8) Dsw

− 0.6 8.0 Re sw

] (10)

]

0.32 Dratio , sw

Dsw Do

ù é δD 2 − δD ú ê 4 ú = 4⋅ ê ê δD − δ + D ú úû êë 2

(11)

MCHF=

critical heat flux (MW/m2)

MTB =

transition boiling heat flux (MW/m2)

TCHF=

wall temperature at local CHF (°C)

Tsat =

saturation temperature at liquid bulk pressure (°C)

D0 =

reference inner diameter (0.0127 m)

Dsw =

swirl tape tube modified-diameter (m)

wall temperature (°C)

fsw =

swirl tape tube modified friction factor

Resw =

swirl tape tube modified Reynolds number

Y

swirl tape twist ratio

Tw =

B. Swirl Tape Mockup When the cooling channel features a swirl tape insert, swirl tape factors must be applied to the previously described heat transfer correlations that were originally defined for bare

Re sw =

=

GDsw µb

6. Transition Boiling. Same.

III. COMPARISON WITH EXPERIMENTAL DATA Data for two prototypical divertor channels were published by Marshall in 1998. He used a rastered electron beam to generate one-sided heating of oxygen-free, highconductivity copper (OFHC-Cu) square monoblock mockups that were water-cooled. One mockup featured a bare coolant channel and the other featured a swirl tape with a twist ratio of 2. The inlet water conditions were chosen to coincide with the design specification of the International Thermonuclear Experimental Reactor. The data from these experiments were used to validate the predictions of the proposed model. The experimental data were thermocouple temperatures that were located 0.6 mm beneath the heated surface of the two mockups. However, the developed heat transfer model predicts heat transfer coefficients, heat fluxes, and temperatures at the wetted perimeter of the coolant channel. In order to calculate the corresponding thermal temperatures of the mockup, finite element analyses (FEAs) were required. These FEAs were accomplished through the use of the commercially available ABAQUS software [20]. ABAQUS was used to perform transient thermal analyses using a nonsymmetric Jacobian matrix and the Petrov-Galerkin method of discretization to analyze the heat transfer process by solving the basic Green and Naghdi energy balance [5]. Since ABAQUS uses the finite volume method of the energy balance, energy is automatically conserved both locally and globally. The evolution of the wetted perimeter temperature during the FEA was governed by the two-dimensional form of Fourier’s law of heat conduction in cylindrical coordinates [21][22]. For each of the mockups, an individual finite element mesh (FEM) was created. These FEMs were constructed to be as close as possible to the physical mockups. For the bare channel mockup, the FEM had a heated width of 15.7 mm and a coolant channel inner diameter (ID) of 7.7 mm. For the swirl tape mockup, the FEM had a heated width of 15.7 mm and an ID of 7.3 mm. The mockups’ ID differed simply because of the machining process. Both FEMs used 2619 nodes and 770 elements to model the 2 cm2 cross-sectional area of the mockups. To generate the FEA-predicted temperatures, the inlet water conditions of the experiments were used with the heat transfer model to create a Nukiyama curve. The heat fluxes, heat transfer coefficients, and wall temperatures from this curve were used in the FEA to predict heat transfer at the wetted perimeter for the incident heat fluxes of the experiments. Temperatures at the FEM nodes that corresponded to the physical location of the mockups’ thermocouples were used for comparison with the experimentally measured temperatures. The results of the comparison are presented below.

A. Bare Channel Mockup Figure 2 shows the comparison between the FEApredicted and experimentally-measured temperatures. The listed water parameter are at the inlet while Hlength and Hwidth are the heated length and width, respectively. Figure 3 presents the model-predicted Nukiyama curve. B. Swirl Tape Mockup Figure 4 shows the comparison between the FEApredicted and experimentally-measured temperatures. The listed water parameter are at the inlet while Hlength and Hwidth are the heated length and width, respectively. Figure 5 presents the model-predicted Nukiyama curve. IV. DISCUSSION A. Heat Transfer Model Figures 2 and 4 show that the developed heat transfer model does an excellent job of matching the experimental data in all observed regimes of heat transfer. There are three noteworthy conclusions to be drawn from the two figures. Firstly, the successful application of the heat transfer correlations did not require any correction factors in response to the one-sided heating. This is due to the wall temperature being a function of the angular position around the wetted perimeter and the FEA correspondingly calculating individual heat transfer properties for each node encompassing the wetted perimeter. Second, the model works equally well for both the bare channel and the swirl tape mockups. This is attributed to the optimum choice of correlations that were created using physical properties. The third noteworthy point is the performance of the model at the local CHF and in the transition boiling regime. As of the writing of this paper, there is no other existing model that has demonstrated equally accurate predictions at and above the local CHF. Figures 3 and 5 present the predicted Nukiyama curves based on the chosen correlations. Prior to the local CHF, the curves are similar to those that have been previously published in fusion-related heat transfer studies. Above the local CHF, it was only required to model the transition boiling regime. This was based on the assumption that the maximum achievable temperature at the wetted perimeter of the OFHC-Cu would be too low to establish wide-spread and steady-state film boiling. This is a benefit of the non-uniform heat flux distribution at the wetted perimeter that is caused by the one-sided heating. B. FILM-2000 Software The correct application of the developed model requires an intimate integration of the selected heat transfer correlations. For example, the transition from one regime of heat transfer to the subsequent regime was accomplished with a maximum flux difference of 0.005 W/cm2. Such stringent tolerances are required for an accurate representation of the Nukiyama curve.

800 700

Temperature (C)

600 500

Pressure Temperature Velocity ID Hlength

MPa C m/s mm

1 70 1 7.7

mm

40

Hwidth

mm

15.7

Y

400

Thermocouple Predicted

0

300 200 100 0

0

1

2

3

4

5

6

7

8

9

10

11

2

Incident Heat Flux ( MW/m )

Figure 2: Experimental and predicted thermocouple temperatures for bare channel mockup.

9 8 Wall Heat Flux ( MW/m2 )

7 6 5 MPa C m/s mm

1 70 1 7.7

2

Pressure Temperature Velocity ID Length

mm

40

1

Widthheated

mm

15.7

4 3

heated

Y 0

0

100

200

300

0 400

500

600

700

800

900

Wall Temperature (C)

Figure 3: Predicted Nukiyama curve for bare channel mockup.

1000

800 700

Temperature (C)

600

Pressure Temperature Velocity ID H

MPa C m/s mm

1 70 1 7.3

mm

30

Hwidth

mm

15.7

length

500

Y

400

Thermocouple Predicted

2

300 200 100 0

0

2

4

6

8

10

12

14

16

18

20

2

Incident Heat Flux ( MW/m )

Figure 4: Experimental and predicted thermocouple temperatures for swirl tape mockup.

22 20

2

Wall Heat Flux ( MW/m )

18 16 14 12 10 8 6 4 2 0

Pressure Temperature Velocity ID Lengthheated

MPa C m/s mm

1 70 1 7.3

mm

30

Widthheated

mm

15.7

Y 0

100

200

300

2 400

500

600

700

800

Wall Temperature (C)

Figure 5: Predicted Nukiyama curve for swirl tape mockup.

900

1000

To assist the thermal engineer apply the developed model, a Microsoft Windows-based software program has been created. This program accepts input from the user and produces realtime plots and ASCII data files of the predicted heat transfer properties. The program is available at http://film2000.free.fr. V. CONCLUSIONS A physical model has been developed for predicting the heat transfer properties of divertor channels that are onesided heated and water-cooled. The model includes all regimes of the Nukiyama curve, minus film boiling, and has exhibited excellent agreement with fusion-specific experimental data. The model is especially noteworthy because it works equally well with bare channel and swirl tape mockups. Based on this observation, the model is considered as an excellent heat transfer prediction tool for the thermalhydraulics requirements of the fusion community. REFERENCES 1. S. Nukiyama, “Maximum and Minimum Values of Heat Transmitted from Metal to Boiling Water Under Atmospheric Pressure,” Journal of the Japanese Society of Mechanical Engineers, 37, pg 367 (1934). 2. E. N. Sieder and G. E. Tate, “Heat Transfer and Pressure Drop of Liquids in Tubes,” Industrial and Engineering Chemistry, 28, 12, pgs. 1429-1435 (1936). 3. J. Schlosser and J. Boscary, Final Report: PDT 2.4 Task. Manufacture and Tests of Divertor Samples and Mock-ups: Thermalhydraulic Tests on Divertor Targets Using Swirl Tubes, CEA-Cadarache Report P/Co 94/03 (1994). 4. J. Schlosser and J. Boscary, “Thermalhydraulic Tests at NET/ITER Relevant Conditions on Divertor Targets Using Swirl Tubes,” Proceedings of the 6th International Conference on Nuclear Reactor Thermalhydraulics, pgs. 815-824, Grenoble, France (1993). 5. H. Kinoshita and T. Yoshida, “Study on the Mechanism of Critical Heat Flux Enhancement for Subcooled Flow Boiling in a Tube with Internal Twisted Tape under Nonuniform Heating Conditions,” Heat Transfer Japanese Research, 25, 5, pgs. 293-307 (1996). 6. T. D. Marshall, R. D. Watson, J. M. McDonald, and D. L. Youchison, "Post-Critical Heat Flux Behavior of a Prototypical ITER Divertor Mockup," Proceedings of the 7th International Conference on Nuclear Reactor Thermal-Hydraulics, Saratoga Springs, NY (1995). 7. J. Schlosser, Design Task (D 67) - Update of EC Thermalhydraulic Database: Thermalhydraulic Correlations for ITER High Heat Flux Components, CEA-Cadarache, Report P/Co 96 (1996). 8. A. E. Bergles and W. M. Rohsenow, "The Determination of Forced Convection Surface Boiling Heat Transfer," Transactions of the American Society of Mechanical Engineers, Series C, 86, pgs. 365-372 (1964). 9. E. J. Davis and G. H. Anderson, "The Incipience of Nucleate Boiling in Forced Convection Flow," Transactions of the American Institute of Chemical Engineers, 12, 4, pgs. 774-780 (1966). 10. M. Araki, M. Ogawa, T. Kunugi, S. Ikeda, K. Satoh, and S. Suzuki, "Experiment on Heat Transfer of Smooth and Swirl Tubes Under One-Sided Heating Conditions," International Journal of Heat and Mass Transfer, 39, 14, pgs. 3045-3055 (1996). 11. L. S. Tong, “A Phenomenological Study of Critical Heat Flux,” Proceedings of the American Institute of Chemical Engineers - American Society of Mechanical Engineers Heat Transfer Conference, San Francisco, California, ASME Paper 75-HT-68 (1975).

12. S. T. Yin, A. Cardella, A. H. Abdelmessih, Z. Jin, and B. F. Bromley, “Assessment of a Heat Transfer Correlations Package for Water-Cooled Plasma-Facing Components in fusion Reactors," Proceedings of the 5th Nuclear Reactor Thermal Hydraulics Conference, Salt Lake City, Utah (1992). 13. J. A. Koski, A. G. Beattie, J. B. Whitley, and C. D. Croessman, “Experimental Verification of Subcooled flow Boiling for Tokamak Pump Limiter Designs,” American Society of Mechanical Engineers, paper 87HT-45 (1987). 14. J. A. Koski, R. D. Watson, A. M. Hassanein, P. L. Goranson, and J. C. Salmonson, “Thermal-Hydraulic Design Issues and Analysis for the ITER Divertor,” Fusion Technology, 19, pgs. 1729 - 1735 (1991). 15. T. D. Marshall, Experimental Examination of the Post-Critical Heat Flux and Loss of Flow Accident Phenomena for Prototypical ITER Divertor Channels, Doctoral Thesis, Rensselaer Polytechnic Institute, Troy, New York (1998). 16. D. C. Groeneveld, S. C. Cheng, L. K. H. Leung, and C. Nguyen, “Computation of Single and Two-Phase Heat Transfer Rates Suitable for Water-Cooled Tubes and Subchannels,” Nuclear Engineering and Design, 114, pgs. 61-77 (1989). 17. S. Grigoriev and V. Divavin, “Cardinal Features of About Critical Temperature State Calculation for the Divertor Elements at One-Side Heat Loading”, Proceedings of the ITER Thermal-Hydraulic Technical Meeting, Focus: High Heat Flux Cooling, JET, Abingdon, United Kingdom (1996). 18. D. Schroeder-Richter, “Analytical Modeling of Complete Nukiyama Curves Corresponding to Expected Low Void Fraction at High Subcooling and Flow Rate,” Fusion Technology, 29, pgs. 468-486 (1996). 19. H. S. Ragheb, S. C. Cheng, and D. C. Groeneveld, “Observations in Transition Boiling of Subcooled Water Under Forced Convective Conditions,” International Journal of Heat and Mass Transfer, 24, 7, pgs. 1127-1137 (1981). 20. P. Weber and K. Johannsen, “Temperature-Controlled Measurement of Boiling Curves for Forced Upflow of Subcooled Water in a Circular tube at 0.1 to 1 MPa,” Heat Transfer Equipment Fundamentals, Design, Applications, and Operating Problems, ed. R. K. Shah, ASME HTD-Vol. 108, proceedings of the 1989 National Heat Transfer Conference, Philadelphia, Pennsylvania, pgs. 13-21 (1989). 21. ABAQUS/Standard User’s Manual, Hibbitt, Karlsson & Sorensen, Incorporated, Pawtucket, Rhode Island (1998). 22. S. L. Milora, S. K. Combs, and C. A. Foster, “A Numerical Model for Swirl Flow Cooling in High-Heat-Flux Particle Beam Targets and the Design of a Swirl-Flow-Based Plasma Limiter,” Nuclear Engineering and Design, 3, pgs. 301-308 (1986).