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Film boiling on a vertical plate in subcooled helium II
Cryogenics 47 (2007) 209–219 www.elsevier.com/locate/cryogenics
Film boiling on a vertical plate in subcooled helium II K. Hama *, M. Shiotsu Graduate School of Energy Science, Kyoto University, Uji, Kyoto 611-0011, Japan Received 28 November 2005; received in revised form 26 December 2006; accepted 27 December 2006
Abstract Film boiling heat transfer coefficients were measured on 10, 30 and 50 mm long vertical plates in subcooled He II for bulk liquid temperatures from 1.8 to 2.1 K. A film boiling model on a vertical plate in subcooled He II was presented based on convection heat transport in the vapor film, radiation heat transport, and heat transport in He II. The numerical solutions of the model were obtained and an equation which can express the numerical solutions within ±5% difference was derived. The equation predicted well the experimental data for lower DT range but significantly under-predicted the data for higher DT. A correlation of film boiling heat transfer including radiation contribution was presented by modifying the equation based the experimental data. This correlation can describe the experimental data within ±20% difference. Ó 2007 Published by Elsevier Ltd. Keywords: Film boiling (C); Heat transfer (C); Superfluid helium (He II) (B)
1. Introduction There are many experimental researches of film boiling on a horizontal cylinder in He II, but research on film boiling heat transfer from a vertical plate in subcooled He II is hardly done, as far as the authors are concerned. Some workers presented analytical model of film boiling heat transfer from a vertical plate in saturated He II. Rivers and McFadden [1] analyzed the vertical problem by the same method as they used for horizontal cylinder. However, there have been few theoretical works for a vertical plate in subcooled He II. The purposes of this study are fourfold. First is to obtain the experimental data of film boiling heat transfer from vertical plates with various lengths in subcooled He II for the liquid temperature from 1.8 to 2.1 K. Second is to present a film boiling model for a vertical plate in subcooled He II. Third is to present a film boiling heat transfer equation based on the numerical solutions for the model and the experimental data. Fourth is to make clear the *
Corresponding author. Tel.: +81 774 38 4453; fax: +81 774 38 4451. E-mail address: [email protected] (K. Hama).
0011-2275/$ - see front matter Ó 2007 Published by Elsevier Ltd. doi:10.1016/j.cryogenics.2006.12.004
availability of the equation by comparing the experimental data and the predicted values. 2. Apparatus and method The experimental apparatus is shown schematically in Fig. 1. The cryostat is the Claudet type, 45 cm in inner diameter and 157 cm in height. About 150 l of liquid helium is initially contained in the vessel. The inner bath is divided by a glass-epoxy-separator (3) with He I in the upper section and He II in the lower section. The bath above the separator contains saturated He I of about 74 l with a free surface. The bath below the separator contains subcooled He II of about 76 l with no free surface. Four vertical test heaters can be mounted (1) in the lower bath (He II bath) and there is a ring shaped helium boiler (4) and a heat exchanger (6). Liquid helium is provided from the upper bath (He I bath) through the heat exchanger and a motor-controlled Joule–Thomson valve (JT valve) (5) to the ring shaped boiler. By evacuating liquid helium in the boiler with a vacuum pump (10,000 l/min), the saturated liquid helium in the heat exchanger can be cooled below k-point. The liquid helium in the lower part
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Nomenclature ai bi cp Gr g Hi h hc hr k L l Nu Pr Qb q qb qb qc qr R
coefficients of 4th degree velocity profile polynomial, Eq. (17) coefficients of 4th degree temperature profile polynomial, Eq. (18) specific heat at constant pressure (J/kg K) Gr ¼ gl3 qv ðqb qv Þ=l2v ; Grashof number gravity acceleration (m/s2) Hi = L/(cpvDTsat), interface enthalpy parameter h = q/DTsat, average film boiling heat transfer coefficient (W/m2 K) hc = qc/DTsat, conduction–convection heat transfer coefficient (W/m2 K) hr = qr/DTsat, radiation heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) latent heat of vaporization (J/kg) length of test heater (m) Nu = hcl/kv, Nusselt number Pr = lvcpv/kv, Prandtl number Qb = l(qb qr)/(kvDTsat), interface heat flux parameter q = qc + qr, surface heat flux (W/m2) heat flux on the vapor–liquid interface (W/m2) heat flux on the He I–He II interface (W/m2) conduction–convection contribution of surface heat flux (W/m2) radiation of surface heat flux (W/m2) contribution h 7 1 1 R ¼ Qb Gr4 þ 0:9fðH i þ 0:25ÞPrg4 i7 47 1 , non-dimensional þ0:28 Qb Gr4 parameter
of bath will be cooled to a desired temperature below kpoint by the boiler and a spiral heat exchanger below the boiler. The liquid temperature in the He II bath is measured by calibrated Ge resistance thermometer. Liquid level in the heat exchanger boiler is measured by superconducting level sensor and is automatically kept constant at a desired value by the motor-controlled JT valve. The He II bath is connected to the He I bath by the super leak of He II because the superfluid component with no viscosity can flow through the narrow gap between the glass-epoxy separator and the wall of the inner bath. Heat loss through the gap is little, because normal component cannot go though the gap. Therefore the pressure in the He I bath can be propagated into the He II bath independent of the He II temperature. As a consequence, the cryostat shown in Fig. 1 can realize subcooled He II and be kept constant temperature when there is some heat penetration under subcooled conditions. Four test heaters called (a)–(d) were used in this experiment. The test heaters (a)–(c) are made of Au–Mn
T u v w x y
temperature (K) velocity component (m/s) velocity component (m/s) width of flat plate heater (m) tangential coordinate (m) perpendicular coordinate (m)
Greek symbols a absorptivity d thickness of laminar vapor layer (m) dI thickness of liquid He I layer (m) DT DT = Tw Tb (K) DTsat DTsat = Tw Tsat, surface superheat (K) ew radiation emissivity l viscosity (N s/m2) q density (kg/m3) rs Stefan–Boltzman constant Subscripts b bulk I He I layer sat saturation v vapor w wall d at vapor–liquid interface k at lambda point
(0.25 wt.%) alloy of thickness 0.03 mm, width 2.5 mm, and about 20, 30 and 50 mm in length, respectively. Test heater (d) is made of 316# stainless steel (SUS 316) with 3.3 mm in width, 50 mm in length, and 0.1 mm thickness. Two fine 0.05 mm diameter platinum wires were spotwelded as potential taps at about 5 mm from each end of the test heaters of (a), (b) and (d). To investigate the local heat transfer characteristics along the heater length, four same platinum wires were spot-welded on the test heaters (c) as potential taps. The first one was positioned at 7 mm from the leading edge, and the second to fourth ones were spot-welded at every 10 mm. The gradient of the electrical resistivity versus temperature curve for the Au–Mn alloy is positive from He II temperature to several hundred Kelvin. The gradient of the electrical resistivity versus temperature curve for SUS 316 is positive for the film boiling range from around 20 K to several hundred Kelvin. The test plates were annealed in an inert atmosphere and their electrical resistance versus temperature relations were calibrated in a temperature-
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211
Fig. 1. Schematic illustration of experimental apparatus.
controlled vessel by immersion in liquid helium and liquid nitrogen for temperatures up to 100 K. The calibrations for the temperatures from 290 K to 373 K and from 373 K to 453 K were performed in water and Glycerin baths, respectively. The calibration accuracy was estimated to be ±0.1 K for the temperature below 10 K for the Au–Mn alloy, and ±0.5% for higher temperatures for the Au–Mn alloy and SUS 316. Test heaters were heated by direct current from a power amplifier. The average temperature of the test heater between potential taps were measured by the resistance thermometry using a double bridge circuit including the test heater as a branch. The voltage drops across the potential taps of the test heater and across a standard resistance were amplified and passed to the analog to digital converters of a digital computer. The temperature and heat flux were obtained with the aid of previously calibrated resistance– temperature relation. The heater surface temperature was calculated from the measured average temperature and heat generation rate for each interval between the potential taps by solving the thermal conduction equation for the heater assuming uniform heat flux between the intervals. Experimental error of each interval is estimated to be ±(0.1 K + 0.8% of the measured value) for the surface temperature and to be ±2% for the heat flux of both material.
Fig. 2. Steady-state heat transfer for l = 47.3 mm type (c) test plate.
ture Tb as a parameter. We can see from this figure that in the region where DT is between critical heat flux and around 100 K, heat flux for each Tb increases very gradually, which is followed by the region where heat flux increases rapidly. The heat flux in the former region increases significantly with the decrease in Tb from 2.1 K to 2.0 K and increases gradually with further decrease in Tb. The difference in the value of the heat flux of various Tb becomes smaller as DT increases in the latter region. 3.2. Film boiling heat transfer coefficients
3. Experimental results 3.1. Heat transfer curve Fig. 2 shows typical example of steady-state heat transfer curves obtained from potential taps 1–4 of the type (c) vertical plate in subcooled He II with bulk liquid tempera-
The average film boiling heat transfer coefficients, h, were measured for bulk liquid temperatures, Tb, from 1.8 to 2.1 K under atmospheric pressure. The data for the type (a) test plate are shown versus surface superheat, DTsat in Fig. 3 with bulk liquid temperature as a parameter. The heat transfer coefficients for a constant
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Fig. 3. Average heat transfer coefficients versus heater surface superheat for l = 18 mm type (a) test plate.
Fig. 5. Average heat transfer coefficients versus heater surface superheat for l = 47.3 mm type (c) test plate.
DTsat are higher at lower value of Tb. The heat transfer coefficients for each Tb significantly decrease first and then become almost constant as the surface superheat increases. The data for longer test plates, type (b), (c) and (d), are shown versus surface superheat in Figs. 4–6, respectively, with liquid temperature as a parameter. The trend of dependence on the surface superheat and liquid temperature is similar to that for the type (a): the heat transfer coef-
Fig. 6. Average heat transfer coefficients versus heater surface superheat forl = 47 mm type (d) test plate.
Fig. 4. Average heat transfer coefficients versus heater surface superheat for l = 30 mm type (b) test plate.
ficients are higher for lower values of DTsat and Tb. Systematic effect of heater length on heat transfer coefficient cannot be seen for the range of heater length tested. Fig. 7 shows the heat transfer coefficients for the type (c) test plate in 1.8 K He II measured by the potential taps between 1–2, 2–3, 3–4 and 1–4. The first potential tap is positioned at 7 mm from the leading edge, and the second to fourth ones are fixed every 10 mm. A variation of local heat transfer coefficients cannot be seen in this figure.
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Fig. 7. Local heat transfer coefficients versus heater surface superheat for type (c) test plate.
4. Numerical analysis
Fig. 8. Physical model and coordinate system.
fluxes, qcr, on a flat plate with length l and width w in subcooled He II
4.1. Film boiling model qb ¼ A model for film boiling on a vertical plate in subcooled He II was considered based on the following assumptions: (1) Surface temperature of the vertical plate, Tw, is uniform. (2) Thickness of vapor film is negligible compared with the width of the plate. (3) The convection, which occurs within the film, is steady two-dimensional and boundary layer type flow. According to these assumptions, the model for subcooled He II film boiling on a vertical plate is illustrated schematically in Fig. 8. In this case, the vapor layer is surrounded by a thin He I layer and bulk He II outside it exists. Although the temperature gap due to non-equilibrium vaporization and condensation suggested by Labuntzov and Ametistov [2] may exist at the vapor–He I interface, there would be no temperature gap at the He I–He II interface with the temperature of Tk. Then, the heat flux, qb , removed from the He I–He II interface to the bulk He II is considered to be equal to the critical heat flux. As the thickness of the He I layer, shown very much enlarged in the figure, is estimated from dI ¼ k I ðT sat T k Þ=qb to be thinner than 1 lm for the bulk temperatures lower than 2.1 K, the heat flux qb removed from the vapor–He I interface can be dealt with equal to qb . Therefore the value of qb is given by the following equation presented by Tatsumoto et al. [3] for the critical heat
213
qb
2 ¼ 0:58 lw=f2ðl þ wÞg 1
where f ðT Þ
Z
Tk
1
13
f ðT Þ dT
;
ð1Þ
Tb 3
6:8 ¼ gðT k Þ½T 6:8 R ð1 T R Þ
gðT k Þ ¼ q2 s4k T 3k =Ak ; T R ¼ T =T k ; sk ¼ 1559 J=ðkg KÞ; Ak ffi 1150 ms=kg: Surface heat flux in film boiling is expressed by q ¼ qc þ q r ;
ð2Þ
where qc is the conduction–convection contribution and qr is the radiation contribution of the total film boiling heat flux. The radiation heat flux qr is given by rs ðT 4 T 4sat Þ; qr ¼ ð3Þ 1=ew þ 1=a 1 w where ew and a are the emissivity and absorptivity of radiation. The ew for the Au–Mn plate is 0.03 and that for SUS316 is 0.4, and a is taken to be unity. The laws of conservation of mass, momentum and energy are expressed as follows in the form of boundary layer equations: ou ov þ ¼ 0; ox oy
ð4Þ
u
ou ou 1 l o2 u þ v ¼ gðqb qv Þ þ v 2 ; ox oy qv qv oy
ð5Þ
u
oT oT k v o2 T þv ¼ : ox oy qv cpv oy 2
ð6Þ
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The boundary conditions are expressed in the following form: At the vertical plate surface (y = 0), u ¼ v ¼ 0;
ð7Þ
T ¼ T w:
ð8Þ
At the vapor liquid interface (y = d), u ¼ ud ; v ¼ vd ;
ð9Þ ð10Þ
T ¼ T sat ;
ð11Þ
dd ; qb vb ¼ qv vd ud dx ou ¼ qb vb ud ; lv oy d oT kv ¼ qb vb L þ qb qr : oy d
ð12Þ ð13Þ ð14Þ
Boundary layer equations (Eqs. (4)–(6)) were transformed into the following integrodifferential form by first integrating partial differential equations from 0 to d and then substituting the interface boundary conditions into the result Z d d qv u dy ¼ qb vb ; ð15Þ dx 0 Z d Z d d ou 2 qv u dy ¼ gðqb qv Þ dy lv ; ð16Þ dx 0 oy w 0 Z d d qv cpv uðT T sat Þ dy dx 0 oT ¼ qb vb L qb þ qr k v : ð17Þ oy w The velocity and temperature profiles are given approximately by the following fourth degree polynomials: uðx; yÞ ¼ a1 þ a2 y þ a3 y 2 þ a4 y 3 þ a5 y 4 ;
ð18Þ
T ðx; yÞ ¼ b1 þ b2 y þ b3 y 2 þ b4 y 3 þ b5 y 4 :
ð19Þ
Expressions for the coefficients ai and bi are obtained as functions of x by applying the boundary conditions of u and T in the y-direction. The integral equations, the profile expressions and the boundary conditions described above were non-dimensionalized. The characteristic of the phenomena is dependent on the following five dimensionless parameters: the Prandtl number, Pr = lvcpv/kv, average Nusselt number of characteristic dimension l for plate, Nu = hcl/kv, Grashof number, Gr ¼ gl3 qv ðqb qv Þ=l2v , the ‘‘interface enthalpy’’ parameter, Hi = L/(cpvDTsat), and the interface heat flux parameter, Qb = l(qb qr)/(kvDTsat). The Qb and Gr always appear together in the form of QbGr1/4. Using the profile expressions with the formulated coefficients ai and bi, the internal equations were reduced to three first-order ordinary differential equations. Numerical solutions for the differential equations were obtained for the QbGr1/4 ranging from 0.01 to 10 by using Runge–Kutta method. Calculated results are shown in Fig. 9 on Nu Gr1/4 versus Qb Gr1/4 graph with Hi as a parameter. Details on the derivation of the ordinary differential equations with the boundary conditions are described in Appendix A. It seems that the results for a fixed value of Hi are very weakly dependent on Qb for QbGr1/4 lower than 0.1, and they approach a line of Nu = Qb with increasing QbGr1/4. The former is the convection dominant region where vapor film thickness is relatively thick, and the latter is the conduction dominant region where vapor film thickness is relatively thin. The results for Hi ranging from 0.01 to 100 obtained here were fitted by the following equation: Nu ¼ ðGr RÞ
1=4
;
where h 1 1 7 R ¼ Qb Gr4 þ 0:9fðH i þ 0:25ÞPrg4 i7 47 14 : þ0:28 Qb Gr
Following boundary conditions were introduced to solve the problem as done by Rivers and McFadden [1]. At y = 0, 2 ou lv ¼ gðqb qv Þ; ð20Þ oy 2 w 2 oT ¼ 0: ð21Þ kv oy 2 w At y = d, 2 ou l2v ¼ q2b v2b ud lv gðqb qv Þ; oy 2 d 2 2 o T ¼ cpv qb vb ðqb vb L qb þ qr Þ: kv oy 2 d
ð22Þ ð23Þ Fig. 9. Calculated results of average heat transfer coefficients.
ð24Þ
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215
The values given by Eq. (24) are shown in Fig. 9 in comparison with the numerical solutions. The equation expressed the numerical solutions within ±5% accuracy. The values of q predicted from Eq. (1) combined with Eqs. (2) and (3) and (24) are compared with the corresponding experimental data in Figs. 3–6. It can be seen that this equation predicts the experimental data in good accuracy for lower DTsat range but under-predicts the data with the increase in DTsat.
Gr R. This difference may be due to the heat transfer from both sides of vapor film neglected in the assumption (2) in the model.
5. Correlation of film boiling heat transfer from a vertical plate in subcooled He II
The curve of Eq. (25) is shown in Figs. 10 and 11 for comparison. It seems that the curve fitting by the least square method is reasonable. Experimental data of film boiling heat transfer coefficients for the liquid temperature of 1.8, 1.9, 2.0, 2.1 K previously shown in Figs. 3–6 are compared with the values predicted from Eq. (25). It can be seen from these figures that all the experimental data on four different sized heaters in subcooled He II for wide range of surface superheat obtained here are in good agreement with the predicted values within ±20%.
5.1. Comparison of the experimental data with the analytical solution Figs. 10 and 11 show the experimental data for various liquid temperatures on type (a) and (d) test heaters, respectively, plotted on log(Nu) versus log(Gr R) graph. It can be seen from these figures that the experimental data are on a single straight line almost independent of the bulk liquid temperature. The values given by Eq. (24) are also shown on these graphs as a straight line with the gradient of 1/4 for comparison. The data for high Gr R range agree well with the predicted values, but gradually they become higher than the predicted values with the decrease in
5.2. Derivation of the film boiling heat transfer correlation The Eq. (24) was modified as follows based on the experimental data for the test heaters. Nu ¼ 2:5ðGr RÞ
0:23
ð25Þ
5.3. Radiation effect To see the radiation effect, the predicted values of film boiling heat transfer coefficients for the vertical plates with several ew and with the same dimension as type (d) heater in 2.0 K He II are shown on a linear graph in Fig. 12 against DTsat up to 1100 K. The experimental data for the type (d) test heater with ew of 0.4 are also shown in the figure for comparison. We can see that the predicted heat transfer coefficient at DTsat = 800 K for the plate with ew of 1.0 is only 4.2% higher than that for ew of 0. Experimental datum
Fig. 10. Comparison of the experimental data for type (a) test plate with the values given by Eqs. (24) and (25).
Fig. 11. Comparison of the experimental data for type (d) test plate with the values given by Eqs. (24) and (25).
Fig. 12. Predicted values of h for vertical plates with several ew in 2.0 K He II in comparison with the experimental data for type (d) test plate.
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Table 1 Calculated components in film boiling heat transfer coefficient for various radiation emissivity TB = 2.0 K, qb = 6.74E4 (W/m2) ew
DTsat (K)
hr (W/m2 K)
hc (W/m2 K)
h (W/m2 K)
0
500 800 1100
0 0 0
368 364 368
368 364 368
0.4
500 800 1100
2.9 11.3 30.4
366 358 353
369 370 383
500 800 1100
7.2 29.3 76.0
364 350 331
371 379 407
1.0
at DTsat = 760 K is within the small range. The predicted heat transfer coefficients at DTsat = 1100 K for the plates with ew of 0.4 and 1.0 are 3.3% and 8.2% higher than that for ew of 0. This does not mean the hr are such percentage of h. Table 1 shows the calculated values of hr , hc and h on plates with ew of 0, 0.4 and 1.0 in 2.0 K He II at DTsat of 500, 800 and 1100 K. At DTsat = 1100 K, for instance, the hr on the plate with ew of 1.0 is about 19% of the h. The value of hc becomes smaller with the increase of ew (namely, with the increase of hr) due to the increased thickness of vapor film. The film boiling heat transfer correlation for a vertical plate in He II including the radiation effect was derived firstly by modifying slightly the solution of numerical model based on our experimental data. This correlation is necessary to be built into a computer code in order to analyze dynamic stability of superconducting magnets. Initial local spot of thermal disturbance on the superconductor due to the conductor motion, etc. would be the order of dimension of the test plate in this experiment. As there are few experimental data by other workers, comparison with them cannot be made. Further study is necessary to clarify the applicability range of this correlation. 6. Conclusions Film boiling heat transfer coefficients were measured on 10, 30 and 50 mm long vertical plates in subcooled He II for bulk liquid temperature ranging from 1.8 to 2.1 K. It was confirmed that little dependence on heater length and width exists on the film boiling heat transfer from a vertical plate under the present experimental conditions. A model for film boiling on a vertical plate in subcooled He II was presented based on convection heat transport in the vapor film, radiation heat transport to the vapor–liquid interface, and the heat transport in the He II. The numerical solutions of the model were obtained for several fixed parameters and an equation which can express the numerical solutions within ±5% difference was derived. The equation predicted well the experimental data for lower DT range but under-predicted the data for higher DT.
A correlation of film boiling heat transfer including radiation effect was presented firstly based on the numerical solutions of the model and the experimental data. It was confirmed that this correlation can describe our experimental data in subcooled He II within ±20% difference. Further study is necessary to clarify the applicability range of this correlation. Appendix A. Transformation of the fundamental equations to ordinary differential equations A.1. Non-dimensionalization of the system The following dimensionless variables were introduced to non-dimensionalize this mathematical system: n ¼ x=l; y ¼ y=l; q lu u ¼ v ; lv T T sat ; h¼ T w T sat d d ¼ ; l q lud ud ¼ v ; lv q lvb vb ¼ b : lv
ðA:1Þ ðA:2Þ ðA:3Þ ðA:4Þ ðA:5Þ ðA:6Þ ðA:7Þ
Substitutions of the above definitions into Eqs. (15)–(17) leads to Eqs. (A.8)–(A.10), respectively. d dn d dn d dn
Z
d
u dy ¼ vb ;
ðA:8Þ
0
d ou 2 u dy ¼ Grd ; oy w 0 Z d 1 oh PrH i vb Qb u h dy ¼ ; Pr oy w 0 Z
ðA:9Þ ðA:10Þ
where u* and h are velocity and temperature profile polynomials, u ¼ a1 þ a2 y þ a3 y 2 þ a4 y 3 þ a5 y 4 ; h¼
b1
þ
b2 y
þ
b3 y 2
þ
b4 y 3
þ
b5 y 4 ;
ðA:11Þ ðA:12Þ
where ai and bi are non-dimensional forms of ai and bi. The boundary conditions at the wall y = 0, u ¼ 0; h ¼ 1; 2 ou ¼ Gr; oy 2 w 2 oh ¼0 oy 2 w
ðA:13Þ ðA:14Þ ðA:15Þ ðA:16Þ
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The boundary conditions at the interface y = d,
ðu Þd ¼
ud ;
where ðA:17Þ
ðhÞd ¼ 0; ou ¼ vb ud ; oy d oh ¼ PrH i vb Qb ; oy d 2 ou ¼ ud v2 b Gr; oy 2 d 2 oh ¼ Prvb ðPrH i vb Qb Þ: oy 2 d
ðA:18Þ ðA:19Þ ðA:20Þ ðA:21Þ ðA:22Þ
ðA:23Þ ðA:24Þ ðA:25Þ ðA:26Þ
ðA:30Þ
1 dd 1 dX ¼ d dn 4X dn
ðA:31Þ ðA:32Þ
ðA:35Þ
4
X ðnÞ ¼ Grd ;
ðA:36Þ
Y ðnÞ ¼ d vb ; ZðnÞ ¼ d2 ud ; SðnÞ ¼ Qb d
ðA:37Þ ðA:38Þ ðA:39Þ
are defined and used to transform the profile expressions and the integral equations into a more easily manipulated mathematical system. New velocity profile is, 2 3 4 1ðn; gÞ ¼ a 2 g þ a3 g þ a4 g þ a 5 g ;
Substitution of 1 and g into Eq. (A.8) yields, Z 1 d 1 1 dg ¼ vb : dn d 0
Substitution of Y and the relation
ðA:34Þ
hðn; gÞ ¼ hðn; y Þ;
ðA:41Þ
1 b 2 ¼ ½ðPrH i Y SÞðPrY 6Þ 12; 6 1 b4 ¼ ½ðPrH i Y SÞðPrY 4Þ 4; 2 1 b5 ¼ ½ðPrH i Y SÞðPrY 3Þ 3: 3
ðA:29Þ
and a set of dependent variables,
3 4 hðn; gÞ ¼ 1 þ b 2 g þ b4 g þ b5 g ;
ðA:28Þ
ðA:27Þ
ðA:33Þ
1ðn; gÞ ¼ d2 u ;
New temperature profile is,
Differentiation and multiplication by d* yields, Z 1 Z d 1 dd 1 1 dg 1 dg ¼ d vb : dn 0 d dn 0
In order to simplify the analysis, a new independent variable, g ¼ y =d
1 2 a 2 ¼ ½ZðY 6Y þ 12Þ; 6 1 a 3 ¼ ½X ; 2 1 2 a 4 ¼ ½2X ZðY 4Y þ 4Þ; 2 1 2 a 5 ¼ ½3X 2ZðY 3Y þ 3Þ: 6
where
The velocity and temperature profile polynomials given by Eqs. (A.11) and (A.12) are substituted into the above listed non-dimensional boundary conditions and the coefficients ai ; bi are determined as follows: a1 ¼ 0; 1 a2 ¼ ½ud ðd2 v2 b 6d vb þ 12Þ=d ; 6 1 a3 ¼ ½Grd2 =d2 ; 2 1 3 a4 ¼ ½2Grd2 ud ðd2 v2 b 4d vb þ 4Þ=d ; 2 1 4 a5 ¼ ½3Grd2 2ud ðd2 v2 b 3d vb þ 3Þ=d ; 6 b1 ¼ 1; 1 b2 ¼ ½d ðPrH i vb Qb ÞðPrd vb 6Þ 12=d ; 6 b3 ¼ 0; 3 1 b4 ¼ ½d ðPrH i vb Qb ÞðPrd vb 4Þ 4=d ; 2 4 1 b5 ¼ ½d ðPrH i vb Qb ÞðPrd vb 3Þ 3=d : 3
217
ðA:40Þ
ðA:42Þ
and multiplication by 4X resulted in the following form of the continuity equation: Z 1 Z 1 d dX 4X 1 dg 1 dg ¼ 4XY ðA:43Þ dn 0 dn 0 Similarly, substitution of 1 and g into Eq. (A.9), differentiation and multiplication by d*, substitution of X and Eq. (A.42), and multiplication by 4X resulted in the following form of the momentum equation: Z 1 Z d dX 1 2 o1 2 4X 1 dg 3 1 dg ¼ 4X X : dn 0 dn 0 og w ðA:44Þ Substitution of 1 and g into Eq. (A.10), differentiation and multiplication by d*, substitution of Y, S, and Eq. (A.42), and multiplication by 4X resulted in the following form of the energy equation: Z 1 Z 1 d dX 4X 1h dg 1h dg dn 0 dn 0 4X oh ðPrH i Y SÞ ¼ : ðA:45Þ Pr og w
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A.2. Reduction of the integral equations to ordinary differential equations
where
A.2.1. The continuity equation Integration of Eq. (A.40) yields Z 1 1 ½2X þ 3ZC1 1 dg ¼ 120 0
A22 ¼ 8X ½3XZC5 Z 2 C6 ;
A21 ¼ 3½30X 2 XZC3 2Z 2 C4 ;
ðA:46Þ
A23 ¼ 4X ½3X C3 4ZC4 ; B2 ¼ 12X ½2520ð6X ZC7 Þ and
where C1 = (Y2 8Y + 28). Differentiation of Eq. (A.46) gives Z 1 d 1 dC1 0 2X 0 þ 3Z 1 dg ¼ Y þ ð3C1 ÞZ 0 : dn 0 120 dY
C5 ¼ ð17Y 66Þ; C6 ¼ ð76Y 3 835Y 2 þ 3810Y 6768Þ; C7 ¼ ðY 2 6Y þ 12Þ: ðA:47Þ
Here, the prime notation indicates differentiation with respect to n. The following ordinary differential form of the continuity equation was then obtained by substituting Eqs. (A.46) and (A.47) into Eq. (A.43) and collecting terms.
A.2.3. The energy equation Differentiation of Eq. (A.41) yields oh 1 ¼ ½ðPrH i Y SÞðPrY 6Þ 12: og w 6
A11 X 0 þ A12 Y 0 þ A13 Z 0 ¼ B1 ;
Multiplication of Eq. (A.40) by Eq. (A.41) and integration of the result gives Z 1 1 ½3X C8 þ 2ZC9 ; 1h dg ¼ ðA:54Þ 90720 0
ðA:48Þ
where A11 ¼ 3½2X þ ZC1 ; A13 ¼ 12X ½C1 ;
A12 ¼ 24X ½ZC2 ;
B1 ¼ 8X ½60Y
and C2 = (Y 4).
where
A.2.2. The momentum equation Differentiation of Eq. (A.40) yields o1 1 ¼ ½ZðY 2 6Y þ 12Þ: og w 6
C8 ¼ ½ðPrH i Y SÞð17PrY 132Þ þ 120;
ðA:53Þ
C9 ¼ ½ðPrH i Y SÞfPrY ð38Y 2 285Y þ 789Þ ð285Y 2 2232Y þ 6768Þg
ðA:49Þ
The squaring of Eq. (A.40) and integration of the result gives Z 1 1 ½16X 2 3X C3 þ 2Z 2 C4 ; 12 dg ¼ ðA:50Þ 45360 0 where
þ ð345Y 2 2304Y þ 5328Þ: Differentiation of Eq. (A.54) yields Z 1 d 1 oC8 oC9 0 ð3C8 ÞX 0 3X 2Z 1h dg ¼ S dn 0 90720 oS oS oC8 oC9 0 3X 2Z Y þ ð2C9 ÞZ 0 : oY oY ðA:55Þ
C3 ¼ ð17Y 2 132Y þ 384Þ; C4 ¼ ð19Y 4 285Y 3 þ 1905Y 2 6768Y þ 13212Þ: Differentiation of Eq. (A.50) yields Z 1 d 1 2 1 dg ¼ 3ð12X ZC3 ÞX 0 dn 0 45360 dC3 dC4 0 2Z 2 Y ð3X C3 4ZC4 ÞZ 0 : 3XZ dY dY ðA:51Þ
The following ordinary differential form of the energy equation was then obtained by substituting Eqs. (A.53)– (A.55) and the relation: S0 ¼
S 0 X 4X
into Eq. (A.45) A31 X 0 þ A32 Y 0 þ A33 Z 0 ¼ B3 ;
The following ordinary differential form of the momentum equation was then obtained by substituting Eqs. (A.49)– (A.51) into Eq. (A.44) and collecting terms:
A31 ¼ ½3X C10 þ 2ZC11 ;
A21 X 0 þ A22 Y 0 þ A23 Z 0 ¼ B2 ;
A32 ¼ 4X ½3X C12 2ZC13 ;
ðA:52Þ
ðA:56Þ
where
ðA:57Þ
K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
A11 X 0 þ A12 Y 0 þ A13 Z 0 ¼ B1 ; A21 X 0 þ A22 Y 0 þ A23 Z 0 ¼ B2 ;
A33 ¼ 8X ½C9 ; B3 ¼ 12X ½5040C14 ; C10 ¼ ½ð3PrH i Y 4SÞð17PrY 132Þ þ 360; C11 ¼ ½PrH i Y fPrY ð38Y 2 285Y þ 789Þ ð285Y 2 2232Y þ 6768Þg; þ ð345Y 2 2304Y þ 5328Þ C12 ¼ Pr½17ð2PrH i Y SÞ 132H i ; 2
219
A31 X 0 þ A32 Y 0 þ A33 Z 0 ¼ B3 :
References 2
C13 ¼ ½PrH i fPrY ð152Y 855Y þ 1578Þ 9ð95Y 496Y þ 752Þg; 3SfPrð38Y 2 190Y þ 263Þ 2ð95Y 372Þg þ 6ð115Y 384Þ C14 ¼ ½ðPrH i Y SÞðPrY 12Þ 12=Pr:
By the above mentioned procedure, the fundamental Eqs. (A.8)–(A.10) were transformed into the following ordinary differential equations:
[1] Rivers WJ, McFadden PW. Film free convection in helium II. ASME J Heat Transfer 1966;88(4):343–50. [2] Labuntzov DA, Ametistov YeV. Analysis of helium II film boiling. Cryogenics 1979;19:401–4. [3] Tatsumoto H, Hata K, Hama K, Shirai Y, Shiotsu M. Critical heat flux on a flat plate in pressurized He II. Cryogenics 2001; 41(1):35–8.
Film boiling on a vertical plate in subcooled helium II K. Hama *, M. Shiotsu Graduate School of Energy Science, Kyoto University, Uji, Kyoto 611-0011, Japan Received 28 November 2005; received in revised form 26 December 2006; accepted 27 December 2006
Abstract Film boiling heat transfer coefficients were measured on 10, 30 and 50 mm long vertical plates in subcooled He II for bulk liquid temperatures from 1.8 to 2.1 K. A film boiling model on a vertical plate in subcooled He II was presented based on convection heat transport in the vapor film, radiation heat transport, and heat transport in He II. The numerical solutions of the model were obtained and an equation which can express the numerical solutions within ±5% difference was derived. The equation predicted well the experimental data for lower DT range but significantly under-predicted the data for higher DT. A correlation of film boiling heat transfer including radiation contribution was presented by modifying the equation based the experimental data. This correlation can describe the experimental data within ±20% difference. Ó 2007 Published by Elsevier Ltd. Keywords: Film boiling (C); Heat transfer (C); Superfluid helium (He II) (B)
1. Introduction There are many experimental researches of film boiling on a horizontal cylinder in He II, but research on film boiling heat transfer from a vertical plate in subcooled He II is hardly done, as far as the authors are concerned. Some workers presented analytical model of film boiling heat transfer from a vertical plate in saturated He II. Rivers and McFadden [1] analyzed the vertical problem by the same method as they used for horizontal cylinder. However, there have been few theoretical works for a vertical plate in subcooled He II. The purposes of this study are fourfold. First is to obtain the experimental data of film boiling heat transfer from vertical plates with various lengths in subcooled He II for the liquid temperature from 1.8 to 2.1 K. Second is to present a film boiling model for a vertical plate in subcooled He II. Third is to present a film boiling heat transfer equation based on the numerical solutions for the model and the experimental data. Fourth is to make clear the *
Corresponding author. Tel.: +81 774 38 4453; fax: +81 774 38 4451. E-mail address: [email protected] (K. Hama).
0011-2275/$ - see front matter Ó 2007 Published by Elsevier Ltd. doi:10.1016/j.cryogenics.2006.12.004
availability of the equation by comparing the experimental data and the predicted values. 2. Apparatus and method The experimental apparatus is shown schematically in Fig. 1. The cryostat is the Claudet type, 45 cm in inner diameter and 157 cm in height. About 150 l of liquid helium is initially contained in the vessel. The inner bath is divided by a glass-epoxy-separator (3) with He I in the upper section and He II in the lower section. The bath above the separator contains saturated He I of about 74 l with a free surface. The bath below the separator contains subcooled He II of about 76 l with no free surface. Four vertical test heaters can be mounted (1) in the lower bath (He II bath) and there is a ring shaped helium boiler (4) and a heat exchanger (6). Liquid helium is provided from the upper bath (He I bath) through the heat exchanger and a motor-controlled Joule–Thomson valve (JT valve) (5) to the ring shaped boiler. By evacuating liquid helium in the boiler with a vacuum pump (10,000 l/min), the saturated liquid helium in the heat exchanger can be cooled below k-point. The liquid helium in the lower part
210
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Nomenclature ai bi cp Gr g Hi h hc hr k L l Nu Pr Qb q qb qb qc qr R
coefficients of 4th degree velocity profile polynomial, Eq. (17) coefficients of 4th degree temperature profile polynomial, Eq. (18) specific heat at constant pressure (J/kg K) Gr ¼ gl3 qv ðqb qv Þ=l2v ; Grashof number gravity acceleration (m/s2) Hi = L/(cpvDTsat), interface enthalpy parameter h = q/DTsat, average film boiling heat transfer coefficient (W/m2 K) hc = qc/DTsat, conduction–convection heat transfer coefficient (W/m2 K) hr = qr/DTsat, radiation heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) latent heat of vaporization (J/kg) length of test heater (m) Nu = hcl/kv, Nusselt number Pr = lvcpv/kv, Prandtl number Qb = l(qb qr)/(kvDTsat), interface heat flux parameter q = qc + qr, surface heat flux (W/m2) heat flux on the vapor–liquid interface (W/m2) heat flux on the He I–He II interface (W/m2) conduction–convection contribution of surface heat flux (W/m2) radiation of surface heat flux (W/m2) contribution h 7 1 1 R ¼ Qb Gr4 þ 0:9fðH i þ 0:25ÞPrg4 i7 47 1 , non-dimensional þ0:28 Qb Gr4 parameter
of bath will be cooled to a desired temperature below kpoint by the boiler and a spiral heat exchanger below the boiler. The liquid temperature in the He II bath is measured by calibrated Ge resistance thermometer. Liquid level in the heat exchanger boiler is measured by superconducting level sensor and is automatically kept constant at a desired value by the motor-controlled JT valve. The He II bath is connected to the He I bath by the super leak of He II because the superfluid component with no viscosity can flow through the narrow gap between the glass-epoxy separator and the wall of the inner bath. Heat loss through the gap is little, because normal component cannot go though the gap. Therefore the pressure in the He I bath can be propagated into the He II bath independent of the He II temperature. As a consequence, the cryostat shown in Fig. 1 can realize subcooled He II and be kept constant temperature when there is some heat penetration under subcooled conditions. Four test heaters called (a)–(d) were used in this experiment. The test heaters (a)–(c) are made of Au–Mn
T u v w x y
temperature (K) velocity component (m/s) velocity component (m/s) width of flat plate heater (m) tangential coordinate (m) perpendicular coordinate (m)
Greek symbols a absorptivity d thickness of laminar vapor layer (m) dI thickness of liquid He I layer (m) DT DT = Tw Tb (K) DTsat DTsat = Tw Tsat, surface superheat (K) ew radiation emissivity l viscosity (N s/m2) q density (kg/m3) rs Stefan–Boltzman constant Subscripts b bulk I He I layer sat saturation v vapor w wall d at vapor–liquid interface k at lambda point
(0.25 wt.%) alloy of thickness 0.03 mm, width 2.5 mm, and about 20, 30 and 50 mm in length, respectively. Test heater (d) is made of 316# stainless steel (SUS 316) with 3.3 mm in width, 50 mm in length, and 0.1 mm thickness. Two fine 0.05 mm diameter platinum wires were spotwelded as potential taps at about 5 mm from each end of the test heaters of (a), (b) and (d). To investigate the local heat transfer characteristics along the heater length, four same platinum wires were spot-welded on the test heaters (c) as potential taps. The first one was positioned at 7 mm from the leading edge, and the second to fourth ones were spot-welded at every 10 mm. The gradient of the electrical resistivity versus temperature curve for the Au–Mn alloy is positive from He II temperature to several hundred Kelvin. The gradient of the electrical resistivity versus temperature curve for SUS 316 is positive for the film boiling range from around 20 K to several hundred Kelvin. The test plates were annealed in an inert atmosphere and their electrical resistance versus temperature relations were calibrated in a temperature-
K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
211
Fig. 1. Schematic illustration of experimental apparatus.
controlled vessel by immersion in liquid helium and liquid nitrogen for temperatures up to 100 K. The calibrations for the temperatures from 290 K to 373 K and from 373 K to 453 K were performed in water and Glycerin baths, respectively. The calibration accuracy was estimated to be ±0.1 K for the temperature below 10 K for the Au–Mn alloy, and ±0.5% for higher temperatures for the Au–Mn alloy and SUS 316. Test heaters were heated by direct current from a power amplifier. The average temperature of the test heater between potential taps were measured by the resistance thermometry using a double bridge circuit including the test heater as a branch. The voltage drops across the potential taps of the test heater and across a standard resistance were amplified and passed to the analog to digital converters of a digital computer. The temperature and heat flux were obtained with the aid of previously calibrated resistance– temperature relation. The heater surface temperature was calculated from the measured average temperature and heat generation rate for each interval between the potential taps by solving the thermal conduction equation for the heater assuming uniform heat flux between the intervals. Experimental error of each interval is estimated to be ±(0.1 K + 0.8% of the measured value) for the surface temperature and to be ±2% for the heat flux of both material.
Fig. 2. Steady-state heat transfer for l = 47.3 mm type (c) test plate.
ture Tb as a parameter. We can see from this figure that in the region where DT is between critical heat flux and around 100 K, heat flux for each Tb increases very gradually, which is followed by the region where heat flux increases rapidly. The heat flux in the former region increases significantly with the decrease in Tb from 2.1 K to 2.0 K and increases gradually with further decrease in Tb. The difference in the value of the heat flux of various Tb becomes smaller as DT increases in the latter region. 3.2. Film boiling heat transfer coefficients
3. Experimental results 3.1. Heat transfer curve Fig. 2 shows typical example of steady-state heat transfer curves obtained from potential taps 1–4 of the type (c) vertical plate in subcooled He II with bulk liquid tempera-
The average film boiling heat transfer coefficients, h, were measured for bulk liquid temperatures, Tb, from 1.8 to 2.1 K under atmospheric pressure. The data for the type (a) test plate are shown versus surface superheat, DTsat in Fig. 3 with bulk liquid temperature as a parameter. The heat transfer coefficients for a constant
212
K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
Fig. 3. Average heat transfer coefficients versus heater surface superheat for l = 18 mm type (a) test plate.
Fig. 5. Average heat transfer coefficients versus heater surface superheat for l = 47.3 mm type (c) test plate.
DTsat are higher at lower value of Tb. The heat transfer coefficients for each Tb significantly decrease first and then become almost constant as the surface superheat increases. The data for longer test plates, type (b), (c) and (d), are shown versus surface superheat in Figs. 4–6, respectively, with liquid temperature as a parameter. The trend of dependence on the surface superheat and liquid temperature is similar to that for the type (a): the heat transfer coef-
Fig. 6. Average heat transfer coefficients versus heater surface superheat forl = 47 mm type (d) test plate.
Fig. 4. Average heat transfer coefficients versus heater surface superheat for l = 30 mm type (b) test plate.
ficients are higher for lower values of DTsat and Tb. Systematic effect of heater length on heat transfer coefficient cannot be seen for the range of heater length tested. Fig. 7 shows the heat transfer coefficients for the type (c) test plate in 1.8 K He II measured by the potential taps between 1–2, 2–3, 3–4 and 1–4. The first potential tap is positioned at 7 mm from the leading edge, and the second to fourth ones are fixed every 10 mm. A variation of local heat transfer coefficients cannot be seen in this figure.
K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
Fig. 7. Local heat transfer coefficients versus heater surface superheat for type (c) test plate.
4. Numerical analysis
Fig. 8. Physical model and coordinate system.
fluxes, qcr, on a flat plate with length l and width w in subcooled He II
4.1. Film boiling model qb ¼ A model for film boiling on a vertical plate in subcooled He II was considered based on the following assumptions: (1) Surface temperature of the vertical plate, Tw, is uniform. (2) Thickness of vapor film is negligible compared with the width of the plate. (3) The convection, which occurs within the film, is steady two-dimensional and boundary layer type flow. According to these assumptions, the model for subcooled He II film boiling on a vertical plate is illustrated schematically in Fig. 8. In this case, the vapor layer is surrounded by a thin He I layer and bulk He II outside it exists. Although the temperature gap due to non-equilibrium vaporization and condensation suggested by Labuntzov and Ametistov [2] may exist at the vapor–He I interface, there would be no temperature gap at the He I–He II interface with the temperature of Tk. Then, the heat flux, qb , removed from the He I–He II interface to the bulk He II is considered to be equal to the critical heat flux. As the thickness of the He I layer, shown very much enlarged in the figure, is estimated from dI ¼ k I ðT sat T k Þ=qb to be thinner than 1 lm for the bulk temperatures lower than 2.1 K, the heat flux qb removed from the vapor–He I interface can be dealt with equal to qb . Therefore the value of qb is given by the following equation presented by Tatsumoto et al. [3] for the critical heat
213
qb
2 ¼ 0:58 lw=f2ðl þ wÞg 1
where f ðT Þ
Z
Tk
1
13
f ðT Þ dT
;
ð1Þ
Tb 3
6:8 ¼ gðT k Þ½T 6:8 R ð1 T R Þ
gðT k Þ ¼ q2 s4k T 3k =Ak ; T R ¼ T =T k ; sk ¼ 1559 J=ðkg KÞ; Ak ffi 1150 ms=kg: Surface heat flux in film boiling is expressed by q ¼ qc þ q r ;
ð2Þ
where qc is the conduction–convection contribution and qr is the radiation contribution of the total film boiling heat flux. The radiation heat flux qr is given by rs ðT 4 T 4sat Þ; qr ¼ ð3Þ 1=ew þ 1=a 1 w where ew and a are the emissivity and absorptivity of radiation. The ew for the Au–Mn plate is 0.03 and that for SUS316 is 0.4, and a is taken to be unity. The laws of conservation of mass, momentum and energy are expressed as follows in the form of boundary layer equations: ou ov þ ¼ 0; ox oy
ð4Þ
u
ou ou 1 l o2 u þ v ¼ gðqb qv Þ þ v 2 ; ox oy qv qv oy
ð5Þ
u
oT oT k v o2 T þv ¼ : ox oy qv cpv oy 2
ð6Þ
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The boundary conditions are expressed in the following form: At the vertical plate surface (y = 0), u ¼ v ¼ 0;
ð7Þ
T ¼ T w:
ð8Þ
At the vapor liquid interface (y = d), u ¼ ud ; v ¼ vd ;
ð9Þ ð10Þ
T ¼ T sat ;
ð11Þ
dd ; qb vb ¼ qv vd ud dx ou ¼ qb vb ud ; lv oy d oT kv ¼ qb vb L þ qb qr : oy d
ð12Þ ð13Þ ð14Þ
Boundary layer equations (Eqs. (4)–(6)) were transformed into the following integrodifferential form by first integrating partial differential equations from 0 to d and then substituting the interface boundary conditions into the result Z d d qv u dy ¼ qb vb ; ð15Þ dx 0 Z d Z d d ou 2 qv u dy ¼ gðqb qv Þ dy lv ; ð16Þ dx 0 oy w 0 Z d d qv cpv uðT T sat Þ dy dx 0 oT ¼ qb vb L qb þ qr k v : ð17Þ oy w The velocity and temperature profiles are given approximately by the following fourth degree polynomials: uðx; yÞ ¼ a1 þ a2 y þ a3 y 2 þ a4 y 3 þ a5 y 4 ;
ð18Þ
T ðx; yÞ ¼ b1 þ b2 y þ b3 y 2 þ b4 y 3 þ b5 y 4 :
ð19Þ
Expressions for the coefficients ai and bi are obtained as functions of x by applying the boundary conditions of u and T in the y-direction. The integral equations, the profile expressions and the boundary conditions described above were non-dimensionalized. The characteristic of the phenomena is dependent on the following five dimensionless parameters: the Prandtl number, Pr = lvcpv/kv, average Nusselt number of characteristic dimension l for plate, Nu = hcl/kv, Grashof number, Gr ¼ gl3 qv ðqb qv Þ=l2v , the ‘‘interface enthalpy’’ parameter, Hi = L/(cpvDTsat), and the interface heat flux parameter, Qb = l(qb qr)/(kvDTsat). The Qb and Gr always appear together in the form of QbGr1/4. Using the profile expressions with the formulated coefficients ai and bi, the internal equations were reduced to three first-order ordinary differential equations. Numerical solutions for the differential equations were obtained for the QbGr1/4 ranging from 0.01 to 10 by using Runge–Kutta method. Calculated results are shown in Fig. 9 on Nu Gr1/4 versus Qb Gr1/4 graph with Hi as a parameter. Details on the derivation of the ordinary differential equations with the boundary conditions are described in Appendix A. It seems that the results for a fixed value of Hi are very weakly dependent on Qb for QbGr1/4 lower than 0.1, and they approach a line of Nu = Qb with increasing QbGr1/4. The former is the convection dominant region where vapor film thickness is relatively thick, and the latter is the conduction dominant region where vapor film thickness is relatively thin. The results for Hi ranging from 0.01 to 100 obtained here were fitted by the following equation: Nu ¼ ðGr RÞ
1=4
;
where h 1 1 7 R ¼ Qb Gr4 þ 0:9fðH i þ 0:25ÞPrg4 i7 47 14 : þ0:28 Qb Gr
Following boundary conditions were introduced to solve the problem as done by Rivers and McFadden [1]. At y = 0, 2 ou lv ¼ gðqb qv Þ; ð20Þ oy 2 w 2 oT ¼ 0: ð21Þ kv oy 2 w At y = d, 2 ou l2v ¼ q2b v2b ud lv gðqb qv Þ; oy 2 d 2 2 o T ¼ cpv qb vb ðqb vb L qb þ qr Þ: kv oy 2 d
ð22Þ ð23Þ Fig. 9. Calculated results of average heat transfer coefficients.
ð24Þ
K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
215
The values given by Eq. (24) are shown in Fig. 9 in comparison with the numerical solutions. The equation expressed the numerical solutions within ±5% accuracy. The values of q predicted from Eq. (1) combined with Eqs. (2) and (3) and (24) are compared with the corresponding experimental data in Figs. 3–6. It can be seen that this equation predicts the experimental data in good accuracy for lower DTsat range but under-predicts the data with the increase in DTsat.
Gr R. This difference may be due to the heat transfer from both sides of vapor film neglected in the assumption (2) in the model.
5. Correlation of film boiling heat transfer from a vertical plate in subcooled He II
The curve of Eq. (25) is shown in Figs. 10 and 11 for comparison. It seems that the curve fitting by the least square method is reasonable. Experimental data of film boiling heat transfer coefficients for the liquid temperature of 1.8, 1.9, 2.0, 2.1 K previously shown in Figs. 3–6 are compared with the values predicted from Eq. (25). It can be seen from these figures that all the experimental data on four different sized heaters in subcooled He II for wide range of surface superheat obtained here are in good agreement with the predicted values within ±20%.
5.1. Comparison of the experimental data with the analytical solution Figs. 10 and 11 show the experimental data for various liquid temperatures on type (a) and (d) test heaters, respectively, plotted on log(Nu) versus log(Gr R) graph. It can be seen from these figures that the experimental data are on a single straight line almost independent of the bulk liquid temperature. The values given by Eq. (24) are also shown on these graphs as a straight line with the gradient of 1/4 for comparison. The data for high Gr R range agree well with the predicted values, but gradually they become higher than the predicted values with the decrease in
5.2. Derivation of the film boiling heat transfer correlation The Eq. (24) was modified as follows based on the experimental data for the test heaters. Nu ¼ 2:5ðGr RÞ
0:23
ð25Þ
5.3. Radiation effect To see the radiation effect, the predicted values of film boiling heat transfer coefficients for the vertical plates with several ew and with the same dimension as type (d) heater in 2.0 K He II are shown on a linear graph in Fig. 12 against DTsat up to 1100 K. The experimental data for the type (d) test heater with ew of 0.4 are also shown in the figure for comparison. We can see that the predicted heat transfer coefficient at DTsat = 800 K for the plate with ew of 1.0 is only 4.2% higher than that for ew of 0. Experimental datum
Fig. 10. Comparison of the experimental data for type (a) test plate with the values given by Eqs. (24) and (25).
Fig. 11. Comparison of the experimental data for type (d) test plate with the values given by Eqs. (24) and (25).
Fig. 12. Predicted values of h for vertical plates with several ew in 2.0 K He II in comparison with the experimental data for type (d) test plate.
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K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
Table 1 Calculated components in film boiling heat transfer coefficient for various radiation emissivity TB = 2.0 K, qb = 6.74E4 (W/m2) ew
DTsat (K)
hr (W/m2 K)
hc (W/m2 K)
h (W/m2 K)
0
500 800 1100
0 0 0
368 364 368
368 364 368
0.4
500 800 1100
2.9 11.3 30.4
366 358 353
369 370 383
500 800 1100
7.2 29.3 76.0
364 350 331
371 379 407
1.0
at DTsat = 760 K is within the small range. The predicted heat transfer coefficients at DTsat = 1100 K for the plates with ew of 0.4 and 1.0 are 3.3% and 8.2% higher than that for ew of 0. This does not mean the hr are such percentage of h. Table 1 shows the calculated values of hr , hc and h on plates with ew of 0, 0.4 and 1.0 in 2.0 K He II at DTsat of 500, 800 and 1100 K. At DTsat = 1100 K, for instance, the hr on the plate with ew of 1.0 is about 19% of the h. The value of hc becomes smaller with the increase of ew (namely, with the increase of hr) due to the increased thickness of vapor film. The film boiling heat transfer correlation for a vertical plate in He II including the radiation effect was derived firstly by modifying slightly the solution of numerical model based on our experimental data. This correlation is necessary to be built into a computer code in order to analyze dynamic stability of superconducting magnets. Initial local spot of thermal disturbance on the superconductor due to the conductor motion, etc. would be the order of dimension of the test plate in this experiment. As there are few experimental data by other workers, comparison with them cannot be made. Further study is necessary to clarify the applicability range of this correlation. 6. Conclusions Film boiling heat transfer coefficients were measured on 10, 30 and 50 mm long vertical plates in subcooled He II for bulk liquid temperature ranging from 1.8 to 2.1 K. It was confirmed that little dependence on heater length and width exists on the film boiling heat transfer from a vertical plate under the present experimental conditions. A model for film boiling on a vertical plate in subcooled He II was presented based on convection heat transport in the vapor film, radiation heat transport to the vapor–liquid interface, and the heat transport in the He II. The numerical solutions of the model were obtained for several fixed parameters and an equation which can express the numerical solutions within ±5% difference was derived. The equation predicted well the experimental data for lower DT range but under-predicted the data for higher DT.
A correlation of film boiling heat transfer including radiation effect was presented firstly based on the numerical solutions of the model and the experimental data. It was confirmed that this correlation can describe our experimental data in subcooled He II within ±20% difference. Further study is necessary to clarify the applicability range of this correlation. Appendix A. Transformation of the fundamental equations to ordinary differential equations A.1. Non-dimensionalization of the system The following dimensionless variables were introduced to non-dimensionalize this mathematical system: n ¼ x=l; y ¼ y=l; q lu u ¼ v ; lv T T sat ; h¼ T w T sat d d ¼ ; l q lud ud ¼ v ; lv q lvb vb ¼ b : lv
ðA:1Þ ðA:2Þ ðA:3Þ ðA:4Þ ðA:5Þ ðA:6Þ ðA:7Þ
Substitutions of the above definitions into Eqs. (15)–(17) leads to Eqs. (A.8)–(A.10), respectively. d dn d dn d dn
Z
d
u dy ¼ vb ;
ðA:8Þ
0
d ou 2 u dy ¼ Grd ; oy w 0 Z d 1 oh PrH i vb Qb u h dy ¼ ; Pr oy w 0 Z
ðA:9Þ ðA:10Þ
where u* and h are velocity and temperature profile polynomials, u ¼ a1 þ a2 y þ a3 y 2 þ a4 y 3 þ a5 y 4 ; h¼
b1
þ
b2 y
þ
b3 y 2
þ
b4 y 3
þ
b5 y 4 ;
ðA:11Þ ðA:12Þ
where ai and bi are non-dimensional forms of ai and bi. The boundary conditions at the wall y = 0, u ¼ 0; h ¼ 1; 2 ou ¼ Gr; oy 2 w 2 oh ¼0 oy 2 w
ðA:13Þ ðA:14Þ ðA:15Þ ðA:16Þ
K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
The boundary conditions at the interface y = d,
ðu Þd ¼
ud ;
where ðA:17Þ
ðhÞd ¼ 0; ou ¼ vb ud ; oy d oh ¼ PrH i vb Qb ; oy d 2 ou ¼ ud v2 b Gr; oy 2 d 2 oh ¼ Prvb ðPrH i vb Qb Þ: oy 2 d
ðA:18Þ ðA:19Þ ðA:20Þ ðA:21Þ ðA:22Þ
ðA:23Þ ðA:24Þ ðA:25Þ ðA:26Þ
ðA:30Þ
1 dd 1 dX ¼ d dn 4X dn
ðA:31Þ ðA:32Þ
ðA:35Þ
4
X ðnÞ ¼ Grd ;
ðA:36Þ
Y ðnÞ ¼ d vb ; ZðnÞ ¼ d2 ud ; SðnÞ ¼ Qb d
ðA:37Þ ðA:38Þ ðA:39Þ
are defined and used to transform the profile expressions and the integral equations into a more easily manipulated mathematical system. New velocity profile is, 2 3 4 1ðn; gÞ ¼ a 2 g þ a3 g þ a4 g þ a 5 g ;
Substitution of 1 and g into Eq. (A.8) yields, Z 1 d 1 1 dg ¼ vb : dn d 0
Substitution of Y and the relation
ðA:34Þ
hðn; gÞ ¼ hðn; y Þ;
ðA:41Þ
1 b 2 ¼ ½ðPrH i Y SÞðPrY 6Þ 12; 6 1 b4 ¼ ½ðPrH i Y SÞðPrY 4Þ 4; 2 1 b5 ¼ ½ðPrH i Y SÞðPrY 3Þ 3: 3
ðA:29Þ
and a set of dependent variables,
3 4 hðn; gÞ ¼ 1 þ b 2 g þ b4 g þ b5 g ;
ðA:28Þ
ðA:27Þ
ðA:33Þ
1ðn; gÞ ¼ d2 u ;
New temperature profile is,
Differentiation and multiplication by d* yields, Z 1 Z d 1 dd 1 1 dg 1 dg ¼ d vb : dn 0 d dn 0
In order to simplify the analysis, a new independent variable, g ¼ y =d
1 2 a 2 ¼ ½ZðY 6Y þ 12Þ; 6 1 a 3 ¼ ½X ; 2 1 2 a 4 ¼ ½2X ZðY 4Y þ 4Þ; 2 1 2 a 5 ¼ ½3X 2ZðY 3Y þ 3Þ: 6
where
The velocity and temperature profile polynomials given by Eqs. (A.11) and (A.12) are substituted into the above listed non-dimensional boundary conditions and the coefficients ai ; bi are determined as follows: a1 ¼ 0; 1 a2 ¼ ½ud ðd2 v2 b 6d vb þ 12Þ=d ; 6 1 a3 ¼ ½Grd2 =d2 ; 2 1 3 a4 ¼ ½2Grd2 ud ðd2 v2 b 4d vb þ 4Þ=d ; 2 1 4 a5 ¼ ½3Grd2 2ud ðd2 v2 b 3d vb þ 3Þ=d ; 6 b1 ¼ 1; 1 b2 ¼ ½d ðPrH i vb Qb ÞðPrd vb 6Þ 12=d ; 6 b3 ¼ 0; 3 1 b4 ¼ ½d ðPrH i vb Qb ÞðPrd vb 4Þ 4=d ; 2 4 1 b5 ¼ ½d ðPrH i vb Qb ÞðPrd vb 3Þ 3=d : 3
217
ðA:40Þ
ðA:42Þ
and multiplication by 4X resulted in the following form of the continuity equation: Z 1 Z 1 d dX 4X 1 dg 1 dg ¼ 4XY ðA:43Þ dn 0 dn 0 Similarly, substitution of 1 and g into Eq. (A.9), differentiation and multiplication by d*, substitution of X and Eq. (A.42), and multiplication by 4X resulted in the following form of the momentum equation: Z 1 Z d dX 1 2 o1 2 4X 1 dg 3 1 dg ¼ 4X X : dn 0 dn 0 og w ðA:44Þ Substitution of 1 and g into Eq. (A.10), differentiation and multiplication by d*, substitution of Y, S, and Eq. (A.42), and multiplication by 4X resulted in the following form of the energy equation: Z 1 Z 1 d dX 4X 1h dg 1h dg dn 0 dn 0 4X oh ðPrH i Y SÞ ¼ : ðA:45Þ Pr og w
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K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
A.2. Reduction of the integral equations to ordinary differential equations
where
A.2.1. The continuity equation Integration of Eq. (A.40) yields Z 1 1 ½2X þ 3ZC1 1 dg ¼ 120 0
A22 ¼ 8X ½3XZC5 Z 2 C6 ;
A21 ¼ 3½30X 2 XZC3 2Z 2 C4 ;
ðA:46Þ
A23 ¼ 4X ½3X C3 4ZC4 ; B2 ¼ 12X ½2520ð6X ZC7 Þ and
where C1 = (Y2 8Y + 28). Differentiation of Eq. (A.46) gives Z 1 d 1 dC1 0 2X 0 þ 3Z 1 dg ¼ Y þ ð3C1 ÞZ 0 : dn 0 120 dY
C5 ¼ ð17Y 66Þ; C6 ¼ ð76Y 3 835Y 2 þ 3810Y 6768Þ; C7 ¼ ðY 2 6Y þ 12Þ: ðA:47Þ
Here, the prime notation indicates differentiation with respect to n. The following ordinary differential form of the continuity equation was then obtained by substituting Eqs. (A.46) and (A.47) into Eq. (A.43) and collecting terms.
A.2.3. The energy equation Differentiation of Eq. (A.41) yields oh 1 ¼ ½ðPrH i Y SÞðPrY 6Þ 12: og w 6
A11 X 0 þ A12 Y 0 þ A13 Z 0 ¼ B1 ;
Multiplication of Eq. (A.40) by Eq. (A.41) and integration of the result gives Z 1 1 ½3X C8 þ 2ZC9 ; 1h dg ¼ ðA:54Þ 90720 0
ðA:48Þ
where A11 ¼ 3½2X þ ZC1 ; A13 ¼ 12X ½C1 ;
A12 ¼ 24X ½ZC2 ;
B1 ¼ 8X ½60Y
and C2 = (Y 4).
where
A.2.2. The momentum equation Differentiation of Eq. (A.40) yields o1 1 ¼ ½ZðY 2 6Y þ 12Þ: og w 6
C8 ¼ ½ðPrH i Y SÞð17PrY 132Þ þ 120;
ðA:53Þ
C9 ¼ ½ðPrH i Y SÞfPrY ð38Y 2 285Y þ 789Þ ð285Y 2 2232Y þ 6768Þg
ðA:49Þ
The squaring of Eq. (A.40) and integration of the result gives Z 1 1 ½16X 2 3X C3 þ 2Z 2 C4 ; 12 dg ¼ ðA:50Þ 45360 0 where
þ ð345Y 2 2304Y þ 5328Þ: Differentiation of Eq. (A.54) yields Z 1 d 1 oC8 oC9 0 ð3C8 ÞX 0 3X 2Z 1h dg ¼ S dn 0 90720 oS oS oC8 oC9 0 3X 2Z Y þ ð2C9 ÞZ 0 : oY oY ðA:55Þ
C3 ¼ ð17Y 2 132Y þ 384Þ; C4 ¼ ð19Y 4 285Y 3 þ 1905Y 2 6768Y þ 13212Þ: Differentiation of Eq. (A.50) yields Z 1 d 1 2 1 dg ¼ 3ð12X ZC3 ÞX 0 dn 0 45360 dC3 dC4 0 2Z 2 Y ð3X C3 4ZC4 ÞZ 0 : 3XZ dY dY ðA:51Þ
The following ordinary differential form of the energy equation was then obtained by substituting Eqs. (A.53)– (A.55) and the relation: S0 ¼
S 0 X 4X
into Eq. (A.45) A31 X 0 þ A32 Y 0 þ A33 Z 0 ¼ B3 ;
The following ordinary differential form of the momentum equation was then obtained by substituting Eqs. (A.49)– (A.51) into Eq. (A.44) and collecting terms:
A31 ¼ ½3X C10 þ 2ZC11 ;
A21 X 0 þ A22 Y 0 þ A23 Z 0 ¼ B2 ;
A32 ¼ 4X ½3X C12 2ZC13 ;
ðA:52Þ
ðA:56Þ
where
ðA:57Þ
K. Hama, M. Shiotsu / Cryogenics 47 (2007) 209–219
A11 X 0 þ A12 Y 0 þ A13 Z 0 ¼ B1 ; A21 X 0 þ A22 Y 0 þ A23 Z 0 ¼ B2 ;
A33 ¼ 8X ½C9 ; B3 ¼ 12X ½5040C14 ; C10 ¼ ½ð3PrH i Y 4SÞð17PrY 132Þ þ 360; C11 ¼ ½PrH i Y fPrY ð38Y 2 285Y þ 789Þ ð285Y 2 2232Y þ 6768Þg; þ ð345Y 2 2304Y þ 5328Þ C12 ¼ Pr½17ð2PrH i Y SÞ 132H i ; 2
219
A31 X 0 þ A32 Y 0 þ A33 Z 0 ¼ B3 :
References 2
C13 ¼ ½PrH i fPrY ð152Y 855Y þ 1578Þ 9ð95Y 496Y þ 752Þg; 3SfPrð38Y 2 190Y þ 263Þ 2ð95Y 372Þg þ 6ð115Y 384Þ C14 ¼ ½ðPrH i Y SÞðPrY 12Þ 12=Pr:
By the above mentioned procedure, the fundamental Eqs. (A.8)–(A.10) were transformed into the following ordinary differential equations:
[1] Rivers WJ, McFadden PW. Film free convection in helium II. ASME J Heat Transfer 1966;88(4):343–50. [2] Labuntzov DA, Ametistov YeV. Analysis of helium II film boiling. Cryogenics 1979;19:401–4. [3] Tatsumoto H, Hata K, Hama K, Shirai Y, Shiotsu M. Critical heat flux on a flat plate in pressurized He II. Cryogenics 2001; 41(1):35–8.