- Email: [email protected]
Assessment of thermal–hydraulic correlations for narrow rectangular channels with high heat flux and coolant velocity
International Journal of Heat and Mass Transfer 99 (2016) 344–356
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Assessment of thermal–hydraulic correlations for narrow rectangular channels with high heat flux and coolant velocity Alberto Ghione a,b,⇑, Brigitte Noel a, Paolo Vinai b, Christophe Demazière b a b
Commissariat à l’Énergie Atomique et aux énergies alternatives, CEA, DEN/DM2S/STMF/LATF, 17 rue des Martyrs, Grenoble, France Chalmers University of Technology, Division of Subatomic and Plasma Physics, Department of Physics, Gothenburg, Sweden
a r t i c l e
i n f o
Article history: Received 3 November 2015 Received in revised form 15 March 2016 Accepted 26 March 2016 Available online 16 April 2016 Keywords: Narrow rectangular channels Friction coefficient Single-phase forced convection Fully developed boiling
a b s t r a c t The focus of the paper is on the evaluation of the correlations for predicting single-phase friction, singleand two-phase forced convection heat transfer coefficients in rectangular narrow channels, where the wall heat flux and the coolant flow can reach relatively high values. For this purpose, several correlations are reviewed and assessed against the SULTAN-JHR experiments. These tests were performed at CEA-Grenoble with upward water flow in two vertical uniformly heated narrow rectangular channels with gap of 1.51 and 2.16 mm. The experimental conditions range between 0.2 and 0.9 MPa for the pressure; 0.5–18 m/s for the coolant velocity and between 0.5 and 7.5 MW/m2 for the heat flux. The use of an appropriate turbulent friction factor leads to good comparison with the experimental data. The analysis of the single-phase turbulent heat transfer coefficient shows that the standard correlations (e.g. Dittus–Boelter) significantly under-estimate the heat transfer coefficient, especially at high Reynolds number. Therefore, new best-fitting correlations are derived. It is also observed that a reduction in gap size may lead to the enhancement of the heat transfer. The heat transfer is also under-estimated in two-phase flow if standard correlations (e.g. Jens–Lottes) are employed; however, good comparison with the experimental data are obtained with more appropriate models for fully developed boiling, such as the Forster–Greif correlation. The global accuracy associated to these correlations is also quantified in a rigorous manner. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Narrow rectangular channels can be an attractive solution for several engineering applications that require high-performance cooling capabilities within compact volumes, such as research nuclear reactors, electromagnets, and electronic equipment. In order to use this type of channels in an efficient and reliable manner, accurate simulation models based on appropriate correlations for single- and two-phase flows and heat transfer are needed. The case of heated vertical narrow channels, with an upward forced flow of water, is of particular interest for nuclear research reactors, where high power densities are produced in small core volumes. Several studies are available from the open literature (e.g. [1–3]), however the data are often limited to relatively small ⇑ Corresponding author at: Chalmers University of Technology, Division of Subatomic and Plasma Physics, Department of Physics, Gothenburg, Sweden. E-mail addresses: [email protected] (A. Ghione), [email protected] (B. Noel), [email protected] (P. Vinai), [email protected] (C. Demazière). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.03.099 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
ranges of conditions, with relatively low heat and mass fluxes. In this paper, selected correlations for single-phase turbulent friction factor, for the single-phase turbulent heat transfer coefficient, and for the wall superheat under fully developed boiling (FDB) are evaluated over wider conditions. In particular, the assessment relies on experiments with high heat flux at the wall (up to 7.5 MW/m2) and with high velocities of the coolant flow (up to 18 m/s), in channels with gaps of 1.509 and 2.161 mm. These experiments were carried out in connection to the design of the Jules Horowitz Reactor [4], at CEA-Grenoble, within the SULTAN-JHR campaign [5,6]. Preliminary analyses of the SULTAN-JHR data were presented in [6–8]. The current study improves and completes the previous results; identifies suitable correlations that can be applied to the range of conditions of interest; and, also, includes a rigorous estimation of the accuracy associated to those correlations. Several methods for the quantification of the uncertainty are applied and compared, so that reliable quantitative information can be obtained.
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
345
Nomenclature h Reynolds number Re ¼ GD l temperature °C temperature difference °C thermodynamic steam quality axial distance m
A cp Db Dh f g
flow area m2 specific heat capacity J/kg/K bubble diameter m hydraulic diameter Dh ¼ P4Aw m friction factor acceleration of gravity g = 9.8066 m/s2
Re T DT x z
G
2 _ mass flux G ¼ m A kg/m /s
h i ilg k _ m Nu p pcrit Dp Pr Ph Pr Pw
heat transfer coefficient W/m2/K specific enthalpy J/kg latent heat J/kg thermal conductivity W/m/K mass flow rate kg/s Nusselt number Nu ¼ hDk h pressure Pa critical pressure (for water: pcrit = 22.064106) Pa pressure drop Pa lc Prandtl number Pr ¼ k p heated perimeter m reduced pressure P r ¼ pp crit wet perimeter m
Greek symbols l dynamic viscosity kg/m/s q density kg/m3 r surface tension kg/s2 / heat flux W/m2
The paper is organized as follows: in the next section a brief description of the SULTAN-JHR experiments is given; in Section 3 the modeling of the experiments and the methods used for the quantification of the accuracy of the correlations are explained; in Sections 4,5 and 6 the results for the single-phase turbulent friction factor, the single-phase turbulent heat transfer coefficient, and the wall superheat under fully developed boiling are presented, respectively; in Section 7 conclusions are drawn. 2. The Sultan-JHR experiments The SULTAN-JHR experimental campaign was conducted at CEA Grenoble (France) during the years 2001–2008, with the objective of providing a reliable set of data for system code validation. The test section consisted of a narrow vertical rectangular channel that is uniformly electrically heated and where demineralized and degassed water flows upward. About 300 steady-state tests were carried out. The experimental conditions, which are summarized in Table 1, were selected to be representative of the ones in the Jules Horowitz Reactor. 2.1. Test section geometry Two different test sections were used: Section 3 (SE3) and Section 4 (SE4) with channel gap equal to 1.509 and 2.161 mm, respectively. As shown in Fig. 1, the channel is delimited by two Inconel-600 plates that are approximatively 1 mm thick. The power was supplied via direct electrical heating of the plates. The extremities of the walls are thinner in order to avoid heat concentration effects that may cause early boiling and potential thermal crisis at the corners.
Subscripts g gas l liquid sat saturation sub subcooled w wall
The test section is encapsulated in an electrical mica-based insulation (CogethermÒ) and two pressure steel plates which maintain the channel gap and geometry reasonably constant during all tests. In fact, the gap size was proven to be quite constant along the channel, by comparing the different pressure drops measured in the isothermal tests. On the external side, the test section is thermally insulated with 200 mm of rock wool so that heat losses could be reduced. The dimensions of the test section with the associated nomenclature are reported in Table 2. The axial geometry and instrumentation layout of the test section is shown in Fig. 2. The central part of the channel is heated with an approximately uniform heat flux, while two 70 mm-long adiabatic zones are present at the extremities of the test section. A smooth entrance in the test section was used in order to minimize the entrance effects. 2.2. Instrumentation Several quantities were measured during the experiments, including the mass flow rate, the electrical voltage and current, the water and dry wall temperatures, the absolute pressures and the pressure drops along the channel. The signals from the sensors were integrated over a 20 ms time range and the final measurements were obtained as an average of 100 acquisitions, in order to increase the stability and reliability of the measurements. The voltage DV between the pressure taps P3 and P6 and the electrical current I were measured, so that the electrical power supplied to the test section could be estimated according to the formula P = DV I. The water temperatures at the inlet (TE1 and TE2) and at the outlet (TS1 and TS2) were measured with a platinum probe.
Table 1 SULTAN-JHR experimental conditions. Outlet pressure [MPa] Inlet water temperature [°C] Mass flow rate [kg/s] Flow velocity [m/s] Uniform heat flux [MW/m2]
0.2–0.9 25–160 0.05–2.0 0.5–18 0.5–7.5 Fig. 1. Geometry of the SULTAN-JHR test section (top view).
346
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Table 2 Test section geometry (dimensions in mm).
Gap size (ech) Heated height (Hch) Adiabatic zone height (Had) Plate width (lpl) Length of the corners (lcor) Thickness of the corners (ecor) Averaged thickness of plate 1 (e¯pl,1) Averaged thickness of plate 2 (e¯pl,2)
Table 3 Measurement uncertainties. SE3
SE4
1.509 ± 0.040 599.8 ± 0.1 70.0 ± 0.1 47.2 ± 0.1 3.15 ± 0.1 0.5 1.0087 ± 0.006 0.9818 ± 0.018
2.161 ± 0.050 599.7 ± 0.1 70.0 ± 0.1 47.15 ± 0.1 2.85 ± 0.1 0.5 1.003 ± 0.002 1.004 ± 0.002
Flow rate Absolute pressure Fluid temperature
±1% ±0.8% ±0.25 °C
Electrical power Differential pressure Dry wall temperature
±1.4% ±0.8% ±1.5 °C
the simplified scheme in Fig. 3. The glue was introduced to guarantee the direct contact between the thermocouple and the insulation layer. The estimated uncertainties on the measurements are reported in Table 3. A more detailed description of the experimental campaign and facility may be found in [9]. 2.3. Experimental data reduction The wet wall temperatures Tw are derived from the dry wall temperatures Tdw, which are measured quantities (see Fig. 3). As described in [9], the one-dimensional Fourier’s law of conduction is applied, so the wet wall temperatures read as:
Tw ¼
1 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b aðepl ð/ 2/loss Þ aT 2iw 2bT iw Þ b
ð1Þ
In Eq. (1), /loss is the experimental heat loss and epl is the thickness of the Inconel-600 plates. The parameters a and b come from the relationship of the Inconel conductivity (kinconel [W/m°C] = b + a T [°C]) and they are equal to 0.0178 W/m°C2 and 12.12 W/m°C, respectively. The temperature at the interface between the electrical insulation and the heated plate Tiw is calculated from the relationship:
T iw ¼ T dw þ /loss
Fig. 2. SULTAN-JHR axial geometry and instrumentation layout.
eco egl þ kco kgl
ð2Þ
where eco, egl are the thicknesses of the CogethermÒ and glue layer respectively, and kco, kgl are the thermal conductivities. The Corsan conductivity [10] for Inconel 600 is used. The errors on the wet wall temperatures were obtained with a propagation of the experimental uncertainties, and they vary between ±1.6 at low values of heat flux and wall temperature, and between ±6.3 °C for high heat flux and wall temperature. 3. Methodology
The arrangement of the thermocouples and pressure taps on the two Inconel-600 plates is displayed in Fig. 2. The 8 pressure taps (0.5 mm in diameter) were placed on plate 1 at different locations: at the entrance (PE1) and the exit (PS8) of the test section, in the adiabatic zones (P2, P3, P6 and P7) and in the heated zone (P4 and P5). The following pressure differentials were measured: P2–P3; P6–P7; P3–P6; P3–P4; P4–P5; P5–P6. The dry wall temperatures were measured with insulated K-thermocouples (1 mm in diameter) centrally located on both the plates at 42 axial locations along the heated channel. The thermocouples were placed in the insulation layer, according to
3.1. Simulation of the experiments For the prediction of the experimental points, different computational capabilities are applied and/or combined. The experiments were simulated with the thermal–hydraulic code CATHARE [11]. This code is based on a transient 2-fluid 6equation model, complemented by proper closure laws for single-phase and two-phase flows, and wall heat transfer. Accordingly, the SULTAN-JHR test sections are modeled as a 1-D channel with a hydraulic diameter Dh, evaluated from the data reported in
Fig. 3. Schematic of a thermocouple layout.
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Table 2. The heated length of the channel is discretized with 150 computational volumes of 4 mm each. This nodalization is chosen in order to have the center of the volumes in the same positions of the thermocouples. The heated walls are modeled according to the layout in Fig. 3. The mesh independence of the results was proven. This part of the work is useful for several reasons. Those parameters that are not experimentally available, but may be needed in the correlations, can be reasonably determined from the CATHARE calculations. Correlations of interest can be implemented and tested within the framework of CATHARE. The uncertainty of some correlations can be quantified with the CIRCE tool that is connected to CATHARE (for details about CIRCE, see next Section 3.2). To simplify the procedure for the assessment of the singlephase turbulent heat transfer coefficient (see Section 5), a specific model was developed in Matlab. The nodalization scheme for the test sections is analogous to the CATHARE one, but a uniform heat flux boundary condition replaces the CATHARE modeling of the heated walls. The estimation of the bulk liquid temperature is based on a heat balance, which gives the fluid enthalpy il as a function of the axial distance z:
il ðzÞ ¼ il ðz ¼ 0Þ þ
/Pw z _ m
ð3Þ
The pressure drops are calculated according to the following expression:
Dp ¼ Dpgrav þ Dpfric þ Dpacc
ð4Þ
Since only single-phase flows were analyzed with this model, the acceleration term in Eq. (4) can be neglected. In fact, the latter is much smaller than the total pressure drop (Dpacc < 104 Dp). The gravity and the friction terms read respectively:
Dpgrav ¼ qg Dz Dpfric ¼ f
Dz G2 D h 2q
ð5Þ ð6Þ
where the friction factor f is optimized for the SULTAN-JHR tests, as discussed in Section 4.
The accuracy of a correlation can be estimated by comparing the measurements from proper experiments and the simulation results. For this purpose, it is first necessary to identify a suitable quantity, a so-called response, which can be related to the model output as well as to the measurements (e.g. the pressure drop for the friction factor). From the experimental and the calculated values of the response, one can obtain a sample of discrepancies (or residuals). These discrepancies can be then statistically analyzed to provide appropriate measures of the overall uncertainty associated to the correlation. In the current work, the sample of residuals ri for a selected correlation was obtained as a relative difference in percentage between the appropriate set of experimental points hexp,i extracted from the SULTAN-JHR database and the corresponding predictions hcalc,i:
hcalc;i hexp;i hexp;i
Std ¼
N 1X ri N i¼1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN jr Meanj2 i¼1 i N1
ð8Þ
ð9Þ
2. The bias and standard deviation of the correlation, estimated according to the CIRCE methodology [13]; 3. The probability density function (pdf) of the sample, evaluated with the density estimator available in the uncertainty methodology for physical models developed at the Paul Scherrer Institute (PSI) [14]. The CIRCE methodology was developed at CEA for the quantification of the uncertainty of the closure laws in CATHARE, and it was embedded in the CEA platform URANIE [15] for uncertainty and sensitivity analysis. In CIRCE, the sample of the differences between experimental and calculated responses is processed with an Expectation–Maximization (E–M) algorithm, so that the bias and the standard deviation of the related correlation can be determined. The algorithm is based on the hypothesis of normality for the distribution of the discrepancies. The PSI methodology evaluates the pdf that underlies the sample of residuals, using a non-parametric density estimator of kernel-type:
N X ri R ^f ðRÞ ¼ 1 K KE;h Nh i¼1 h
ð10Þ
where R is the random variable; and the kernel K is chosen as a Gaussian function. Thus, the pdf is expressed as a series expansion of Gaussian functions centered on the sample elements. The bandwidth h is selected by minimizing the integrated square differences between the Kernel Estimator and a second estimator, namely the Universal Orthogonal Estimator (which consists in a Fourier expansion). In this case, no assumptions for the distribution of the discrepancies are required. 4. Single-phase turbulent friction factor
3.2. Evaluation of the accuracy of the correlations
r i ¼ 100
Mean ¼
347
ð7Þ
with i = 1, . . ., N (being N the total number of experimental points). The accuracy is then quantified in terms of: 1. Mean and standard deviation of the sample, respectively [12]:
The single-phase turbulent friction factor has been modeled as:
f ¼ F cor f iso
ð11Þ
The friction factor fiso is optimized over the isothermal SULTANJHR experiments and it reads as:
f iso ¼ 0:202Re0:196
ð12Þ
The results for these isothermal experiments are shown in Fig. 4, and point out that Eq. (12) provides higher values (and better agreement) in comparison to the conventional Blasius correlation [16]. The corrective factor F cor for the diabatic tests takes into account the influence of the heat flux on the friction. It is expressed by:
F cor ¼ 1
Ph 0:0085ðT w T l Þ h i1:5 Pw þT l 1 þ 2 T w200
ð13Þ
Eq. (13) is a modified version of the Costa correlation, developed internally at CEA [17]. The implementation of this friction factor in CATHARE leads to good predictions of the pressure drops at the center of the channel (i.e. Dp45) of all the heated tests (both SE3 and SE4) under singlephase turbulent flow (i.e. Re > 10,000), as shown in Fig. 5. A total number of 95 tests were analyzed with Reynolds numbers between 1.0 104 and 3.1 105.
348
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Fig. 4. Experimental isothermal friction factor as a function of the Reynolds number.
Fig. 5. Comparison of the experimental pressure drops with CATHARE results obtained using the friction factor in Eq. (11).
The evaluation of the uncertainty associated to the diabatic friction factor is shown in Fig. 6. The mean and standard deviation of the discrepancies between the experimental and calculated pressure drops are equal to 0.35% and 5.09%, respectively. The bias and the standard deviation com-
puted with CIRCE, under the assumption that the pdf of the residuals is a Gaussian, are equal to 0.22% and 5.64%, respectively (dashed line in Fig. 6). The CIRCE normal distribution is very similar to the normal distribution with mean and standard deviation calculated with Eqs. (8) and (9) (see dashed line against
Fig. 6. Estimated pdfs for the uncertainty associated to the friction factor (Eq. (11)).
349
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
dash-dotted line). The pdf estimated with the kernel estimator introduced in Section 3.2 is consistent with the previous findings and demonstrates that the residuals are approximately normally distributed (full line). In the figure, the x-axis label R indicates the continuous random variable (whose possible realizations are the discrete values of the residuals ri). The ‘x’ marks represents the discrete distribution of the calculated residuals. The y-axis has therefore no meaning for these data. 5. Single-phase turbulent heat transfer coefficient In this section, the single-phase turbulent heat transfer coefficient is assessed. The experimental Nusselt number is calculated as:
Nuexp ¼
hexp Dh k
ð14Þ
where the experimental heat transfer coefficient hexp is defined as the ratio of the imposed local heat flux / to the difference between the experimental wet wall temperature and the calculated liquid bulk temperature (Tw Tl):
hexp ¼
/ ðT w T l Þ
ð15Þ
To assure that the experimental points selected from the SULTAN-JHR database and used for the analysis are related to single-phase turbulent conditions, specific criteria were assumed. Accordingly, only the points corresponding to a wall temperature at least 5 degrees smaller than the saturation temperature (i.e. Tw < Tsat 5 °C) and to a Reynold number greater than 10,000 (i.e. Re > 10,000), were considered. Furthermore, the measurements of the thermocouples at the beginning and at the end of the test section were not employed in order to rule out the possible influence of the entrance effects
and of the axial conduction. Thus, a total number of 1723 and 1036 experimental points were retrieved for SE3 and SE4, respectively. The Reynolds number varies between 1.0 104 and 2.69 105 and the Prandtl number is between 1.18 and 5.94. 5.1. Comparison with selected correlations from the literature The modeling of the single-phase turbulent heat transfer coefficient in narrow rectangular channels at high Reynolds number has been investigated in a limited number of publications. These studies (i.e. [18,19]) were carried out in the 1960 s and led to contradictory results. In the open literature, it is usually suggested to utilize standard correlations for circular pipes (e.g. the Dittus–Boelter correlation) with the use of the hydraulic diameter [20]. In this study, the experimental values of the Nusselt number were compared with the predictions from several correlations. The selected models along with their validity ranges are listed in Table 4. A summary of the performances of these correlations is presented in Table 5. This table contains the properties of the residuals (calculated with Eq. (7)) of the Nusselt number, for SE4 and SE3, in terms of: average value (Eq. (8)), standard deviation (Eq. (9)), minimum and maximum values. The analysis shows that the commonly-used correlations developed from circular pipes significantly under-estimate the experimental heat transfer coefficient, especially when applied to SE3. As an example, the results of the Dittus–Boelter correlation are provided in Fig. 7. Also, the correlations of Liang and Ma, that were specifically developed for bilaterally heated narrow rectangular channels in a limited range of validity (i.e. Re < 13,000), bring improvements but they still cannot predict closely the heat transfer. On the one hand the correlation of Ma adequately estimates the points for SE4, while on the other hand it does not for SE3. This may be
Table 4 Selected correlations for single-phase turbulent heat transfer. Correlations
Geometry
Re
Pr
Formula
Dittus–Boelter [21]
Circular
104–1.24105
0.7–120
Nu ¼ 0:023Re0:8 Pr 0:4
Seider–Tate [22]
Circular
>104
0.7–120
0:14 Nu ¼ 0:027Re0:8 Pr 0:33 ll
Gnielinski [23]
Circular
2300–5106
0.5–2000
Nu ¼
Popov–Petukhov [24]
Circular [24] with Siman-Tov correction for rectangular geometry [25]
104–1.25105
2.0–140
f G ¼ ð1:58lnRe 3:28Þ 0:11
w
0:5f G ðRe1000ÞPr 0:5 1þ12:7ðf G =2Þ ðPr2=3 1Þ 2
Nu ¼ fP ¼
l
0:125f P RePr l w
h
ð1þ3:4f P Þþ
i
11:7þ 1:8 Pr 1=3
0:5
ðf P =8Þ
ðPr2=3 1Þ
½1:08750:1125ðech =lch Þ ð1:82log 10 Re1:64Þ2
Ricque-Siboul [26]
Circular, Dh = 2–4 mm
104–1.47105
2.5–9.2
0:14 Nu ¼ 0:0092Re0:88 Pr 0:5 ll
Liang [1]
Rectangular, 1.8 50 mm
2300–6150
–
Nu ¼ 0:00666Re0:933 Pr0:4
Ma [2]
Rectangular, 2 40 mm
4000–13,000
–
Nu ¼ 0:00354Re1:0 Pr 0:4
w
Table 5 Comparison of the experimental data with correlations for single-phase turbulent heat transfer (values in %). Correlations
Dittus–Boelter Seider–Tate Gnielinski Popov–Petukhov Ricque-Siboul Liang Ma
SE4
SE3
Mean
Std
Min
Max
Mean
Std
Min
Max
30.2 17.0 23.9 16.4 19.6 12.2 1.8
9.5 10.7 8.4 9.3 5.9 7.6 10.1
48.5 38.5 44.7 41.0 38.3 31.3 25.2
0.5 19.0 2.7 16.9 2.0 19.8 40.5
38.1 27.0 33.8 27.9 32.2 25.3 18.3
11.4 13.1 10.7 12.1 8.9 9.3 9.4
63.5 58.5 58.7 56.5 57.4 51.8 45.9
0.5 19.8 0.6 11.0 7.1 0.6 19.9
350
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Fig. 7. Comparison of the experimental data with Dittus–Boelter correlation.
due to the fact that such a correlation was obtained with experiments in a geometry similar to SE4 (although at smaller Reynolds number and lower heat fluxes). As regards the correlation of Liang, it performs poorly for both test sections because the SULTAN-JHR conditions are outside of the validity range of the relationship. 5.2. Development of a correlation based on the SULTAN-JHR data To the authors’ knowledge, no correlation is available in the open literature for highly turbulent single-phase heat transfer in narrow rectangular channels. Moreover, as discussed in the previous subsection, the existing correlations significantly underestimate the SULTAN-JHR experimental data. Therefore, new correlations were developed from the best-fitting of the SULTAN-JHR data. The Seider–Tate form of the correlation was assumed:
Nu ¼ aRebRe Pr cPr
l lw
0:14 ð16Þ
This form was preferred to the Dittus–Boelter one since better predictions could be achieved at high heat fluxes, thanks to the presence of the viscosity ratio. The coefficients of the correlation were optimized using a multiple linear regression approach. The procedure was applied to the data for SE3 and SE4 separately. The best-fitting correlation for the test section SE4 reads as:
Nu ¼ 0:0044Re0:960 Pr0:568
l lw
0:14 ð17Þ
The coefficient of determination R2 is equal to 0.995 and the correlation is valid for Reynolds numbers between 1.0 104 and 2.69 105 and Prandtl numbers between 1.20 and 5.94. The behavior of this correlation with respect to the experiments is summarized in Fig. 8. The best-fitting correlation for SE3 is:
Nu ¼ 0:00184Re1:056 Pr0:618
l lw
0:14 ð18Þ
with a coefficient of determination equal to 0.985, Reynolds numbers between 1.0 104 and 1.77 105 and Prandtl numbers between 1.18 and 5.70. Fig. 9 displays the performance of Eq. (18). The statistical analysis of the accuracy of the optimized correlations gives the following results. The mean and standard deviation of the discrepancies between the experimental and calculated Nusselt numbers are respectively: 0.13% and 5.02% for correlation (17); 0.35% and 8.36% for correlation (18). Similar values for the quantification of the uncertainty are computed with CIRCE, as reported in Table 6.
Fig. 8. Best-fitting correlation (17) vs SE4 experimental data.
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
351
Fig. 9. Best-fitting correlation (18) vs SE3 experimental data.
Table 6 Bias and standard deviation estimated with CIRCE for correlation (17) and (18).
S-T SE4 (17) S-T SE3 (18)
a
bRe
cPr
R2
Mean [%]
Std [%]
0.0044 0.00184
0.960 1.056
0.568 0.618
0.995 0.985
0.27 0.05
5.17 8.99
In Figs. 10 and 11, the normal distributions based on the above values of mean and standard deviation are compared to the pdf determined with the kernel estimator given in Eq. (10). The results are in close agreement with each other, so the assumption of normality is reasonable and the CIRCE results may be retained.
Fig. 10. Estimated pdfs for the uncertainty associated to correlation (17).
Fig. 11. Estimated pdfs for the uncertainty associated to correlation (18).
352
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Fig. 12. Nusselt number as a function of the Reynolds number.
5.3. Influence of the channel geometry As already pointed out in [6], the channel geometry can affect the heat transfer coefficient (see the comparison of the experimental data for SE3 and SE4 in Fig. 12). In fact, the heat transfer coefficient is higher in SE3 that has the smaller channel gap. This suggests that an enhancement of the heat transfer may occur with the decrease of the gap size. Furthermore, the differences between the two test sections become larger and larger with the increase of the Reynolds number. In the figure, the experimental Nusselt number is divided by the Prandtl number to the power of the corresponding cPr coefficient (see Table 6) in order to better visualize the influence of the Reynolds number on the heat transfer. For the sake of completeness, the lines representing the correlations (17), (18) and of Dittus– Boelter are also included in the plot. The comparison between the experimental data and Dittus–Boelter also emphasizes how this correlation may be considered to be accurate at low Reynolds number (i.e. up to Re 25,000), but again the discrepancies increase with the Reynolds number. From the analysis of Fig. 12 and Table 6, the change of the channel gap seems to impact mainly the proportionality coefficient a, and the power coefficient bRe associated to the Reynolds number, rather than the contribution related to the Prandtl number. A generalized correlation for narrow rectangular channels could not however be derived using the SULTAN-JHR database. Additional experimental data for test sections with other gap sizes and channel widths would be required for modeling the possible geometrical effects. 6. Correlations for fully developed boiling The flow boiling in vertical heated channels is a complex phenomenon which was studied for many decades [27]. The boiling starts in sub-cooled flow conditions at the Onset of Nucleate Boiling (ONB) and the heat transfer mechanism gradually changes from single-phase forced convection, through the so-called partial boiling, to fully developed boiling (FDB) [28]. The focus of this section is on the evaluation of possible FDB correlations with respect to the flow boiling tests available in the SULTAN-JHR database. 6.1. Sultan-JHR experimental data under fully developed boiling The most influential parameters in FDB are the system pressure and the heat flux, but other quantities may also play a role, such as
the channel geometry, the surface properties (e.g. nucleation sites, roughness), the fluid properties, the dissolved gases content, etc. In this study, the predictions of the FDB correlations are compared to the experimental wall superheat, which is defined as:
DT sat ¼ T w T sat
ð19Þ
The saturation temperature Tsat was estimated with CATHARE, whose single-phase friction and heat transfer correlations were optimized according to the findings of Sections 4 and 5. CATHARE was used to estimate the pressure (and, thus, the saturation temperature) at the exact locations of the thermocouples, which are different from the locations of the pressure taps (see Fig. 2). This approach is justified by the good performance of the code to predict the pressure along the SULTAN-JHR test section. From the comparison between the calculated pressure profiles and the pressure measurements at the positions P5 and P6 (i.e. where most of the experimental points were taken for this work), a standard deviation of 2.5% and a bias of 0.46% were found. A careful review of the SULTAN-JHR database led to the selection of 32 tests, where FDB could be clearly identified. All these tests were carried out in test section SE4. As shown in Fig. 13, the experimental FDB region is assumed to start when the experimental wall temperature profile becomes flat (or slightly decreasing), which indicates that the contribution of the single-phase forced convection heat transfer mechanism becomes negligible. The dash-dotted curve in the plot represents the wall temperature Tw,FG predicted with the Forster–Greif correlation in the form reported in Table 8. It has no physical meaning before the actual FDB occurrence and it is only intended to facilitate the visualization of the FDB region. A total of 227 points are then collected. The range of variation of the physical parameters is reported in Table 7. 6.2. Assessment of FDB correlations Several FDB correlations were assessed against the experimental data (see list in Table 8). The results of the comparison are summarized in Table 9. This table contains the average value (Eq. (8)), the standard deviation (Eq. (9)) and the minimum and maximum values of the residuals, calculated from the experimental and computed wall superheat according to Eq. (7). The correlations of Jens–Lottes and Thom are frequently used for the modeling of conventional nuclear reactors and were developed for circular pipes, mainly at high pressure conditions. The two correlations significantly under-predict the FDB heat transfer and lead to higher wall superheat in comparison with the experiments,
353
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Fig. 13. Identification of the experimental FDB region, based on the experimental temperature profile.
Table 7 Range of physical parameters in the selected tests. Mass flux G [kg/m2s] Pressure p [MPa] Steam quality x Liquid sub-cooling DTsub [°C] Heat flux / [MW/m2]
500–5364 0.23–0.9 0.08 to 0.08 0–38.5 0.46–4.41
Table 8 Selected FDB heat transfer correlations. Correlations
Geometry
/ [MW/m2]
p [MPa]
Jens–Lottes [29]
Circular Dh = 3.6–5.7 mm
0.8–7.8
0.6–17.2
Thom [30]
Circular Dh = 12.7 mm
Qiu [31]
0.3–1.6
Annulus Gap = 1.0–1.5 mm
<0.1
5.2–13.8 1.2–4.0
Belhadj [3]
Rectangular Gap = 2.0–4.0 mm
<0.12
0.139–0.145
Gorenflo [32]
Pool boiling
–
0.02–2.2
Forster–Greif [33]
Pool boiling
0.16–6.3
0.1–0.8
Fabrega [34]
Circular Dh = 6 mm
–
0.8
Table 9 Comparison of the SE4 experimental data with FDB correlations. Correlation
Mean [%]
Std [%]
Min [%]
Max [%]
Jens–Lottes Thom Qiu Belhadj Gorenflo Forster–Greif Fabrega
39.0 52.8 51.2 12.9 0.01 1.3 18.4
19.7 25.4 6.9 16.1 11.9 10.1 12.5
2.6 5.6 65.8 48.8 24.6 22.6 9.5
78.7 112.2 37.3 18.6 23.5 19.2 41.8
as highlighted by the largely positive mean values of the residuals (i.e. 39.0% and 52.8%, respectively) in Table 9. The correlations of Qiu and Belhadj were specifically and respectively developed for narrow annuli and rectangular chan-
Formula
DT sat ¼ 25
/ 106
DT sat ¼ 22:65 DT sat ¼ A
/ 106
0:25
/ 106
0:5
0:25
e
1 62
e
p 105
1 87
e
p 105
1 62
p
105
0:26 b DT sat ¼ 0:484ð/Þ0:25 ech 1:13D ech h i0:5 ql C pl T sat 5=4 Db ¼ 1:5 104 gðq rq Þ qg hlg l g n 0:133 Rp hPB ¼ h0 F P // Rp0 0 n ¼ 0:9 0:3P 0:15 r 0:68 F P ¼ 1:73P 0:27 þ 6:1 þ 1P P 2r r r 0:23 0:35 p / DT sat ¼ 4:57 105 104 0:23 0:385 p / DT sat ¼ 4:44 105 104
nels. They both take into account the influence of the narrow gap on the boiling mechanisms: an enhancement of the heat transfer, compared to the one predicted by the correlation of Jens–Lottes, was observed. However, the SULTAN-JHR tests are poorly predicted probably because these correlations are not applied within their range of validity (compare Tables 7 and 8). The mean value of the residuals for Qiu correlation is equal to 51.2%, indicating a strong over-estimation of the heat transfer. A possible source of inaccuracy for Qiu correlation may come from the constant A that depends on the gap size of the narrow channel. For the current case, the value of A for SE4 was set equal to 8.77 as a result of a linear extrapolation. The Belhadj correlation gives smaller discrepancies (as reported in Table 9, the mean value is 12.9% and the standard deviation is 16.1%). Despite this, it fails to reproduce the correct dependency of
354
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
the wall superheat from the pressure, when applied to the present tests (see Fig. 14). The reason for this may be that the range of pressure in the experiments used by Belhadj is relatively limited (0.139–0.145 MPa) in comparison with SULTAN-JHR. On the other hand, good comparisons are obtained if the correlation of Gorenflo or the simplified correlation of Forster–Greif is used (see again Table 9). The Gorenflo correlation [32] is a pool boiling correlation based on a reference heat transfer coefficient h0 = 5600 W/m2 K. This reference value is obtained with a surface roughness Rp0 = 0.4 lm, a heat flux /0 = 20,000 W/m2, and a reduced pressure Pr0 = 0.1. The surface roughness is set to 0.4 lm as suggested in [27,35]. The relationship approximately has no bias with respect to the SULTANJHR experimental points (the mean value of the residuals is 0.01%) and the discrepancies are distributed in a relatively narrow band (the standard deviation is 11.9%). The Forster–Greif correlation was originally developed from theoretical considerations complemented with a limited set of experimental pool boiling data for water at 1 and 50 atm [33]. To overcome the complexity of the correlation, simplified formulations have been derived for the implementation in system codes. The origin of the simplified form of the Forster–Greif correlation, that is included in Table 8, could not be identified exactly (more details can be found in [8]). However, this version has been employed in the thermal–hydraulic modeling of research reactors
with fuel flat plates (e.g. in [36]) and it was verified for FDB in small-diameter tubes (between 2 and 4 mm), at high heat fluxes (between 5.6 and 20.5 MW/m2), and low pressures (approximately between 0.13 and 0.5 MPa) [26]. The comparison with the SULTANJHR experimental data gives excellent results (mean value = 1.3%; standard deviation = 10.1%), as summarized in Fig. 15 and Table 9. The largest discrepancies for the correlations of Gorenflo and of Forster–Greif with respect to the experiments are observed at low pressures (0.2–0.4 MPa) and low heat fluxes (0.5 MW/m2), as shown in Fig. 16. This effect is to be expected because, at low pressures, it is challenging to stabilize the boiling phenomenon, and larger uncertainties are associated to the estimation of the saturation temperature. Other simplified versions of the Forster–Greif correlation may provide worse results. For instance, the one suggested by Fabrega [34] was found to predict higher wall superheat than experimentally observed (see Table 9). The accuracy of the Forster–Greif correlation was also investigated in details. As mentioned, the analysis relies on a significant number of experimental points (i.e. 227), though from only 32 SULTAN-JHR tests. In Fig. 17, the normal distribution with mean value and standard deviation equal to the ones reported in Table 9, is plotted together with the results of the non-parametric estimator discussed in Section 3.2. From the comparison, the normal distribution can be considered as a reasonable approximation for
Fig. 14. Relationship between pressure and wall superheat according to Belhadj correlation and to the SULTAN-JHR experiments.
Fig. 15. Comparison of the experimental data with the simplified Forster–Greif correlation.
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
355
Fig. 16. Relationship between pressure and wall superheat according to the SULTAN-JHR experiments, and to the correlations of Forster–Greif and of Gorenflo.
Fig. 17. Estimated pdfs for the uncertainty associated to the Forster–Greif correlation.
describing the uncertainty associated to the correlation. However the pdf based on Eq. (10) provides additional details, such as: over-prediction of the wall superheat (i.e. positive residuals) may occur with higher probabilities; a relative peak in probability may characterize large negative discrepancies, namely between 15% and 20%. The properties of this pdf are strictly related to the system conditions of the 32 experiments used, so including other tests at different conditions may impact the distribution. CIRCE could not be used for this case, because it would have required the Forster–Greif correlation to be evaluated together with the whole sub-cooled flow boiling modeling in CATHARE. Thus, the procedure led to issues due to the model complexity of the code algorithm and the special numerical treatment of the boundaries between the different heat transfer regions needed to assure the robustness of the calculation scheme.
7. Summary and conclusions In this paper, correlations for single-phase friction factors, single-phase and two-phase heat transfer were investigated against the SULTAN-JHR experimental database. The experiments were performed in two vertical narrow rectangular channels with gap sizes of 1.51 and 2.16 mm at low pressures (between 0.2 and 0.9 MPa), high heat fluxes (between 0.5 and 7.5 MW/m2) and high velocities (between 0.5 and 18 m/s).
To quantify the accuracy of the correlations different approaches were used. The mean value and the standard deviation of the samples of the differences between the experimental and calculated points were determined. The probability density functions underlying the samples of discrepancies were derived by applying the CIRCE methodology (assuming a normal distribution of the discrepancies) and by a non-parametric density estimator. The analysis of the single-phase turbulent friction factor for Reynolds numbers between 1.0 104 and 3.1 105 highlighted that the conventional correlation of Blasius under-estimates the experimental data. A relationship optimized over the SULTAN-JHR experiments was therefore used and the associated uncertainty may be approximately described with a normal distribution with relatively small bias (0.22%) and standard deviation (5.64%). The evaluation of the turbulent heat transfer coefficient proved that the standard correlations (e.g. Dittus–Boelter) significantly under-estimate the heat transfer, especially at high Reynolds number. Therefore, a best-fitting correlation for each test section was proposed with a validity range that consists in: Reynolds number between 1.0 104 and 2.69 105 and Prandtl number between 1.18 and 5.94. The uncertainties were shown to be normally distributed with small biases (<0.27%) and standard deviations of 5.17% and 8.99%, respectively for the two channels with gap equal to 2.16 and 1.51 mm. The study also confirmed that a reduction of the channel gap may lead to an enhancement of the heat transfer coefficient.
356
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
The assessment of the heat transfer correlations for fully developed boiling showed that the ones of Jens–Lottes, Thom, Belhadj, Qiu and Fabrega poorly predict the experimental wall superheat. On the contrary, the relationship of Gorenflo and the simplified relationship of Forster–Greif lead to accurate results. In particular, the accuracy of the simplified Forster–Greif correlation was reasonably approximated with a Gaussian with mean value of 1.3% and standard deviation of 10.1%, even if deviations from normality were observed. In addition to the good performance, the simplified Forster–Greif correlation is simple to implement in thermal– hydraulic codes and it is independent of the surface conditions and roughness (that may not be accurately known for complex systems). Acknowledgements The current research project is conducted within a cooperation agreement between the French Alternative Energies and Atomic Energy Commission (CEA) and the Swedish Research Council (VR). The authors would like to acknowledge the financial support from the Swedish Research Council (Research contract No. B0774701). References [1] Z.H. Liang, Y. Wen, C. Gao, W.X. Tian, Y.W. Wu, G.H. Su, S.Z. Qiu, Experimental investigation on flow and heat transfer characteristics of single-phase flow with simulated neutronic feedback in narrow rectangular channel, Nucl. Eng. Des. 248 (2012) 82–92. [2] J. Ma, L. Li, Y. Huang, X. Liu, Experimental studies on single-phase flow and heat transfer in a narrow rectangular channel, Nucl. Eng. Des. 241 (2011) 2865– 2873. [3] M. Belhadj, T. Aldemir, R.N. Christensen, Determining wall superheat under fully developed nucleate boiling in plate-type research reactor cores with lowvelocity upwards flows, Nucl. Technol. 95 (1991) 95–102. [4] D. Iracane, The JHR, a new material testing reactor in Europe, Nucl. Eng. Technol. 38 (2006) 437–442. [5] G. Willermoz, A. Aggery, D. Blanchet, S. Cathalau, C. Chichoux, J. Di Salvo, C. Döderlein, D. Gallo, F. Gaudier, N. Huot, S. Loubière, B. Noël, H. Servière, Horus3D code package development and validation for the JHR modeling, in: Proceedings of the Physics of Reactors (PHYSOR) conference, Chicago, USA, 2004. [6] C. Chichoux, J. Delhaye, P. Clement, T. Chataing, Experimental study on heat transfer and pressure drop in rectangular narrow channel at low pressure, in: Proceedings of the 13th International Heat Transfer Conference (IHTC13), Sidney, Australia, 2006. [7] A. Ghione, B. Noel, P. Vinai, C. Demazière, On the prediction of single-phase forced convection heat transfer in narrow rectangular channels, in: Proceedings of the 10th International Topical Meeting on Nuclear Thermal Hydraulics, Operation and Safety (NUTHOS-10), Okinawa, Japan, 2014. [8] A. Ghione, B. Noel, P. Vinai, C. Demazière, Wall superheat prediction in narrow rectangular channels under fully developed boiling of water at low pressures, in: Proceedings of the 16th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-16), Chicago, USA, 2015. [9] A. Ghione, Improvement of the Nuclear Safety Code CATHARE based on Thermal–Hydraulic Experiments for the Jules Horowitz Reactor, Licentiate thesis CTH-NT-306, Chalmers University of Technology, Gothenburg, Sweden, 2015. [10] J.M. Corsan, N.J. Budd, An intercomparison involving PTB and NPL of thermal conductivity measurements on stainless steel, Inconel and Nimonic alloy reference materials, and an iron alloy, in: Proceedings of the 12th European Conference on Thermo-physical Properties, Vienna, Austria, vol. 23, 1990, pp. 119–128.
[11] G. Geffraye, O. Antoni, M. Farvacque, D. Kadri, G. Lavialle, B. Rameau, A. Ruby, CATHARE 2 V2.5_2: a single version for various applications, Nucl. Eng. Des. 241 (2011) 4456–4463. [12] J.A. Rice, Mathematical Statistics and Data Analysis, third ed., Duxbury Press, Pacific Grove, USA, 2007. [13] A. de Crècy, Determination of the uncertainties of the constitutive relationships of the CATHARE 2 code, in: Proceedings of the M&C conference, Salt Lake City, Utah, USA, 2001. [14] P. Vinai, R. Macian-Juan, R. Chawla, A statistical methodology for quantification of uncertainty in best estimate code physical models, Ann. Nucl. Energy 34 (2007) 628–640. [15] F. Gaudier, URANIE: the CEA/DEN uncertainty and sensitivity platform, Procedia – Social and Behavioral Science: Special issue on the Sixth International Conference on Sensitivity Analysis of Model Output 2 (2010) 7660–7661. [16] H. Blasius, Das Ähnlichkeitsgesetz bei reibungsvorgängen in flüssigkeiten, Forschg. Arb. Ing.-Wes. 131 (1913). Obtained from secondary reference [20]. [17] J. Costa, Mesure du coefficient de frottement en écoulement turbulent simple phase liquide avec transfert de chaleur, dans un canal rectangulaire, Technical Report: CEA-N-1142, CEA-Grenoble, France, 1969. [18] W.R. Gambill, R.D. Bundy, Heat-transfer studies of water flow in thin rectangular channels, Part I – Heat Transfer, Burnout and Friction for water in turbulent forced convection, Nuclear Science and Engineering 18 (1964) 69– 79. [19] S. Levy, R.A. Fuller, R.O. Niemi, Heat transfer to water in thin rectangular channels, ASME J. Heat Transfer (1959) 129–143. [20] S. Kakaç, R.K. Shah, W. Aung, Handbook of Single-phase Convective heat Transfer, John Wiley and Sons, USA, 1987. [21] F.W. Dittus, L.M.K. Boelter, Heat transfer in automobile radiators of the tubular type, Univ. Calif. Publ. Eng. 2 (1930) 3–22. [22] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, in: Proceedings of the Heat Transfer Symposium, Division of Industrial and Engineering Chemistry, Yale University, USA, 1936, pp. 1429–1435. [23] V. Gnielinski, New equations for heat and mass transfer in turbulent pipe and channel flow, Int. Chem. Eng. 16 (1976) 359–368. [24] B.S. Petukhov, Heat transfer and friction in turbulent pipe flow with variable physical properties, Adv. Heat Transfer 6 (1970) 503–564. [25] M. Siman-Tov, W.R. Gambill, W.R. Nelson, A.E. Ruggles, G.L. Yoder, Thermalhydraulic correlations for the advanced neutron source reactor fuel element design and analysis, in: Proceedings of the ASME Heat Transfer Division, vol. 190, ASME, 1991, pp. 63–78. [26] R. Ricque, R. Siboul, Etude expérimentale de l’échange thermique à flux élevé avec l’eau en convection forcée à grande vitesse dans des tubes de petit diamètre, avec et sans ébullition, Int. J. Heat Mass Transf. 15 (1972) 579–592. [27] J.G. Collier, J.R. Thome, Convective Boiling and Condensation, Clarendon Press, Oxford, 1994. [28] S.G. Kandlikar, Heat transfer characteristics in partial boiling, fully developed boiling and significant void flow regions of subcooled flow boiling, ASME J. Heat Transfer 120 (1998) 395–401. [29] W.H. Jens, P.A. Lottes, Analysis of Heat Transfer, Burnout, Pressure Drop and Density Data for High Pressure Water, Technical Report: ANL-4627, Argonne National Laboratory, USA, 1951. [30] J.R.S. Thom, W.M. Walker, T.A. Fallon, G.F.S. Reising, Boiling in subcooled water during flow up heated tubes or annuli, in: Proceedings of Symposium on Boiling Heat Transfer in Steam Generation Units and Heat Exchangers, 1965 (Manchester, UK). [31] S. Qiu, M. Takahashi, G. Su, D. Jia, Experimental study on heat transfer of single-phase flow and boiling two-phase flow in vertical narrow annuli, in: Proceedings of the 10th International Conference on Nuclear Engineering (ICONE-10), 2002 (Arlington, Virginia, USA). [32] D. Gorenflo, Pool boiling, in: VDI Heat Transfer Atlas, Springer, 2010. [33] K. Engelberg-Forster, R. Greif, Heat transfer to a boiling liquid: mechanism and correlations, ASME J. Heat Transfer 81 (1959) 43–53. [34] S. Fabrega, Etude expérimentale des instabilités hydrodynamiques survenant dans les réacteurs nucléaires à ébullition (Ph.D. thesis), University of Grenoble, France, 1966. [35] U.S. Nuclear Regulatory Commission, TRACE V5.0 Theory Manual: Field Equations, Solution Methods and Physical Models, Washington, USA, 2007. [36] A.P. Olson, K. Kalimullah, A user’s Guide to the PLTEMP/ANL Code (Version 4.1), Technical Report: ANL/RERTR/TM-11-22, Argonne National Laboratory, USA, 2011.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Assessment of thermal–hydraulic correlations for narrow rectangular channels with high heat flux and coolant velocity Alberto Ghione a,b,⇑, Brigitte Noel a, Paolo Vinai b, Christophe Demazière b a b
Commissariat à l’Énergie Atomique et aux énergies alternatives, CEA, DEN/DM2S/STMF/LATF, 17 rue des Martyrs, Grenoble, France Chalmers University of Technology, Division of Subatomic and Plasma Physics, Department of Physics, Gothenburg, Sweden
a r t i c l e
i n f o
Article history: Received 3 November 2015 Received in revised form 15 March 2016 Accepted 26 March 2016 Available online 16 April 2016 Keywords: Narrow rectangular channels Friction coefficient Single-phase forced convection Fully developed boiling
a b s t r a c t The focus of the paper is on the evaluation of the correlations for predicting single-phase friction, singleand two-phase forced convection heat transfer coefficients in rectangular narrow channels, where the wall heat flux and the coolant flow can reach relatively high values. For this purpose, several correlations are reviewed and assessed against the SULTAN-JHR experiments. These tests were performed at CEA-Grenoble with upward water flow in two vertical uniformly heated narrow rectangular channels with gap of 1.51 and 2.16 mm. The experimental conditions range between 0.2 and 0.9 MPa for the pressure; 0.5–18 m/s for the coolant velocity and between 0.5 and 7.5 MW/m2 for the heat flux. The use of an appropriate turbulent friction factor leads to good comparison with the experimental data. The analysis of the single-phase turbulent heat transfer coefficient shows that the standard correlations (e.g. Dittus–Boelter) significantly under-estimate the heat transfer coefficient, especially at high Reynolds number. Therefore, new best-fitting correlations are derived. It is also observed that a reduction in gap size may lead to the enhancement of the heat transfer. The heat transfer is also under-estimated in two-phase flow if standard correlations (e.g. Jens–Lottes) are employed; however, good comparison with the experimental data are obtained with more appropriate models for fully developed boiling, such as the Forster–Greif correlation. The global accuracy associated to these correlations is also quantified in a rigorous manner. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Narrow rectangular channels can be an attractive solution for several engineering applications that require high-performance cooling capabilities within compact volumes, such as research nuclear reactors, electromagnets, and electronic equipment. In order to use this type of channels in an efficient and reliable manner, accurate simulation models based on appropriate correlations for single- and two-phase flows and heat transfer are needed. The case of heated vertical narrow channels, with an upward forced flow of water, is of particular interest for nuclear research reactors, where high power densities are produced in small core volumes. Several studies are available from the open literature (e.g. [1–3]), however the data are often limited to relatively small ⇑ Corresponding author at: Chalmers University of Technology, Division of Subatomic and Plasma Physics, Department of Physics, Gothenburg, Sweden. E-mail addresses: [email protected] (A. Ghione), [email protected] (B. Noel), [email protected] (P. Vinai), [email protected] (C. Demazière). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.03.099 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
ranges of conditions, with relatively low heat and mass fluxes. In this paper, selected correlations for single-phase turbulent friction factor, for the single-phase turbulent heat transfer coefficient, and for the wall superheat under fully developed boiling (FDB) are evaluated over wider conditions. In particular, the assessment relies on experiments with high heat flux at the wall (up to 7.5 MW/m2) and with high velocities of the coolant flow (up to 18 m/s), in channels with gaps of 1.509 and 2.161 mm. These experiments were carried out in connection to the design of the Jules Horowitz Reactor [4], at CEA-Grenoble, within the SULTAN-JHR campaign [5,6]. Preliminary analyses of the SULTAN-JHR data were presented in [6–8]. The current study improves and completes the previous results; identifies suitable correlations that can be applied to the range of conditions of interest; and, also, includes a rigorous estimation of the accuracy associated to those correlations. Several methods for the quantification of the uncertainty are applied and compared, so that reliable quantitative information can be obtained.
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
345
Nomenclature h Reynolds number Re ¼ GD l temperature °C temperature difference °C thermodynamic steam quality axial distance m
A cp Db Dh f g
flow area m2 specific heat capacity J/kg/K bubble diameter m hydraulic diameter Dh ¼ P4Aw m friction factor acceleration of gravity g = 9.8066 m/s2
Re T DT x z
G
2 _ mass flux G ¼ m A kg/m /s
h i ilg k _ m Nu p pcrit Dp Pr Ph Pr Pw
heat transfer coefficient W/m2/K specific enthalpy J/kg latent heat J/kg thermal conductivity W/m/K mass flow rate kg/s Nusselt number Nu ¼ hDk h pressure Pa critical pressure (for water: pcrit = 22.064106) Pa pressure drop Pa lc Prandtl number Pr ¼ k p heated perimeter m reduced pressure P r ¼ pp crit wet perimeter m
Greek symbols l dynamic viscosity kg/m/s q density kg/m3 r surface tension kg/s2 / heat flux W/m2
The paper is organized as follows: in the next section a brief description of the SULTAN-JHR experiments is given; in Section 3 the modeling of the experiments and the methods used for the quantification of the accuracy of the correlations are explained; in Sections 4,5 and 6 the results for the single-phase turbulent friction factor, the single-phase turbulent heat transfer coefficient, and the wall superheat under fully developed boiling are presented, respectively; in Section 7 conclusions are drawn. 2. The Sultan-JHR experiments The SULTAN-JHR experimental campaign was conducted at CEA Grenoble (France) during the years 2001–2008, with the objective of providing a reliable set of data for system code validation. The test section consisted of a narrow vertical rectangular channel that is uniformly electrically heated and where demineralized and degassed water flows upward. About 300 steady-state tests were carried out. The experimental conditions, which are summarized in Table 1, were selected to be representative of the ones in the Jules Horowitz Reactor. 2.1. Test section geometry Two different test sections were used: Section 3 (SE3) and Section 4 (SE4) with channel gap equal to 1.509 and 2.161 mm, respectively. As shown in Fig. 1, the channel is delimited by two Inconel-600 plates that are approximatively 1 mm thick. The power was supplied via direct electrical heating of the plates. The extremities of the walls are thinner in order to avoid heat concentration effects that may cause early boiling and potential thermal crisis at the corners.
Subscripts g gas l liquid sat saturation sub subcooled w wall
The test section is encapsulated in an electrical mica-based insulation (CogethermÒ) and two pressure steel plates which maintain the channel gap and geometry reasonably constant during all tests. In fact, the gap size was proven to be quite constant along the channel, by comparing the different pressure drops measured in the isothermal tests. On the external side, the test section is thermally insulated with 200 mm of rock wool so that heat losses could be reduced. The dimensions of the test section with the associated nomenclature are reported in Table 2. The axial geometry and instrumentation layout of the test section is shown in Fig. 2. The central part of the channel is heated with an approximately uniform heat flux, while two 70 mm-long adiabatic zones are present at the extremities of the test section. A smooth entrance in the test section was used in order to minimize the entrance effects. 2.2. Instrumentation Several quantities were measured during the experiments, including the mass flow rate, the electrical voltage and current, the water and dry wall temperatures, the absolute pressures and the pressure drops along the channel. The signals from the sensors were integrated over a 20 ms time range and the final measurements were obtained as an average of 100 acquisitions, in order to increase the stability and reliability of the measurements. The voltage DV between the pressure taps P3 and P6 and the electrical current I were measured, so that the electrical power supplied to the test section could be estimated according to the formula P = DV I. The water temperatures at the inlet (TE1 and TE2) and at the outlet (TS1 and TS2) were measured with a platinum probe.
Table 1 SULTAN-JHR experimental conditions. Outlet pressure [MPa] Inlet water temperature [°C] Mass flow rate [kg/s] Flow velocity [m/s] Uniform heat flux [MW/m2]
0.2–0.9 25–160 0.05–2.0 0.5–18 0.5–7.5 Fig. 1. Geometry of the SULTAN-JHR test section (top view).
346
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Table 2 Test section geometry (dimensions in mm).
Gap size (ech) Heated height (Hch) Adiabatic zone height (Had) Plate width (lpl) Length of the corners (lcor) Thickness of the corners (ecor) Averaged thickness of plate 1 (e¯pl,1) Averaged thickness of plate 2 (e¯pl,2)
Table 3 Measurement uncertainties. SE3
SE4
1.509 ± 0.040 599.8 ± 0.1 70.0 ± 0.1 47.2 ± 0.1 3.15 ± 0.1 0.5 1.0087 ± 0.006 0.9818 ± 0.018
2.161 ± 0.050 599.7 ± 0.1 70.0 ± 0.1 47.15 ± 0.1 2.85 ± 0.1 0.5 1.003 ± 0.002 1.004 ± 0.002
Flow rate Absolute pressure Fluid temperature
±1% ±0.8% ±0.25 °C
Electrical power Differential pressure Dry wall temperature
±1.4% ±0.8% ±1.5 °C
the simplified scheme in Fig. 3. The glue was introduced to guarantee the direct contact between the thermocouple and the insulation layer. The estimated uncertainties on the measurements are reported in Table 3. A more detailed description of the experimental campaign and facility may be found in [9]. 2.3. Experimental data reduction The wet wall temperatures Tw are derived from the dry wall temperatures Tdw, which are measured quantities (see Fig. 3). As described in [9], the one-dimensional Fourier’s law of conduction is applied, so the wet wall temperatures read as:
Tw ¼
1 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b aðepl ð/ 2/loss Þ aT 2iw 2bT iw Þ b
ð1Þ
In Eq. (1), /loss is the experimental heat loss and epl is the thickness of the Inconel-600 plates. The parameters a and b come from the relationship of the Inconel conductivity (kinconel [W/m°C] = b + a T [°C]) and they are equal to 0.0178 W/m°C2 and 12.12 W/m°C, respectively. The temperature at the interface between the electrical insulation and the heated plate Tiw is calculated from the relationship:
T iw ¼ T dw þ /loss
Fig. 2. SULTAN-JHR axial geometry and instrumentation layout.
eco egl þ kco kgl
ð2Þ
where eco, egl are the thicknesses of the CogethermÒ and glue layer respectively, and kco, kgl are the thermal conductivities. The Corsan conductivity [10] for Inconel 600 is used. The errors on the wet wall temperatures were obtained with a propagation of the experimental uncertainties, and they vary between ±1.6 at low values of heat flux and wall temperature, and between ±6.3 °C for high heat flux and wall temperature. 3. Methodology
The arrangement of the thermocouples and pressure taps on the two Inconel-600 plates is displayed in Fig. 2. The 8 pressure taps (0.5 mm in diameter) were placed on plate 1 at different locations: at the entrance (PE1) and the exit (PS8) of the test section, in the adiabatic zones (P2, P3, P6 and P7) and in the heated zone (P4 and P5). The following pressure differentials were measured: P2–P3; P6–P7; P3–P6; P3–P4; P4–P5; P5–P6. The dry wall temperatures were measured with insulated K-thermocouples (1 mm in diameter) centrally located on both the plates at 42 axial locations along the heated channel. The thermocouples were placed in the insulation layer, according to
3.1. Simulation of the experiments For the prediction of the experimental points, different computational capabilities are applied and/or combined. The experiments were simulated with the thermal–hydraulic code CATHARE [11]. This code is based on a transient 2-fluid 6equation model, complemented by proper closure laws for single-phase and two-phase flows, and wall heat transfer. Accordingly, the SULTAN-JHR test sections are modeled as a 1-D channel with a hydraulic diameter Dh, evaluated from the data reported in
Fig. 3. Schematic of a thermocouple layout.
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Table 2. The heated length of the channel is discretized with 150 computational volumes of 4 mm each. This nodalization is chosen in order to have the center of the volumes in the same positions of the thermocouples. The heated walls are modeled according to the layout in Fig. 3. The mesh independence of the results was proven. This part of the work is useful for several reasons. Those parameters that are not experimentally available, but may be needed in the correlations, can be reasonably determined from the CATHARE calculations. Correlations of interest can be implemented and tested within the framework of CATHARE. The uncertainty of some correlations can be quantified with the CIRCE tool that is connected to CATHARE (for details about CIRCE, see next Section 3.2). To simplify the procedure for the assessment of the singlephase turbulent heat transfer coefficient (see Section 5), a specific model was developed in Matlab. The nodalization scheme for the test sections is analogous to the CATHARE one, but a uniform heat flux boundary condition replaces the CATHARE modeling of the heated walls. The estimation of the bulk liquid temperature is based on a heat balance, which gives the fluid enthalpy il as a function of the axial distance z:
il ðzÞ ¼ il ðz ¼ 0Þ þ
/Pw z _ m
ð3Þ
The pressure drops are calculated according to the following expression:
Dp ¼ Dpgrav þ Dpfric þ Dpacc
ð4Þ
Since only single-phase flows were analyzed with this model, the acceleration term in Eq. (4) can be neglected. In fact, the latter is much smaller than the total pressure drop (Dpacc < 104 Dp). The gravity and the friction terms read respectively:
Dpgrav ¼ qg Dz Dpfric ¼ f
Dz G2 D h 2q
ð5Þ ð6Þ
where the friction factor f is optimized for the SULTAN-JHR tests, as discussed in Section 4.
The accuracy of a correlation can be estimated by comparing the measurements from proper experiments and the simulation results. For this purpose, it is first necessary to identify a suitable quantity, a so-called response, which can be related to the model output as well as to the measurements (e.g. the pressure drop for the friction factor). From the experimental and the calculated values of the response, one can obtain a sample of discrepancies (or residuals). These discrepancies can be then statistically analyzed to provide appropriate measures of the overall uncertainty associated to the correlation. In the current work, the sample of residuals ri for a selected correlation was obtained as a relative difference in percentage between the appropriate set of experimental points hexp,i extracted from the SULTAN-JHR database and the corresponding predictions hcalc,i:
hcalc;i hexp;i hexp;i
Std ¼
N 1X ri N i¼1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN jr Meanj2 i¼1 i N1
ð8Þ
ð9Þ
2. The bias and standard deviation of the correlation, estimated according to the CIRCE methodology [13]; 3. The probability density function (pdf) of the sample, evaluated with the density estimator available in the uncertainty methodology for physical models developed at the Paul Scherrer Institute (PSI) [14]. The CIRCE methodology was developed at CEA for the quantification of the uncertainty of the closure laws in CATHARE, and it was embedded in the CEA platform URANIE [15] for uncertainty and sensitivity analysis. In CIRCE, the sample of the differences between experimental and calculated responses is processed with an Expectation–Maximization (E–M) algorithm, so that the bias and the standard deviation of the related correlation can be determined. The algorithm is based on the hypothesis of normality for the distribution of the discrepancies. The PSI methodology evaluates the pdf that underlies the sample of residuals, using a non-parametric density estimator of kernel-type:
N X ri R ^f ðRÞ ¼ 1 K KE;h Nh i¼1 h
ð10Þ
where R is the random variable; and the kernel K is chosen as a Gaussian function. Thus, the pdf is expressed as a series expansion of Gaussian functions centered on the sample elements. The bandwidth h is selected by minimizing the integrated square differences between the Kernel Estimator and a second estimator, namely the Universal Orthogonal Estimator (which consists in a Fourier expansion). In this case, no assumptions for the distribution of the discrepancies are required. 4. Single-phase turbulent friction factor
3.2. Evaluation of the accuracy of the correlations
r i ¼ 100
Mean ¼
347
ð7Þ
with i = 1, . . ., N (being N the total number of experimental points). The accuracy is then quantified in terms of: 1. Mean and standard deviation of the sample, respectively [12]:
The single-phase turbulent friction factor has been modeled as:
f ¼ F cor f iso
ð11Þ
The friction factor fiso is optimized over the isothermal SULTANJHR experiments and it reads as:
f iso ¼ 0:202Re0:196
ð12Þ
The results for these isothermal experiments are shown in Fig. 4, and point out that Eq. (12) provides higher values (and better agreement) in comparison to the conventional Blasius correlation [16]. The corrective factor F cor for the diabatic tests takes into account the influence of the heat flux on the friction. It is expressed by:
F cor ¼ 1
Ph 0:0085ðT w T l Þ h i1:5 Pw þT l 1 þ 2 T w200
ð13Þ
Eq. (13) is a modified version of the Costa correlation, developed internally at CEA [17]. The implementation of this friction factor in CATHARE leads to good predictions of the pressure drops at the center of the channel (i.e. Dp45) of all the heated tests (both SE3 and SE4) under singlephase turbulent flow (i.e. Re > 10,000), as shown in Fig. 5. A total number of 95 tests were analyzed with Reynolds numbers between 1.0 104 and 3.1 105.
348
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Fig. 4. Experimental isothermal friction factor as a function of the Reynolds number.
Fig. 5. Comparison of the experimental pressure drops with CATHARE results obtained using the friction factor in Eq. (11).
The evaluation of the uncertainty associated to the diabatic friction factor is shown in Fig. 6. The mean and standard deviation of the discrepancies between the experimental and calculated pressure drops are equal to 0.35% and 5.09%, respectively. The bias and the standard deviation com-
puted with CIRCE, under the assumption that the pdf of the residuals is a Gaussian, are equal to 0.22% and 5.64%, respectively (dashed line in Fig. 6). The CIRCE normal distribution is very similar to the normal distribution with mean and standard deviation calculated with Eqs. (8) and (9) (see dashed line against
Fig. 6. Estimated pdfs for the uncertainty associated to the friction factor (Eq. (11)).
349
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
dash-dotted line). The pdf estimated with the kernel estimator introduced in Section 3.2 is consistent with the previous findings and demonstrates that the residuals are approximately normally distributed (full line). In the figure, the x-axis label R indicates the continuous random variable (whose possible realizations are the discrete values of the residuals ri). The ‘x’ marks represents the discrete distribution of the calculated residuals. The y-axis has therefore no meaning for these data. 5. Single-phase turbulent heat transfer coefficient In this section, the single-phase turbulent heat transfer coefficient is assessed. The experimental Nusselt number is calculated as:
Nuexp ¼
hexp Dh k
ð14Þ
where the experimental heat transfer coefficient hexp is defined as the ratio of the imposed local heat flux / to the difference between the experimental wet wall temperature and the calculated liquid bulk temperature (Tw Tl):
hexp ¼
/ ðT w T l Þ
ð15Þ
To assure that the experimental points selected from the SULTAN-JHR database and used for the analysis are related to single-phase turbulent conditions, specific criteria were assumed. Accordingly, only the points corresponding to a wall temperature at least 5 degrees smaller than the saturation temperature (i.e. Tw < Tsat 5 °C) and to a Reynold number greater than 10,000 (i.e. Re > 10,000), were considered. Furthermore, the measurements of the thermocouples at the beginning and at the end of the test section were not employed in order to rule out the possible influence of the entrance effects
and of the axial conduction. Thus, a total number of 1723 and 1036 experimental points were retrieved for SE3 and SE4, respectively. The Reynolds number varies between 1.0 104 and 2.69 105 and the Prandtl number is between 1.18 and 5.94. 5.1. Comparison with selected correlations from the literature The modeling of the single-phase turbulent heat transfer coefficient in narrow rectangular channels at high Reynolds number has been investigated in a limited number of publications. These studies (i.e. [18,19]) were carried out in the 1960 s and led to contradictory results. In the open literature, it is usually suggested to utilize standard correlations for circular pipes (e.g. the Dittus–Boelter correlation) with the use of the hydraulic diameter [20]. In this study, the experimental values of the Nusselt number were compared with the predictions from several correlations. The selected models along with their validity ranges are listed in Table 4. A summary of the performances of these correlations is presented in Table 5. This table contains the properties of the residuals (calculated with Eq. (7)) of the Nusselt number, for SE4 and SE3, in terms of: average value (Eq. (8)), standard deviation (Eq. (9)), minimum and maximum values. The analysis shows that the commonly-used correlations developed from circular pipes significantly under-estimate the experimental heat transfer coefficient, especially when applied to SE3. As an example, the results of the Dittus–Boelter correlation are provided in Fig. 7. Also, the correlations of Liang and Ma, that were specifically developed for bilaterally heated narrow rectangular channels in a limited range of validity (i.e. Re < 13,000), bring improvements but they still cannot predict closely the heat transfer. On the one hand the correlation of Ma adequately estimates the points for SE4, while on the other hand it does not for SE3. This may be
Table 4 Selected correlations for single-phase turbulent heat transfer. Correlations
Geometry
Re
Pr
Formula
Dittus–Boelter [21]
Circular
104–1.24105
0.7–120
Nu ¼ 0:023Re0:8 Pr 0:4
Seider–Tate [22]
Circular
>104
0.7–120
0:14 Nu ¼ 0:027Re0:8 Pr 0:33 ll
Gnielinski [23]
Circular
2300–5106
0.5–2000
Nu ¼
Popov–Petukhov [24]
Circular [24] with Siman-Tov correction for rectangular geometry [25]
104–1.25105
2.0–140
f G ¼ ð1:58lnRe 3:28Þ 0:11
w
0:5f G ðRe1000ÞPr 0:5 1þ12:7ðf G =2Þ ðPr2=3 1Þ 2
Nu ¼ fP ¼
l
0:125f P RePr l w
h
ð1þ3:4f P Þþ
i
11:7þ 1:8 Pr 1=3
0:5
ðf P =8Þ
ðPr2=3 1Þ
½1:08750:1125ðech =lch Þ ð1:82log 10 Re1:64Þ2
Ricque-Siboul [26]
Circular, Dh = 2–4 mm
104–1.47105
2.5–9.2
0:14 Nu ¼ 0:0092Re0:88 Pr 0:5 ll
Liang [1]
Rectangular, 1.8 50 mm
2300–6150
–
Nu ¼ 0:00666Re0:933 Pr0:4
Ma [2]
Rectangular, 2 40 mm
4000–13,000
–
Nu ¼ 0:00354Re1:0 Pr 0:4
w
Table 5 Comparison of the experimental data with correlations for single-phase turbulent heat transfer (values in %). Correlations
Dittus–Boelter Seider–Tate Gnielinski Popov–Petukhov Ricque-Siboul Liang Ma
SE4
SE3
Mean
Std
Min
Max
Mean
Std
Min
Max
30.2 17.0 23.9 16.4 19.6 12.2 1.8
9.5 10.7 8.4 9.3 5.9 7.6 10.1
48.5 38.5 44.7 41.0 38.3 31.3 25.2
0.5 19.0 2.7 16.9 2.0 19.8 40.5
38.1 27.0 33.8 27.9 32.2 25.3 18.3
11.4 13.1 10.7 12.1 8.9 9.3 9.4
63.5 58.5 58.7 56.5 57.4 51.8 45.9
0.5 19.8 0.6 11.0 7.1 0.6 19.9
350
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Fig. 7. Comparison of the experimental data with Dittus–Boelter correlation.
due to the fact that such a correlation was obtained with experiments in a geometry similar to SE4 (although at smaller Reynolds number and lower heat fluxes). As regards the correlation of Liang, it performs poorly for both test sections because the SULTAN-JHR conditions are outside of the validity range of the relationship. 5.2. Development of a correlation based on the SULTAN-JHR data To the authors’ knowledge, no correlation is available in the open literature for highly turbulent single-phase heat transfer in narrow rectangular channels. Moreover, as discussed in the previous subsection, the existing correlations significantly underestimate the SULTAN-JHR experimental data. Therefore, new correlations were developed from the best-fitting of the SULTAN-JHR data. The Seider–Tate form of the correlation was assumed:
Nu ¼ aRebRe Pr cPr
l lw
0:14 ð16Þ
This form was preferred to the Dittus–Boelter one since better predictions could be achieved at high heat fluxes, thanks to the presence of the viscosity ratio. The coefficients of the correlation were optimized using a multiple linear regression approach. The procedure was applied to the data for SE3 and SE4 separately. The best-fitting correlation for the test section SE4 reads as:
Nu ¼ 0:0044Re0:960 Pr0:568
l lw
0:14 ð17Þ
The coefficient of determination R2 is equal to 0.995 and the correlation is valid for Reynolds numbers between 1.0 104 and 2.69 105 and Prandtl numbers between 1.20 and 5.94. The behavior of this correlation with respect to the experiments is summarized in Fig. 8. The best-fitting correlation for SE3 is:
Nu ¼ 0:00184Re1:056 Pr0:618
l lw
0:14 ð18Þ
with a coefficient of determination equal to 0.985, Reynolds numbers between 1.0 104 and 1.77 105 and Prandtl numbers between 1.18 and 5.70. Fig. 9 displays the performance of Eq. (18). The statistical analysis of the accuracy of the optimized correlations gives the following results. The mean and standard deviation of the discrepancies between the experimental and calculated Nusselt numbers are respectively: 0.13% and 5.02% for correlation (17); 0.35% and 8.36% for correlation (18). Similar values for the quantification of the uncertainty are computed with CIRCE, as reported in Table 6.
Fig. 8. Best-fitting correlation (17) vs SE4 experimental data.
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
351
Fig. 9. Best-fitting correlation (18) vs SE3 experimental data.
Table 6 Bias and standard deviation estimated with CIRCE for correlation (17) and (18).
S-T SE4 (17) S-T SE3 (18)
a
bRe
cPr
R2
Mean [%]
Std [%]
0.0044 0.00184
0.960 1.056
0.568 0.618
0.995 0.985
0.27 0.05
5.17 8.99
In Figs. 10 and 11, the normal distributions based on the above values of mean and standard deviation are compared to the pdf determined with the kernel estimator given in Eq. (10). The results are in close agreement with each other, so the assumption of normality is reasonable and the CIRCE results may be retained.
Fig. 10. Estimated pdfs for the uncertainty associated to correlation (17).
Fig. 11. Estimated pdfs for the uncertainty associated to correlation (18).
352
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Fig. 12. Nusselt number as a function of the Reynolds number.
5.3. Influence of the channel geometry As already pointed out in [6], the channel geometry can affect the heat transfer coefficient (see the comparison of the experimental data for SE3 and SE4 in Fig. 12). In fact, the heat transfer coefficient is higher in SE3 that has the smaller channel gap. This suggests that an enhancement of the heat transfer may occur with the decrease of the gap size. Furthermore, the differences between the two test sections become larger and larger with the increase of the Reynolds number. In the figure, the experimental Nusselt number is divided by the Prandtl number to the power of the corresponding cPr coefficient (see Table 6) in order to better visualize the influence of the Reynolds number on the heat transfer. For the sake of completeness, the lines representing the correlations (17), (18) and of Dittus– Boelter are also included in the plot. The comparison between the experimental data and Dittus–Boelter also emphasizes how this correlation may be considered to be accurate at low Reynolds number (i.e. up to Re 25,000), but again the discrepancies increase with the Reynolds number. From the analysis of Fig. 12 and Table 6, the change of the channel gap seems to impact mainly the proportionality coefficient a, and the power coefficient bRe associated to the Reynolds number, rather than the contribution related to the Prandtl number. A generalized correlation for narrow rectangular channels could not however be derived using the SULTAN-JHR database. Additional experimental data for test sections with other gap sizes and channel widths would be required for modeling the possible geometrical effects. 6. Correlations for fully developed boiling The flow boiling in vertical heated channels is a complex phenomenon which was studied for many decades [27]. The boiling starts in sub-cooled flow conditions at the Onset of Nucleate Boiling (ONB) and the heat transfer mechanism gradually changes from single-phase forced convection, through the so-called partial boiling, to fully developed boiling (FDB) [28]. The focus of this section is on the evaluation of possible FDB correlations with respect to the flow boiling tests available in the SULTAN-JHR database. 6.1. Sultan-JHR experimental data under fully developed boiling The most influential parameters in FDB are the system pressure and the heat flux, but other quantities may also play a role, such as
the channel geometry, the surface properties (e.g. nucleation sites, roughness), the fluid properties, the dissolved gases content, etc. In this study, the predictions of the FDB correlations are compared to the experimental wall superheat, which is defined as:
DT sat ¼ T w T sat
ð19Þ
The saturation temperature Tsat was estimated with CATHARE, whose single-phase friction and heat transfer correlations were optimized according to the findings of Sections 4 and 5. CATHARE was used to estimate the pressure (and, thus, the saturation temperature) at the exact locations of the thermocouples, which are different from the locations of the pressure taps (see Fig. 2). This approach is justified by the good performance of the code to predict the pressure along the SULTAN-JHR test section. From the comparison between the calculated pressure profiles and the pressure measurements at the positions P5 and P6 (i.e. where most of the experimental points were taken for this work), a standard deviation of 2.5% and a bias of 0.46% were found. A careful review of the SULTAN-JHR database led to the selection of 32 tests, where FDB could be clearly identified. All these tests were carried out in test section SE4. As shown in Fig. 13, the experimental FDB region is assumed to start when the experimental wall temperature profile becomes flat (or slightly decreasing), which indicates that the contribution of the single-phase forced convection heat transfer mechanism becomes negligible. The dash-dotted curve in the plot represents the wall temperature Tw,FG predicted with the Forster–Greif correlation in the form reported in Table 8. It has no physical meaning before the actual FDB occurrence and it is only intended to facilitate the visualization of the FDB region. A total of 227 points are then collected. The range of variation of the physical parameters is reported in Table 7. 6.2. Assessment of FDB correlations Several FDB correlations were assessed against the experimental data (see list in Table 8). The results of the comparison are summarized in Table 9. This table contains the average value (Eq. (8)), the standard deviation (Eq. (9)) and the minimum and maximum values of the residuals, calculated from the experimental and computed wall superheat according to Eq. (7). The correlations of Jens–Lottes and Thom are frequently used for the modeling of conventional nuclear reactors and were developed for circular pipes, mainly at high pressure conditions. The two correlations significantly under-predict the FDB heat transfer and lead to higher wall superheat in comparison with the experiments,
353
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
Fig. 13. Identification of the experimental FDB region, based on the experimental temperature profile.
Table 7 Range of physical parameters in the selected tests. Mass flux G [kg/m2s] Pressure p [MPa] Steam quality x Liquid sub-cooling DTsub [°C] Heat flux / [MW/m2]
500–5364 0.23–0.9 0.08 to 0.08 0–38.5 0.46–4.41
Table 8 Selected FDB heat transfer correlations. Correlations
Geometry
/ [MW/m2]
p [MPa]
Jens–Lottes [29]
Circular Dh = 3.6–5.7 mm
0.8–7.8
0.6–17.2
Thom [30]
Circular Dh = 12.7 mm
Qiu [31]
0.3–1.6
Annulus Gap = 1.0–1.5 mm
<0.1
5.2–13.8 1.2–4.0
Belhadj [3]
Rectangular Gap = 2.0–4.0 mm
<0.12
0.139–0.145
Gorenflo [32]
Pool boiling
–
0.02–2.2
Forster–Greif [33]
Pool boiling
0.16–6.3
0.1–0.8
Fabrega [34]
Circular Dh = 6 mm
–
0.8
Table 9 Comparison of the SE4 experimental data with FDB correlations. Correlation
Mean [%]
Std [%]
Min [%]
Max [%]
Jens–Lottes Thom Qiu Belhadj Gorenflo Forster–Greif Fabrega
39.0 52.8 51.2 12.9 0.01 1.3 18.4
19.7 25.4 6.9 16.1 11.9 10.1 12.5
2.6 5.6 65.8 48.8 24.6 22.6 9.5
78.7 112.2 37.3 18.6 23.5 19.2 41.8
as highlighted by the largely positive mean values of the residuals (i.e. 39.0% and 52.8%, respectively) in Table 9. The correlations of Qiu and Belhadj were specifically and respectively developed for narrow annuli and rectangular chan-
Formula
DT sat ¼ 25
/ 106
DT sat ¼ 22:65 DT sat ¼ A
/ 106
0:25
/ 106
0:5
0:25
e
1 62
e
p 105
1 87
e
p 105
1 62
p
105
0:26 b DT sat ¼ 0:484ð/Þ0:25 ech 1:13D ech h i0:5 ql C pl T sat 5=4 Db ¼ 1:5 104 gðq rq Þ qg hlg l g n 0:133 Rp hPB ¼ h0 F P // Rp0 0 n ¼ 0:9 0:3P 0:15 r 0:68 F P ¼ 1:73P 0:27 þ 6:1 þ 1P P 2r r r 0:23 0:35 p / DT sat ¼ 4:57 105 104 0:23 0:385 p / DT sat ¼ 4:44 105 104
nels. They both take into account the influence of the narrow gap on the boiling mechanisms: an enhancement of the heat transfer, compared to the one predicted by the correlation of Jens–Lottes, was observed. However, the SULTAN-JHR tests are poorly predicted probably because these correlations are not applied within their range of validity (compare Tables 7 and 8). The mean value of the residuals for Qiu correlation is equal to 51.2%, indicating a strong over-estimation of the heat transfer. A possible source of inaccuracy for Qiu correlation may come from the constant A that depends on the gap size of the narrow channel. For the current case, the value of A for SE4 was set equal to 8.77 as a result of a linear extrapolation. The Belhadj correlation gives smaller discrepancies (as reported in Table 9, the mean value is 12.9% and the standard deviation is 16.1%). Despite this, it fails to reproduce the correct dependency of
354
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
the wall superheat from the pressure, when applied to the present tests (see Fig. 14). The reason for this may be that the range of pressure in the experiments used by Belhadj is relatively limited (0.139–0.145 MPa) in comparison with SULTAN-JHR. On the other hand, good comparisons are obtained if the correlation of Gorenflo or the simplified correlation of Forster–Greif is used (see again Table 9). The Gorenflo correlation [32] is a pool boiling correlation based on a reference heat transfer coefficient h0 = 5600 W/m2 K. This reference value is obtained with a surface roughness Rp0 = 0.4 lm, a heat flux /0 = 20,000 W/m2, and a reduced pressure Pr0 = 0.1. The surface roughness is set to 0.4 lm as suggested in [27,35]. The relationship approximately has no bias with respect to the SULTANJHR experimental points (the mean value of the residuals is 0.01%) and the discrepancies are distributed in a relatively narrow band (the standard deviation is 11.9%). The Forster–Greif correlation was originally developed from theoretical considerations complemented with a limited set of experimental pool boiling data for water at 1 and 50 atm [33]. To overcome the complexity of the correlation, simplified formulations have been derived for the implementation in system codes. The origin of the simplified form of the Forster–Greif correlation, that is included in Table 8, could not be identified exactly (more details can be found in [8]). However, this version has been employed in the thermal–hydraulic modeling of research reactors
with fuel flat plates (e.g. in [36]) and it was verified for FDB in small-diameter tubes (between 2 and 4 mm), at high heat fluxes (between 5.6 and 20.5 MW/m2), and low pressures (approximately between 0.13 and 0.5 MPa) [26]. The comparison with the SULTANJHR experimental data gives excellent results (mean value = 1.3%; standard deviation = 10.1%), as summarized in Fig. 15 and Table 9. The largest discrepancies for the correlations of Gorenflo and of Forster–Greif with respect to the experiments are observed at low pressures (0.2–0.4 MPa) and low heat fluxes (0.5 MW/m2), as shown in Fig. 16. This effect is to be expected because, at low pressures, it is challenging to stabilize the boiling phenomenon, and larger uncertainties are associated to the estimation of the saturation temperature. Other simplified versions of the Forster–Greif correlation may provide worse results. For instance, the one suggested by Fabrega [34] was found to predict higher wall superheat than experimentally observed (see Table 9). The accuracy of the Forster–Greif correlation was also investigated in details. As mentioned, the analysis relies on a significant number of experimental points (i.e. 227), though from only 32 SULTAN-JHR tests. In Fig. 17, the normal distribution with mean value and standard deviation equal to the ones reported in Table 9, is plotted together with the results of the non-parametric estimator discussed in Section 3.2. From the comparison, the normal distribution can be considered as a reasonable approximation for
Fig. 14. Relationship between pressure and wall superheat according to Belhadj correlation and to the SULTAN-JHR experiments.
Fig. 15. Comparison of the experimental data with the simplified Forster–Greif correlation.
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
355
Fig. 16. Relationship between pressure and wall superheat according to the SULTAN-JHR experiments, and to the correlations of Forster–Greif and of Gorenflo.
Fig. 17. Estimated pdfs for the uncertainty associated to the Forster–Greif correlation.
describing the uncertainty associated to the correlation. However the pdf based on Eq. (10) provides additional details, such as: over-prediction of the wall superheat (i.e. positive residuals) may occur with higher probabilities; a relative peak in probability may characterize large negative discrepancies, namely between 15% and 20%. The properties of this pdf are strictly related to the system conditions of the 32 experiments used, so including other tests at different conditions may impact the distribution. CIRCE could not be used for this case, because it would have required the Forster–Greif correlation to be evaluated together with the whole sub-cooled flow boiling modeling in CATHARE. Thus, the procedure led to issues due to the model complexity of the code algorithm and the special numerical treatment of the boundaries between the different heat transfer regions needed to assure the robustness of the calculation scheme.
7. Summary and conclusions In this paper, correlations for single-phase friction factors, single-phase and two-phase heat transfer were investigated against the SULTAN-JHR experimental database. The experiments were performed in two vertical narrow rectangular channels with gap sizes of 1.51 and 2.16 mm at low pressures (between 0.2 and 0.9 MPa), high heat fluxes (between 0.5 and 7.5 MW/m2) and high velocities (between 0.5 and 18 m/s).
To quantify the accuracy of the correlations different approaches were used. The mean value and the standard deviation of the samples of the differences between the experimental and calculated points were determined. The probability density functions underlying the samples of discrepancies were derived by applying the CIRCE methodology (assuming a normal distribution of the discrepancies) and by a non-parametric density estimator. The analysis of the single-phase turbulent friction factor for Reynolds numbers between 1.0 104 and 3.1 105 highlighted that the conventional correlation of Blasius under-estimates the experimental data. A relationship optimized over the SULTAN-JHR experiments was therefore used and the associated uncertainty may be approximately described with a normal distribution with relatively small bias (0.22%) and standard deviation (5.64%). The evaluation of the turbulent heat transfer coefficient proved that the standard correlations (e.g. Dittus–Boelter) significantly under-estimate the heat transfer, especially at high Reynolds number. Therefore, a best-fitting correlation for each test section was proposed with a validity range that consists in: Reynolds number between 1.0 104 and 2.69 105 and Prandtl number between 1.18 and 5.94. The uncertainties were shown to be normally distributed with small biases (<0.27%) and standard deviations of 5.17% and 8.99%, respectively for the two channels with gap equal to 2.16 and 1.51 mm. The study also confirmed that a reduction of the channel gap may lead to an enhancement of the heat transfer coefficient.
356
A. Ghione et al. / International Journal of Heat and Mass Transfer 99 (2016) 344–356
The assessment of the heat transfer correlations for fully developed boiling showed that the ones of Jens–Lottes, Thom, Belhadj, Qiu and Fabrega poorly predict the experimental wall superheat. On the contrary, the relationship of Gorenflo and the simplified relationship of Forster–Greif lead to accurate results. In particular, the accuracy of the simplified Forster–Greif correlation was reasonably approximated with a Gaussian with mean value of 1.3% and standard deviation of 10.1%, even if deviations from normality were observed. In addition to the good performance, the simplified Forster–Greif correlation is simple to implement in thermal– hydraulic codes and it is independent of the surface conditions and roughness (that may not be accurately known for complex systems). Acknowledgements The current research project is conducted within a cooperation agreement between the French Alternative Energies and Atomic Energy Commission (CEA) and the Swedish Research Council (VR). The authors would like to acknowledge the financial support from the Swedish Research Council (Research contract No. B0774701). References [1] Z.H. Liang, Y. Wen, C. Gao, W.X. Tian, Y.W. Wu, G.H. Su, S.Z. Qiu, Experimental investigation on flow and heat transfer characteristics of single-phase flow with simulated neutronic feedback in narrow rectangular channel, Nucl. Eng. Des. 248 (2012) 82–92. [2] J. Ma, L. Li, Y. Huang, X. Liu, Experimental studies on single-phase flow and heat transfer in a narrow rectangular channel, Nucl. Eng. Des. 241 (2011) 2865– 2873. [3] M. Belhadj, T. Aldemir, R.N. Christensen, Determining wall superheat under fully developed nucleate boiling in plate-type research reactor cores with lowvelocity upwards flows, Nucl. Technol. 95 (1991) 95–102. [4] D. Iracane, The JHR, a new material testing reactor in Europe, Nucl. Eng. Technol. 38 (2006) 437–442. [5] G. Willermoz, A. Aggery, D. Blanchet, S. Cathalau, C. Chichoux, J. Di Salvo, C. Döderlein, D. Gallo, F. Gaudier, N. Huot, S. Loubière, B. Noël, H. Servière, Horus3D code package development and validation for the JHR modeling, in: Proceedings of the Physics of Reactors (PHYSOR) conference, Chicago, USA, 2004. [6] C. Chichoux, J. Delhaye, P. Clement, T. Chataing, Experimental study on heat transfer and pressure drop in rectangular narrow channel at low pressure, in: Proceedings of the 13th International Heat Transfer Conference (IHTC13), Sidney, Australia, 2006. [7] A. Ghione, B. Noel, P. Vinai, C. Demazière, On the prediction of single-phase forced convection heat transfer in narrow rectangular channels, in: Proceedings of the 10th International Topical Meeting on Nuclear Thermal Hydraulics, Operation and Safety (NUTHOS-10), Okinawa, Japan, 2014. [8] A. Ghione, B. Noel, P. Vinai, C. Demazière, Wall superheat prediction in narrow rectangular channels under fully developed boiling of water at low pressures, in: Proceedings of the 16th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-16), Chicago, USA, 2015. [9] A. Ghione, Improvement of the Nuclear Safety Code CATHARE based on Thermal–Hydraulic Experiments for the Jules Horowitz Reactor, Licentiate thesis CTH-NT-306, Chalmers University of Technology, Gothenburg, Sweden, 2015. [10] J.M. Corsan, N.J. Budd, An intercomparison involving PTB and NPL of thermal conductivity measurements on stainless steel, Inconel and Nimonic alloy reference materials, and an iron alloy, in: Proceedings of the 12th European Conference on Thermo-physical Properties, Vienna, Austria, vol. 23, 1990, pp. 119–128.
[11] G. Geffraye, O. Antoni, M. Farvacque, D. Kadri, G. Lavialle, B. Rameau, A. Ruby, CATHARE 2 V2.5_2: a single version for various applications, Nucl. Eng. Des. 241 (2011) 4456–4463. [12] J.A. Rice, Mathematical Statistics and Data Analysis, third ed., Duxbury Press, Pacific Grove, USA, 2007. [13] A. de Crècy, Determination of the uncertainties of the constitutive relationships of the CATHARE 2 code, in: Proceedings of the M&C conference, Salt Lake City, Utah, USA, 2001. [14] P. Vinai, R. Macian-Juan, R. Chawla, A statistical methodology for quantification of uncertainty in best estimate code physical models, Ann. Nucl. Energy 34 (2007) 628–640. [15] F. Gaudier, URANIE: the CEA/DEN uncertainty and sensitivity platform, Procedia – Social and Behavioral Science: Special issue on the Sixth International Conference on Sensitivity Analysis of Model Output 2 (2010) 7660–7661. [16] H. Blasius, Das Ähnlichkeitsgesetz bei reibungsvorgängen in flüssigkeiten, Forschg. Arb. Ing.-Wes. 131 (1913). Obtained from secondary reference [20]. [17] J. Costa, Mesure du coefficient de frottement en écoulement turbulent simple phase liquide avec transfert de chaleur, dans un canal rectangulaire, Technical Report: CEA-N-1142, CEA-Grenoble, France, 1969. [18] W.R. Gambill, R.D. Bundy, Heat-transfer studies of water flow in thin rectangular channels, Part I – Heat Transfer, Burnout and Friction for water in turbulent forced convection, Nuclear Science and Engineering 18 (1964) 69– 79. [19] S. Levy, R.A. Fuller, R.O. Niemi, Heat transfer to water in thin rectangular channels, ASME J. Heat Transfer (1959) 129–143. [20] S. Kakaç, R.K. Shah, W. Aung, Handbook of Single-phase Convective heat Transfer, John Wiley and Sons, USA, 1987. [21] F.W. Dittus, L.M.K. Boelter, Heat transfer in automobile radiators of the tubular type, Univ. Calif. Publ. Eng. 2 (1930) 3–22. [22] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, in: Proceedings of the Heat Transfer Symposium, Division of Industrial and Engineering Chemistry, Yale University, USA, 1936, pp. 1429–1435. [23] V. Gnielinski, New equations for heat and mass transfer in turbulent pipe and channel flow, Int. Chem. Eng. 16 (1976) 359–368. [24] B.S. Petukhov, Heat transfer and friction in turbulent pipe flow with variable physical properties, Adv. Heat Transfer 6 (1970) 503–564. [25] M. Siman-Tov, W.R. Gambill, W.R. Nelson, A.E. Ruggles, G.L. Yoder, Thermalhydraulic correlations for the advanced neutron source reactor fuel element design and analysis, in: Proceedings of the ASME Heat Transfer Division, vol. 190, ASME, 1991, pp. 63–78. [26] R. Ricque, R. Siboul, Etude expérimentale de l’échange thermique à flux élevé avec l’eau en convection forcée à grande vitesse dans des tubes de petit diamètre, avec et sans ébullition, Int. J. Heat Mass Transf. 15 (1972) 579–592. [27] J.G. Collier, J.R. Thome, Convective Boiling and Condensation, Clarendon Press, Oxford, 1994. [28] S.G. Kandlikar, Heat transfer characteristics in partial boiling, fully developed boiling and significant void flow regions of subcooled flow boiling, ASME J. Heat Transfer 120 (1998) 395–401. [29] W.H. Jens, P.A. Lottes, Analysis of Heat Transfer, Burnout, Pressure Drop and Density Data for High Pressure Water, Technical Report: ANL-4627, Argonne National Laboratory, USA, 1951. [30] J.R.S. Thom, W.M. Walker, T.A. Fallon, G.F.S. Reising, Boiling in subcooled water during flow up heated tubes or annuli, in: Proceedings of Symposium on Boiling Heat Transfer in Steam Generation Units and Heat Exchangers, 1965 (Manchester, UK). [31] S. Qiu, M. Takahashi, G. Su, D. Jia, Experimental study on heat transfer of single-phase flow and boiling two-phase flow in vertical narrow annuli, in: Proceedings of the 10th International Conference on Nuclear Engineering (ICONE-10), 2002 (Arlington, Virginia, USA). [32] D. Gorenflo, Pool boiling, in: VDI Heat Transfer Atlas, Springer, 2010. [33] K. Engelberg-Forster, R. Greif, Heat transfer to a boiling liquid: mechanism and correlations, ASME J. Heat Transfer 81 (1959) 43–53. [34] S. Fabrega, Etude expérimentale des instabilités hydrodynamiques survenant dans les réacteurs nucléaires à ébullition (Ph.D. thesis), University of Grenoble, France, 1966. [35] U.S. Nuclear Regulatory Commission, TRACE V5.0 Theory Manual: Field Equations, Solution Methods and Physical Models, Washington, USA, 2007. [36] A.P. Olson, K. Kalimullah, A user’s Guide to the PLTEMP/ANL Code (Version 4.1), Technical Report: ANL/RERTR/TM-11-22, Argonne National Laboratory, USA, 2011.