Extending thermal-hydraulic modelling capabilities of Apros into coiled geometries

Nuclear Engineering and Design 357 (2020) 110429

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Extending thermal-hydraulic modelling capabilities of Apros into coiled geometries

T

⁎

Joona Leskinena, , Jaakko Ylätalob, Roberto Pettinic,1 a

VTT Technical Research Centre of Finland Ltd, P.O. Box 1000, FI-02044 VTT, Finland Fortum Power and Heat Oy, 02150 Espoo, Finland c Department of Energy, Politecnico di Milano, 20156 Milano, Italy b

A R T I C LE I N FO

A B S T R A C T

Keywords: Helical coils Steam generator SMR Heat transfer Wall friction Dryout

In this work we extend modelling capabilities of the Apros two-fluid thermal hydraulic model into coiled geometries by re-defining the existing closure laws with new correlations valid in coiled tubes. The purpose is to be able to accurately model helically coiled steam generators, often adoped by small modular reactors (SMRs). For that, we identify suitable correlations and implement them into Apros. We also seek to identify areas which can already be accurately described with the existing models and, on the other hand, areas in which further experimental efforts are required for development of more accurate methods. Further, based on open literature, we compile a data base of experimental data for validation and verification purposes. We use the compiled data base for improving the accuracy of some of our calculation models. Specifically, we propose a modified version of an existing correlation for predicting the first dryout quality in helically coiled tubes. The proposed correlation is capable of estimating the first dryout quality with a reasonable accuracy in a range of state parameters and curvature ratios in which helically coiled steam generators generally operate.

1. Introduction Helically coiled tubes are widely utilized in various heat exchangers. As for the nuclear field, heat exchangers with helically coiled tubes are often adopted by alternative reactor concepts as once-through steam generators. Compared to the more conventional steam generators constructed with U-tubes, a helically coiled steam generator offers most of all a very compact solution, making it an interesting option for small modular reactors (SMRs) where the space to be occupied is limited. The helical geometry enhances the heat transfer efficiency, but at the same time increases pressure losses compared to straight tubes (Santini, 2008; Mori and Nakayama, 1967; Guo et al., 1998). The helical geometry also reduces thermomechanical stresses caused by temperature differences, since contrary to conventional U-tube steam generators, in helical heat exchangers the cold secondary fluid generally flows inside the tube (Santini, 2008). With the rising interest of nuclear industry in SMRs, modelling capabilities of helically coiled steam generators become important. It is well known that the helical geometry affects the flow inside the tube in such a way, that conventional methods developed for straight pipes are not accurate (Santini, 2008; Zhao et al., 2003; Cioncolini and Santini,

2006a,b; Xin and Ebadian, 1997; Dravid et al., 1971; Austen and Soliman, 1988; Wang et al., 2019). An especially important parameter when discussing different flow phenomena in helical tubes, is the curvature ratio defined as the ratio of tube inner diameter and the helix diameter. The curvature induces a so-called secondary flow in the pipe, which alters the heat transfer and wall friction. The present work concentrates on developing calculation models of Apros with the goal to accurately model helically coiled steam generators. This is achieved by re-defining the existing closure laws of Apros with correlations valid in coiled geometries. The purpose is to be able to reliably model the behaviour of various light-water SMRs in different situations, for example for purposes of safety analysis. With that, we are mainly interested in nuclear applications of helical coils, restricting our interest in relatively large coils instead of the small ones typically utilized for cooling purposes. In addition, our interest does not extend for other working fluids than water/steam mixture. This work consists, first of all, of identifying reasonable choices of correlations for helical geometries to be used in a system code such as Apros. Available correlations are found in open literature. We identify areas which are already well covered by work done in the previous decades and, on the other hand, the purpose is to identify areas for

⁎

Corresponding author. E-mail addresses: joona.leskinen@vtt.fi (J. Leskinen), [email protected] (J. Ylätalo), [email protected] (R. Pettini). 1 No longer employed by Politecnico di Milano. https://doi.org/10.1016/j.nucengdes.2019.110429 Received 20 September 2019; Received in revised form 31 October 2019; Accepted 31 October 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.

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Nomenclature

ϕ Re

two-phase multiplier Reynolds number

Latin symbols Greek symbols

D d F G g h , h∗ p, P Q q u x

helix diameter tube inside diameter drag force mass flux gravitational constant enthalpy, total enthalpy pressure heat transfer rate heat flux velocity equilibrium quality

α Γ μ ρ Subscripts 1 2 cr c g i k l lo w

Dimensionless numbers

De Nu N Pr

Dean number, Re(d/ D)0.5 Nusselt number number of data points Prandtl number

(3)

with the total enthalpy defined as hk∗ = hk + uk2/2 . Further discussion related to the assumptions made and how the system is solved is found in Siikonen (1987). The system of equations defined above can not be solved unless the equation system is closed with some expressions for the unknowns in the equation system. Specifically, we close the equation system by defining expressions for Fw, k , Fi, k , Qi‴, k , Qw‴, k and Γk . These expressions are in general of empirical origin due to the extremely complicated physics governing the two-phase flow phenomena. The accuracy of the twofluid model relies on accurate description of these expressions. Apros contains a set of closure laws, described in Hänninen et al. (2012). The default closure laws of Apros are, however, intended to be used in straight pipe geometries, or in U-tubes typically used in steam generators of conventional nuclear power plants. In order to accurately model helically coiled steam generators, we need to re-define this set of closure laws with correlations valid for coiled tubes. The results obtained by Apros presented in the following Sections are based on a development version of Apros, but the discussed modifications will be available in upcoming Apros releases.

2. Apros two-fluid thermal-hydraulic model Apros is a simulation software for modelling and dynamic simulation of various types of powers plants, energy systems and industrial processes. In addition to conventional power plants, Apros has been widely utilized for modelling nuclear power plants. Modelling alternative reactor concepts, such as SMRs, by Apros is tempting given that the interest in SMR concepts is rising in the nuclear field. Apros already contains practically all the required calculation models for modelling such reactors. Yet, typically SMRs adopt helically coiled steam generators, which can not accurately be modelled with the existing calculation methods of Apros. The Apros two-fluid model bases on conservation equations for mass, momentum and energy (Hänninen et al., 2012). The conservation equations are separately written for each phase, resulting in a partial differential equation system of six equations. The resulting continuity equation for mass conservation in one-dimensional form for phase k is expressed as

3. Development of Apros modelling capabilities and comparison against experimental data The present Section introduces the calculation methods chosen to be implemented in Apros. Apros results are compared against experimental data available in open literature whenever possible. 3.1. Single phase heat transfer and critical Reynolds number A large number of possible correlations are available for predicting the single-phase heat transfer coefficient in turbulent region. In the laminar region less studies have been performed. An extensive review of the heat transfer models available for helical geometries is provided, e.g., in El-Genk and Schriener (2017) and Gou et al. (2017). For determining the convective single-phase heat transfer coefficient in turbulent region we have chosen the Mori & Nakayama correlation given as (Mori and Nakayama, 1967; Mori and Nakayama,

(1)

The corresponding momentum equation is written as

∂αk ρk uk2 ∂αk ρk uk ∂p + Γk ui, k + αk ρk g + Fw, k + Fi, k. = − αk + ∂z ∂z ∂t

lower boundary upper boundary critical coil gas interface fluid phase liquid liquid-only wall

∂αk ρk hk∗ uk ∂αk ρk hk∗ + ∂z ∂t ∂p + Γk hi∗, k + Qi‴, k + Qw‴, k + Fi, k ui, k + αk ρk uk g , = αk ∂t

which sufficiently accurate calculation models do not yet exist. Given that a limited amount of experimental data related to different phenomena in helically coiled tubes is available in open literature, we collect a data base of experimental data for validation and verification purposes. Suitable correlations are then implemented into Apros, and Apros results are subsequently compared with the available experimental data. Existing calculation methods are improved if found necessary. The following Section provides a short introduction into the Apros simulation software and its two-fluid model. Section 3 discusses the chosen correlations implemented in Apros with comparison of Apros predictions against experimental data. Section 4 provides a further test case describing the functionality of the new correlations as a whole. Finally, Section 5 concludes the work.

∂αk ρk uk ∂αk ρk = Γk. + ∂z ∂t

volume fraction mass transfer rate viscosity density

(2)

The energy conservation can be expressed as 2

Nuclear Engineering and Design 357 (2020) 110429

J. Leskinen, et al.

1967) 1/12

Nu l =

d 1 Pr 2/5 Re 5/6 ⎛ ⎞ 41.0 ⎝D⎠

0.061 ⎤ ⎡1 + ⎥ ⎢ ( Re(d/ D)5/2)1/6 ⎦ ⎣ 1/10

Nu g =

1 Pr d Re 4/5 ⎛ ⎞ 26.2 Pr 2/3 − 0.074 ⎝D⎠

0.098 ⎤, ·⎡1 + ⎢ ( Re(d/ D)2.0)3/2 ⎥ ⎣ ⎦

( (

⎧ Re cr,mild,1 = 0.85·2300 1 + 210 ⎪ ⎨ ⎪ Re cr,mild,2 = 1.15·2300 1 + 210 ⎩

(4)

D ⩽ 24.0 d

−0.57

, if 30 ⩽

⎜

⎟

(7)

D ⩽ 110 d

if

D ⩾ 150, d

(9)

D Re cr, strong = 30000 ⎛ ⎞ ⎝d⎠

, if

−0.57

, if24 ⩽

η = 3ξ 2 − 2ξ 3,

(13)

if Re ⩽ Re cr,1 ⎧ 0, ⎪ Re − Re cr,1 , if Re cr,1 < Re < Re cr,2 . ξ = Re −Re cr,1 ⎨ cr,2 ⎪1, if Re ⩾ Re cr,2 ⎩

(14)

Reference

Guo (Guo et al., 1998) Rogers & Mayhew (Rogers and Mayhew, 1964) Zhao (Zhao et al., 2003) Seban & McLaughlin (Seban and McLaughlin, 1963) Cioncolini (Cioncolini and Santini, 2006a) Xin (Xin and Ebadian, 1997) Dravid (Dravid et al., 1971) Austen (Austen and Soliman, 1988) Wang (Wang et al., 2019)

(10)

−0.31

() ()

D ⎧ ⎪ Re cr,med,1 = 12500 d D ⎨ Re ⎪ cr,med,2 = 120000 d ⎩

D ⩽ 24.0 d

(12)

(15)

Table 1 Available experimental data regarding the single-phase heat transfer in helically coiled tubes and the corresponding ranges of Reynolds numbers and curvature ratios.

where the laminar to turbulent transition occurs point-wise for strong and mild curvature coils. For medium curvature coils the correlation gives a transition region, for which the transition occurs smoothly over the range Re cr,med,1 < Re < Re cr,med,2 . Notably, the correlation as such is not defined for all possible curvature ratios, which is problematic for system code applications. Further, it has been noted that a small transition region is present for mild curvature coils as well (Pettini, 2017). For these reasons, the original correlation is modified by splitting the point-wise transition of mild curvature coils in two and by extending the ranges of medium curvature coils. The modified version of the correlation as implemented in Apros is defined as (Pettini, 2017) −0.47

D > 150. d

The implemented set of correlations is compared against experimental data obtained from open literature. A vast amount of studies regarding the single-phase heat transfer in helically coiled tubes have been performed in the past decades and a good amount of experimental data is available for validation of the different possible correlations. Table 1 presents an overview of the experimental data obtained, with the corresponding ranges of experimental parameters. Please note, that all the presented data is in single-phase liquid conditions. That said, the given comparisons do not verify the validity of Eq. (5). Fig. 1 compares heat transfer coefficients predicted by Apros with experimental values. Given that experimental uncertainties were not available in most of the data sets obtained, the uncertainties are not included. The data set presented in the figure contains data purely in the laminar region, purely in the turbulent region, and data in the transition region. Fig. 1 also presents the results obtained when utilizing the default correlation set of Apros, valid for straight pipes or Utubes. A clear improvement is obtained with the new correlation set, indicating that the curvature does affect the single-phase convective heat transfer. The difference is, however, not large. As expected, the

(8)

D −1.12 ⎞ Re cr, mild = 2300 ⎛1 + 210 ⎛ ⎞ , ⎝d⎠ ⎝ ⎠

if

Nu = (1 − η)·Nu Laminar + η ·Nu Turbulent

(6)

−0.31

() ()

D ⎧ ⎪ Re cr,med,1 = 12500 d D ⎨ Re ⎪ cr,med,2 = 120000 d ⎩

if

), )

With that, the Nusselt number in the transition region would be obtained as

where De represents the Dean number. The correlation is valid within 20 < De < 2000, 0.7 < Pr < 175 and 0.0267 < (d/ D) < 0.0884 . The critical Reynolds number needed for determining whether the flow is laminar or turbulent is determined using a slightly modified version of the Cioncolini-correlation (Cioncolini and Santini, 2006b). The original Cioncolini correlation divides helical coils in three different curvature regions depending on the value of curvature ratio, namely the strong curvature, medium curvature or mild curvature regions. The correlation introduces a transition zone for the medium curvature region. The correlation is expressed as

D −0.47 Re cr, strong = 30000 ⎛ ⎞ , ⎝d⎠

D −1.12 d

The modified Cioncolini correlation bases on a large amount of experimental data from 12 differently sized coils with varying curvature ratios in a wide range of Reynolds numbers. For other available correlations to determine the critical Reynolds number, the reader may refer to El-Genk and Schriener (2017). A smooth interpolation function is utilized in the transition region for determining the heat transfer coefficient based on the laminar and turbulent values. More precisely, the value in the transition region is obtained by weighing the laminar and turbulent values with a weighting factor, defined as

(5)

where the substrict l refers to single-phase liquid and g to single-phase gas. The correlation bases on theoretical analyses performed by Mori & Nakayama. The Reynolds and Prandtl numbers are to be calculated using the bulk average temperature. Due to the assumptions made in the theoretical analysis, Eq. (5) is applicable for Pr ≈ 1.0 and Re(d/ D)2 > 0.1, and Eq. (4) for Pr > 1 and Re(d/ D)2.5 > 0.4 (Mori and Nakayama, 1967). The Mori & Nakayama correlation has been found to perform well in a wide range of state parameters (El-Genk and Schriener, 2017; Gou et al., 2017). Further, also the Seban-McLaughlin correlation could have been a valid option (Seban and McLaughlin, 1963). This correlation does, however, seem to not perform that well for high Reynolds numbers, as noted by Guo et al. (1998). For the laminar region, based on the recommendations in Gou et al. (2017), the Xin-correlation was considered the best option among the available possibilities. The Xin-correlation implemented into Apros is written as

Nu = (2.153 + 0.318De 0.643)Pr 0.177,

D −1.12 d

() ()

D ⩽ 150 d

Combined

(11) 3

Curvature ratio

Reynolds number

[d/D]

[Re]

Data points N

0.043 0.0498–0.0962

65000–180000 12700–108000

33 29

0.0308 0.0588–0.0926

9000–75000 6000–66000

70 24

0.0107–0.0308

2600–58000

348

0.08 0.0537 0.0204

1200–42000 360–8200 200–12000

41 7 29

0.0436

6000–100000

28

0.0107–0.0962

200–180000

609

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Fig. 1. Experimental heat transfer coefficient compared with respective values estimated by Apros. Results obtained using the Apros straight pipe correlation set are included for comparison.

without the respective experimental uncertainties. The available experimental data and the respective ranges of experimental parameters for validation purposes are summarized in Table 2. The experimental data is compared with values predicted by Apros. We note, that the obtained experimental data is somewhat approximative due to the fact that for each experiment only the operating pressure is provided. In case the test section is long, the local pressure might not correspond exactly to the given pressure, causing some error when estimating the same experiment in Apros. In addition, some of the obtained data has rather large uncertainties. Fig. 2 compares the obtained experimental heat transfer coefficients and the predicted values estimated by different correlations implemented in Apros. The Steiner & Taborek correlation predicts the Hwang and Santini data with a reasonable accuracy. The correlation does, however, significantly overestimate both the Zhao and Xiao data, the mean absolute percentage error for the entire data set being 41%. The modified Chen correlation, on the other hand, predicts this data more accurately, but underestimates the Hwang data. The mean absolute percentage error for the modified Chen correlation is 31%. The Hwang data is to the most part dominated by the nucleate boiling term due to the relatively large values of heat flux. Given that the Chen correlation was originally developed using data for which convective part was dominant, the correlation is not capable of accurately estimating the Hwang data set. The Thom correlation provides a reasonable prediction of the experimental data apart from the slight underestimation of the Hwang data. The mean absolute percentage error is 25%. The correlation does, however, not have the correct behaviour as a function of increasing quality. In this case the heat transfer coefficient tends to remain more or less constant as quality increases. Based on the comparison, the Steiner & Taborek or the Thom correlations are recommended. These correlations perform relatively well over a wide range of thermal hydraulic conditions. If it is known that in a certain application the convective term is dominant, that is, in case the heat flux is known to remain low compared to the mass flux, the modified Chen correlation may improve results.

straight pipe correlations underestimate the heat transfer. The chosen set of correlations accurately estimates the heat transfer coefficients in the entire range of data with approximately 96% of data within ± 20 % error with the mean absolute percentage error of 7.7%. The implemented Mori & Nakayama correlation overestimates the Guodata in the high Reynolds number region. Similar behaviour is present for the Wang data as well. In both cases, the straight pipe correlation set estimates the heat transfer coefficient with a good accuracy. This observation agrees with the statement that for higher Reynolds numbers the curvature effect becomes less significant. 3.2. Boiling heat transfer Given that the available correlations are less accurate in the boiling region than for single-phase convective heat transfer, we chose to implement various possible correlations in Apros for the user to choose from. In the boiling region, namely, the available correlations are somewhat more restricted in a certain range of state parameters. On the other hand, literature states that curvature does not significantly affect the boiling heat transfer, and that straight pipe calculation methods could be adopted also in helical geometries (Hwang et al., 2014; Santini et al., 2016; Xiao et al., 2018). We have implemented two separate correlations for boiling heat transfer, namely the Steiner & Taborek correlation (Steiner and Taborek, 1992) and the modified Chen correlation (Chen, 1966; Kim et al., 2013; Chen and Fang, 2014). The Steiner & Taborek correlation is purely a straight-pipe correlation, whereas the modified Chen correlation tries to account for the curvature effect by utilizing the Seban & McLaughlin correlation for the convective term. Given that the curvature effect has been noted to be quite insignificant in the boiling region, the Thom correlation used by default in Apros is included as well (Thom et al., 1965). The implemented correlations are compared against experimental data available in open literature. In total 420 experimental data points have been obtained from four different studies (Zhao et al., 2003; Hwang et al., 2014; Santini et al., 2016; Xiao et al., 2018), although,

Table 2 Available experimental data regarding the boiling heat transfer in helically coiled tubes. Reference

Mass flux [kg/m2s]

Heat flux [kW/m2]

Pressure [MPa]

Curvature ratio [–]

Data points N

Zhao et al. (2003) Hwang et al. (2014) Xiao et al. (2018) Santini et al. (2016)

400–700 175–525 400–1000 200–820

70–470 160–690 200–500 46–200

3.0 1.45–6.0 2.0–7.6 2.0–6.0

0.0308 0.0123–0.0198 0.0329–0.0806 0.0125

66 127 163 64

Combined

175–1000

46–690

2.0–7.6

0.0123–0.0806

420

4

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Fig. 2. Experimental heat transfer coefficients compared with values estimated by correlations implemented in Apros.

gravity affected zone, the zone of entrainment and the zone of redeposition. The zone is determined using the dimensionless parameters x 0 and y0

3.3. Dryout phenomena Drying out of the wall of a helically coiled pipe typically differs significantly from that of a straight pipe. Most importantly, the critical heat flux is affected by the coil curvature. In addition, in helical geometries the dryout occurs over a transition zone of certain length, contrary to the straight pipes where the wall dryout occurs at one single point. With that, for predicting the dryout phenomena in a helically coiled tube accurately, the calculation model requires descriptions for the first and last dryout qualities, and a method for determining the heat transfer coefficient in the transition region where the tube wall is partially dry.

x0 =

y0 =

G ρg gD

(16)

Gd d/0.02 . μl

(17)

Here x 0 characterizes the centrifugal force acting on the gas phase and entrained droplets, and y0 represents the liquid Reynolds number multiplied by a correction factor for the tube diameter (Jayanti and Berthoud, 1990). In the original dryout map, the gravity zone is assumed whenever y0 ⩽ 3.924·105x 0−1.71. The redeposition zone is assumed whenever y0 > 3.924·105x 0−1.71 and y0 ⩽ 6.46·10 4x 00.893 . Otherwise, the zone is the entrainment zone. The Duchatelle correlation bases on work performed by Duchatelle et al. for the Super Phenix project (Duchatelle et al., 1975). They studied four different coils with curvature ratios d/ D = 0.0317, d/ D = 0.0247, d/ D = 0.0111 and d/ D = 0.0074 , and investigated the dryout properties in the pressure range 4.5 < P < 17.5 MPa, mass fluxes 375 < G < 3500 kg/m2s and heat fluxes 310 < q < 1500 kW/m2. The proposed correlation is written as

3.3.1. First dryout quality Due to the very limited validity range of the available correlations for the first dryout quality, we have implemented several correlations in Apros. In total three separate correlations have been chosen for further testing. As the first option, we have implemented the Berthoud & Jayanti correlation (Jayanti and Berthoud, 1990; Berthoud and Jayanti, 1990). This correlation should in principle provide accurate results in the high pressure range, e.g., with P > 10 MPa. The second option is the Duchatelle correlation (Duchatelle et al., 1975), and the third one the Ruffel correlation referred to in Santini. (2008). To ensure numerical stability in Apros, the Apros implementation prevents values below x first = 0.3. Similarly, for stability reasons the maximum value of the first dryout quality is assumed as x first = 0.96. The Berthoud & Jayanti correlation bases on an extensive study related to dryout in helically coiled tubes at high pressures (Jayanti and Berthoud, 1990; Berthoud and Jayanti, 1990). Berthoud & Jayanti studied the critical heat flux, and made an effort to provide prediction methods for the final dryout quality and heat flux in the transition zone between the first and final dryout qualities. Based on a relatively large data set found in literature, they proposed to estimate the first dryout quality based on a so-called dryout map. The idea is to determine which mechanisms are most important based on local system parameters. Three different regions in the dryout map are recognized, namely the

x first = 1.39·10−4q0.732G−0.209e 0.00246P ,

(18)

where the pressure is to be given in bars. Notably, the proposed correlation predicts an increasing behaviour of the first dryout quality as a function of increasing pressure. Various studies regarding the first dryout quality, on the other hand, report a decreasing behaviour of the dryout quality as a function of increasing pressure (Xiao et al., 2018; Hwang et al., 2014; Santini, 2008; Jayanti and Berthoud, 1990; Gou et al., 2017). The Ruffel correlation referred to in Santini. (2008) is expressed as 5

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Xfirst, up = 1 − 0.13 G 1000

n = 0.75

D / (100d) (G / 1000)1.5

q n 100

then re-calculated the empirical coefficients of the correlation by minimizing the square error. We chose to deliberately leave the Santini data out of the regression analysis in order verify the performance of the modified correlation with data that was not used for defining the empirical coefficients. The resulting empirical coefficients are given as

( )

,

(19)

for the first dryout in the uppermost position of the tube. For the tube side the correlation is

Xfirst, side

G ⎞2 D = 1 − 0.0004q − 0.0109 ⎛ . 1000 ⎝ ⎠ d

a1 = 0.266 a2 = 0.157 a3 = 0.013 a4 = −0.003 a5 = −0.059.

(20)

The heat flux q is to be given in kW/m , all the other quantities are in SI-units. The correlation has been developed based on pressures between 6 ⩽ P ⩽ 18 MPa, mass fluxes between 300 ⩽ G ⩽ 1800 kg/m2s, tube diameters within 10.7 ⩽ d ⩽ 18.6 mm and curvature ratios between 0.0054 ⩽ d/D ⩽ 0.16 (Santini, 2008). Although several efforts have been made to develop calculation methods for predicting the first dryout quality, the available correlations at this point are very limited and in general can not estimate the dryout phenomenon with a reasonable accuracy over a wide range of curvatures and system parameters. Literature does, however, agree that the curvature significantly affects the dryout. Table 3 summarizes the available experimental data in open literature and their respective experimental parameter ranges. Fig. 3 compares experimental values of the first dryout quality with corresponding values predicted by the different correlations implemented in Apros. None of the correlations is capable of predicting the first dryout quality with a reasonable accuracy over the entire range of state parameters. The Berthoud & Jayanti correlation predicts well only the high pressure data, on which the correlation is in fact based. Considerable scattering of the data is present with the Duchatelle and Ruffel correlations. Based on the comparison, the Duchatelle correlation could be recommended for modelling helically coiled steam generators, but the accuracy of the correlation is not good enough based on our data set. The mean absolute percentage errors for the Berthoud & Jayanti, Duchatelle and Ruffel correlations are 76%, 38% and 48% respectively. Given that the Duchatelle correlation provided the best correspondence with our set of obtained experimental data, we modified the original Duchatelle correlation to see if the results could be improved. Knowing that the curvature has a significant effect on dryout characteristics in a helical tube, any correlation used for estimating the first dryout quality should include a term which takes the curvature into account. The original Duchatelle correlation, on the other hand, does not include such term. With that, we modified the original Duchatelle correlation by including an additional term for the curvature, resulting in a correlation of the form 2

x first = a1

d ⎞a5 . D ⎝ ⎠

qa2 G a3 e a4 P ⎛

Please note, that with the given coefficients, the heat flux is to be given in kW/m2 and the pressure is to be given in bars. In the original Duchatelle correlation, the heat flux is given as W/m2. This difference results in a much larger value of the a1 coefficient for our modified correlation. In addition, compared to the original Duchatelle correlation given by Eq. (18), the signs of the a3 and a4 coefficients are changed. The negative value of a4 agrees with the statement that the first dryout quality tends to decrease as pressure increases. On the other hand, increasing mass flux could either increase or decrease the first dryout quality (Gou et al., 2017). With that, the change in the sign of the coefficient a3 is justified, although, for further improvements the correlation could be modified by introducing a possibility for an increasing or decreasing dryout quality as a function of increasing mass flux. The modified version of the Duchatelle correlation is compared with the available experimental data in Fig. 4. The proposed correlation provides a reasonable accuracy for most of our data with a mean absolute percentage error of 18%. Considerable overestimation of the predicted first dryout quality is present in case of low dryout qualities, or high pressure. The correlation is capable of predicting the Santini data with a good accuracy, even if the Santini data was not included in the regression analysis. The best correspondence with experimental data is obtained in the region where helically steam generators typically operate. 3.3.2. Transition zone In the transition zone the method proposed by Berthoud & Jayanti is adopted (Jayanti and Berthoud, 1990; Berthoud and Jayanti, 1990). The total heat flux is divided between the two phases and the interface based on the transition parameter fw describing the fraction of wetted wall. The fraction of wetted wall is determined using the normalized quality

δx =

x − x first , xlast − x first

(22)

where x represents the local steam quality. Using the normalized quality, the fraction of wetted surface is obtained by the polynomial equation

(21)

We have not made any effort to predict the first dryout quality with any other working fluid than water. With that, the proposed correlation contains only terms that are not dependent on the working fluid. Based on the combined data set obtained from open literature, we

fw = −13.88(δx )5 + 32.68(δx ) 4 − 29.06(δx )3 + 12.30(δx )2 − 3.04(δx ) (23)

+ 1.0.

Table 3 Experimental data sets related to first dryout quality available in open literature with their respective ranges of state parameters. Data

Mass flux G [kg/m2s]

Heat flux q [kW/m2]

Pressure p [bar]

Curvature d/ D

N

Xiao et al. (2018) Hwang et al. (2014) Ma et al. (1995) Unal (Jayanti and Berthoud, 1990) Breus (Jayanti and Berthoud, 1990) Carver (Jayanti and Berthoud, 1990) Santini et al. (2014)

400–1000 90–520 300–730 620–1500 500–1500 400–1400 200–800

200–400 120–1750 120–1800 190–550 120–1700 390 40–200

20–76 10–61 20–40 150–200 100–150 180 10–60

0.0328–0.0806 0.0093–0.0198 0.0043–0.0758 0.0120–0.0257 0.0160 0.0134 0.0125

54 22 49 21 19 3 12

Combined

90–1500

40–1800

10–200

0.0043–0.0806

180

6

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Fig. 3. The available dryout correlations in Apros compared against different experimental data sets obtained from open literature.

example the behaviour of the total heat transfer coefficient as calculated by Apros, and compares it with coefficients available in Xiao et al. (2018). In this example we used the modified Chen correlation for computing the heat transfer coefficient in the boiling region, and artificially forced the value x first = 0.82 for the first dryout quality. The modified Duchatelle correlation would provide a value of x first = 0.71 in this case, which is within 15% error. This error would, however, affect the behaviour of the heat transfer coefficient in the transition region. In order to properly demonstrate the functionality of the transition parameter fw , the forced value for the first dryout quality was found more representative in this example. The behaviour of the heat transfer coefficient estimated by Apros in the transition zone is correct, apart from the slight overestimation

Fig. 4. The modified Duchatelle correlation compared against different experimental data sets obtained from open literature.

This correlation for the fraction of wetted surface is based on the proposed correlation in Jayanti and Berthoud (1990) and Berthoud and Jayanti (1990) with slight modifications of the polynomial coefficients in order to obtain the correct behaviour at the extreme values of normalized quality, that is, fw = 1 for δx = 0 and fw = 0 for δx = 1. Notably, the correlation does not depend on local pressure, although according to observations of Berthoud & Jayanti, the fraction of wetted surface has a significant pressure effect. The boiling heat transfer coefficient in the transition zone is linearly interpolated between the value computed by the boiling heat transfer correlation and the value obtained for a dry wall. With the small amount of experimental data available regarding the heat transfer coefficient in the transition region, Fig. 5 presents as an

Fig. 5. An example of the Apros prediction of the heat transfer coefficient in partially dried out region compared with experimental values from Xiao et al. (2018). 7

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with values obtained directly from Apros when using the implemented Ito correlation. Corresponding results obtained using the Apros default correlation are shown as well. The Ito-correlation provides a clear improvement. The behaviour of the laminar to turbulent transition as computed by the modified Cioncolini correlation is also visible in the figure.

caused by the overestimation in the boiling region. 3.3.3. Last dryout quality As for the last dryout quality, we decided to use the constant value of xlast = 0.99 as the point for complete dryout. This choice is partially backed up by studies of Jayanti and Berthoud (1990) and Berthoud and Jayanti (1990). Based on their experimental data the complete dryout in general occurs at qualities near to unity. Similar conclusions could be made based on data provided by Santini. (2008) and Xiao et al. (2018). On the other hand, the available experimental data and calculation methods for predicting the last dryout quality are very scarce. The correlation proposed by Berthoud & Jayanti could in principle be utilized, but at this point the constant valued dryout quality for last dryout was considered a better choice due to large uncertainties and possible numerical instabilities caused by the use of the correlation.

3.4.2. Two-phase multipliers Estimating the pressure losses in the two-phase region is more difficult than in the single-phase region. Based on a short literature review, we have implemented the Colombo, Guo and Xiao-correlations into Apros (Colombo et al., 2015; Guo et al., 1998; Xiao et al., 2018). In addition, the Apros default calculation model for straight pipes and Utubes for predicting the pressure loss is available. Colombo et al. combined two different data bases regarding twophase pressure losses in helical coils (Colombo et al., 2015). Based on the combined data base of two curvature ratios d/ D = 0.0125 and d/ D = 0.0308 they proposed a correlation which considers the effect of curvature. The correlation bases on adjusting the Lockhart-Martinelli multiplier and introducing new terms for taking the helical geometry in account. The proposed correlation may be written as

3.4. Wall friction 3.4.1. Single-phase For estimating the single-phase wall friction in the laminar and turbulent region, we have chosen the widely used Ito-correlation (Itō, 1969; Itō, 1959). The Ito correlation may be expressed as

fc, laminar =

344(d/ D)0.5 [1.56 + log10 ( Re d/ D )]5.73

−0.40

ρ 2 De l0.19 ⎜⎛ m ⎟⎞ ϕl = 0.0986ϕLM ⎝ ρl ⎠

(24)

(25)

for turbulent flow. Literature agrees with the effectiveness of the Ito correlations in laminar and turbulent regimes (Cioncolini and Santini, 2006b; El-Genk and Schriener, 2017). Along the widely used Ito-correlation, we mention the correlation proposed quite recently by El-Genk and Schriener (2017). Their correlation is expressed as

fc fs, L

= 1 + 0.00325De m,

(27)

where ρm refers to mixture density, De l to the liquid Dean number and 2 ϕLM to the Lockhart-Martinelli two-phase multiplier computed using the Chisholm equation with an empirical constant of C = 10 . The correlation accurately estimated the combined data base of Santini. (2008) and Zhao et al. (2003). Xiao et al. proposed a correlation for tubes with small coil diameters based on their experimental data (Xiao et al., 2018). The proposed correlation is written as

for laminar flow and

d 0.5 fc, turbulent = 0.076 Re−0.25 + 0.0075 ⎛ ⎞ D ⎝ ⎠

,

ϕlo2= (0.377 + 6.79x − 5.66x 2) ⎡1 + x ⎣ ⎡1 + x ⎛ ρl − 1⎞ ⎤, ⎢ ⎥ ⎝ ρg ⎠⎦ ⎣

(26)

where fc represents the friction factor for helical tubes, fs, L the corresponding friction factor in a straight tube computed as fs, L = 64/Re , and De m the modified Dean number defined as De m = Deδ 0.09 (d/ D)−0.38 , where δ is the coil curvature. In principle, this correlation provides a larger validity range than the Ito-correlation and has the advantage of being able to predict the friction factor both in laminar and turbulent regions. The Ito-correlation was preferred, however, for our purposes since the El-Genk correlation requires the pitch as an input, which in some cases might be unavailable for the modeller, even if introducing such term can result in slightly more accurate results. For a more extensive review of the single-phase pressure loss in helically coiled tubes, the reader may for example refer to El-Genk and Schriener (2017) and Pettini (2017). The same method as descibed in Section 3.1 is used for determining the critical Reynolds number and total friction factor in the transition region. A considerable amount of experimental data regarding the singlephase friction factors in helically coiled tubes is obtained from Cioncolini and Santini (2006b). In the study, twelve differently sized coils were tested with varying coil curvatures at ambient temperature and pressure close to the athmospheric pressure with a wide range of Reynolds numbers. Pressure drops were measured over the tested coils. The experimental data includes data points within curvature ratios 0.0027 ⩽ d/ D ⩽ 0.145and Reynolds numbers 600 < Re < 87340 . Fig. 6 compares friction factors as predicted by Apros with the experimental data obtained for the 12 coils. A very good agreement is obtained with a mean absolute percentage error of 1.86%. Fig. 7 compares the experimental data from chosen experiments

(

μg μl

)

0.25

−1 ⎤ ⎦ (28)

with the single-phase liquid-only friction factor needed for determining the total pressure loss is calculated as

fsp =

0.404 0.3164 ⎡ ⎤. ⎛d⎞ 1 + Re 0.053 lo 0.25 ⎢ ⎥ Re lo ⎣ ⎝D⎠ ⎦

(29)

Good results against their experimental data were obtained with the proposed correlation. The correlation includes the curvature effect in

Fig. 6. Experimental data obtained from Cioncolini and Santini (2006b) compared with values predicted by the Ito-correlation. 8

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Fig. 7. Experimental friction factors from Cioncolini and Santini (2006b) compared with values predicted by Apros.

with low curvature ratio, but as the curvature effects become more important the error increases. The Xiao-correlation is capable of estimating the pressure loss quite accurately in all presented experiments. The Colombo correlation is accurate as well, apart from the underestimation of the Zhao experiment. Notably, this is the experiment with the largest curvature ratio. We also note, that some error is caused in the comparison by the pressure losses in the experiment. The data available in open literature only contains the value of the inlet pressure, but in case the test section is long, the local pressure may differ significantly from the inlet value. This effect is partly taken into account by the Apros model, in which the pressure loss is taken into account as well, but these pressure losses are certainly not equal to the ones in the actual experiment, unless the implemented correlations were to perfectly estimate the wall friction. Based on the comparison above, either the Xiao or Colombo correlation could be recommended. Both of them are capable of estimating the pressure loss with a reasonable accuracy for most curvature ratios and system parameters included in our combined data set. To summarize the performance of these two correlations, Fig. 9 compares the predicted pressure losses with experimental values obtained from Zhao et al. (2003), Xiao et al. (2018) and Kozeki et al. (1970). In this case only the inlet pressure was utilized for estimating the pressure loss, without taking the pressure loss along the tube into account. This certainly causes some slight error in at least the Kozeki data, given that the

the expression for the single-phase friction factor, but not directly in the expression of the two-phase multiplier. The Guo correlation is expressed as (Guo et al., 1998)

p 2 Φlo = 142.2 ⎛⎜ ⎞⎟ ⎝ pcr ⎠ ⎧ ⎪1 + ⎪ ⎪ ψ=

⎨ ⎪ ⎪1 + ⎪ ⎩

x (1 − x )

0.62

D −1.04 ⎛ ⎛ ρ ⎞⎞ ⎛ ⎞ ψ 1 + x⎜ l⎟ ⎜ ρ ⎟ ⎝d⎠ ⎝ g ⎠⎠ ⎝

ρ − 1) ⎛ l ⎞ ( 1000 G ρ ⎝ g⎠ ⎜

⎟

ρ 1 + x ⎛⎜ l − 1⎞⎟ ρ ⎝ g ⎠

x (1 − x )

ρ − 1) ⎛ l ⎞ ( 1000 G ρ ⎝ g⎠ ⎜

ρ 1 + (1 − x ) ⎜⎛ l − 1⎟⎞ ρg

⎝

⎠

(30)

,

for G ⩽ 1000kg/m2s

,

for G ⩾ 1000kg/m2s

⎟

(31)

The Guo-correlation is based on data obtained with several different electrically heated coils oriented in different inclinations. The correlation itself does not include a term for the inclination angle, although, the original study stated that the inclination has a notable effect. The investigated helical diameters were 0.256 m and 0.132 m, with curvature ratios of 0.0430 and 0.0758. The parameter ranges of the Guo 0.5 ⩽ P ⩽ 3.5 MPa experiments were for pressure and 150 ⩽ G ⩽ 1760 kg/m2s for mass flux. The implemented correlations related to the two-phase wall friction are compared with experimental data obtained from Santini. (2008), Zhao et al. (2003), Xiao et al. (2018) and Kozeki et al. (1970). These data sets include a considerable amount of experimental data with different curvature ratios and varying system parameters. The obtained data covers roughly the pressure ranges P = 20 − 60 bar, mass fluxes G = 160 − 900 kg/m2s and curvature ratios in the range d/ D = 0.0125 − 0.0806. Table 4 summarizes the available data. Fig. 8 compares the pressure losses estimated by different correlations against experimental data in selected cases. The selected cases are representative examples for the available data. The Guo correlation does not provide good results in any of the selected cases. The default straight pipe correlation of Apros provides reasonable results for cases

Table 4 Experimental data sets related to two-phase pressure loss available in open literature with their respective ranges of state parameters.

9

Data

Mass flux G [kg/m2s]

Inlet pressure p [bar]

Curvature d/ D

Santini. (2008) Zhao et al. (2003) Xiao et al. (2018) Kozeki et al. (1970)

200–800 400–900 600 161–486

20–60 7.5–30 40 21

0.0125 0.0308 0.0329–0.0806 0.0247

Combined

161–1000

7.5–76

0.0247–0.0806

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Fig. 8. Different available wall friction correlations compared with measured data from selected experiments.

correlation provides a better accuracy in this case, knowing that the Colombo correlation is developed using this same data. The larger inaccuracy of the Xiao-correlation could be caused by the fact that the presented data is based on a coil with a relatively low curvature ratio. We conclude the present subsection by stating that either the correlation proposed by Colombo et al. or the one proposed by Xiao et al. could be utilized for estimating the two-phase pressure loss. Both correlations provide a relatively wide validity range, and most importantly, they seem to function well in a range of state parameters and curvature ratios, in which typical helically coiled steam generators generally operate.

tube was rather long in the original experiment. The Colombo correlation underestimates the Zhao data to the most part. The Xiao-correlation provides a better estimation for this data. The Xiao-correlation is in fact intended for large-curvature coils, like the one used in the Zhao experiments. In case of lower curvatures, the Colombo correlation provides slightly better results. The mean absolute percentage errors for the presented data sets are 25% for the Colombo correlation and 19% for the Xiao correlation. As the last example of the performance of the Colombo and Xiao correlations, Fig. 10 compares predicted values with experimental data obtained from Santini. (2008). Not surprisingly, the Colombo

Fig. 9. Predicted pressure losses by the Colombo and Xiao correlations compared against experimental data available in open literature. 10

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Fig. 10. Performance of the Colombo and Xiao correlations compared against experimental data available in Santini. (2008).

The active reactor part is modeled without neutronics as a 160 MW heat source. The modified Duchatelle correlation was chosen as the dryout correlation in the helical steam generator tubes. The wall friction inside the tubes is determined by the Colombo correlation, and the Steiner & Taborek correlation is utilized for determining the heat transfer coefficient in the boiling region. Otherwise the model uses the calculation methods described in Section 3. Comparison of the NuScale module design values and Apros model is presented in Table 6. Primary natural circulation coolant flow agrees well with the design data with approximately a 0.1% difference. The feedwater flow set as boundary condition was adjusted approximately 1% higher than the design value in order to compensate for missing heat losses in the Apros model. The design values assume 0.6% heat losses from total thermal power. The superheated steam that exits the steam generator is slightly higher than the design value. A likely reason for this would be a different measuring point of this temperature. That is, given that the reported NuScale design value likely corresponds to a value taken later from the steam line, there is a certain decrease in enthalpy that should be taken into account. Overall, Apros is capable of reproducing the reported design values with a very good accuracy, especially when considering all the uncertainties in the geometry of the model. The helical steam generator

3.5. Primary side of the steam generator With the lack of experimental data regarding the heat transfer through a bank of helically coiled tubes, we chose to implement the widely used Zhukauskas correlation for the heat transfer in a tube bank (Žukauskas et al., 1972). We stress, however, that the validity of this choice should be backed up by real experimental data, unavailable at this point. 4. Test case for implemented helical correlations The new correlations were input into a pre-existing Apros model based on the NuScale SMR module (Mays et al., 2015). Validation and comparison of the model was made against the design values of the NuScale 160 MW module. Dimensioning and design of the model is mainly based on the Design and Certification Application of NuScale (NuScale Standard Plant Design Certification Application, 2016a,b). Majority of the detailed design data is omitted from NuScale Standard Plant Design Certification Application (2016a,b) due to intellectual property claims. Such missing design data include steam generator internal diameter and the helix diameter for the helically coiled heat transfer tubes. The model was complemented and scaled with the MASLWR test device dimensions in order to have justified values for the missing data (International Atomic Energy Agency, 2014). Key dimensions and design values are presented in Table 5. The nodalization scheme of the model is presented in Fig. 11. The feedwater flows, module inlet and outlet temperatures and pressures are defined by suitable boundary conditions. The natural circulation in the primary circuit was matched with the design values by adjusting the flow resistance coefficients before and after the active reactor. The pressure difference over the secondary side of the steam generator was similarly matched with the design value by adjusting the flow resistance coefficients, although, it is not exactly known which points were used for obtaining the pressure difference in the NuScale data. With that, accurate comparison of the pressure loss is not possible.

Table 5 NuScale dimensions used in the Apros model. Values based either on data available in literature, or approximated when no data is available. NuScale model internal diameter NuScale model internal height SG heat transfer area Number of SG tubes SG tube outer diameter Vertical height of SG section SG tube thickness (no data) Average helix diameter (no data) SG tube pitch (no data)

11

2.743 m 17.678 m 1.666 m2 1380 15.875 mm 5.0 m 1.27 mm 1.86 m 31.5 mm

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occurs over 4 meters length, boiling and dryout over 8 meters and the superheating over rest of the 12 meters. Based on the model boiling is very efficient in the steam generator tubes. 5. Concluding remarks In this work, we have extended modelling capabilities of Apros to helically coiled steam generators. We have identified suitable correlations valid in helical geometries and implemented these into Apros. We were also interested in any available experimental data for validation purposes, and on the other hand for possible further development of the available calculation models. The present work identifies areas that are already well covered by available calculation models, and on the other hand areas in which further experimental studies would be required for more accurate models. The implemented correlations were compared against experimental data whenever possible. With our data base of experimental data obtained from open literature, we then further improved some of the available correlations for enhanced accuracy. Specifically, we proposed a modified version of the Duchatelle correlation for the first dryout quality by introducing a term for curvature effect, and re-defining the empirical coefficients of the correlation. The resulting correlation was capable of reasonably predicting the first dryout quality in the range of state parameters and curvature ratios in which helically coiled steam generators typically operate. The correlation could easily be further improved when more data becomes available. Similarly, for example the accuracy of the Colombo correlation for two-phase multipliers could most likely be improved especially in the large-curvature region in case more experimental data is available. With the newly implemented calculation models, Apros is now capable of modelling helically coiled steam generators with a reasonable accuracy. We have demonstrated that the implemented models can accurately estimate the single-phase heat transfer and pressure loss. Larger errors are made in the boiling region, but the current results are still reasonable, especially considering that the experimental data against which our models were compared do have relatively high uncertainties. We have demonstrated that the implemented models are capable of predicting the dryout phenomena with a reasonable accuracy as well, at least in the range of state parameters in which helically coiled steam generators typically operate. Yet, we note that the calculation models do have several weak points. First, the correlations for predicting the first and last dryout qualities still have a limited validity range. On the other hand, the implemented model for predicting the fraction of wetted wall in the transition region between the first and last dryout qualities is at this

Fig. 11. Nodalization scheme of the NuScale module used in the test calculation.

Table 6 NuScale design values compared with those estimated by the Apros model.

Primary side Thermal power Hot leg temperature Cold leg temperature Temperature difference Primary flow at 100% Secondary side Steam flow to turbine Steam pressure Feedwater pressure Pressure difference over SG Steam temperature Feedwater temperature Saturation temperature at SG average pressure

NuScale

Apros

Unit

Difference

160 310 258 52 587

160 309.4 257.6 51.8 587.7

MW °C °C °C kg s−1

– 0.2% 0.2% 0.4% 0.1%

67.1 34.0 35.2 1.2 307 149 242

68.0 33.96 35.2 1.24 308.1 149 243

kg s−1 bar bar bar °C °C °C

1% 0% 0% 2% 0.4% 0% 0.3%

model is shown to be accurate with the chosen correlation sets. The temperature profile of the primary and secondary circuit over the steam generator length is displayed in Fig. 12. The pre-heating of the water

Fig. 12. Temperature profiles of the primary and secondary sides over the steam generator length. 12

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point quite approximative. Second, a model for subcooled boiling in helically coiled tubes could improve the results. Further, we note that the implemented Zhukauskas correlation for predicting the heat transfer in the primary side of the steam generator is likely to be a suitable choice, but experimental data would be required for verifying this. The Zhukauskas correlation is developed for a bank of straight tubes, not helical. Last, in a two-fluid thermal hydraulic model such as the one Apros bases on, the total pressure loss estimated by our implemented two-phase multipliers must be divided among the two phases. How the division should be done depends on the flow regime, and specifically of the fraction of wall covered by liquid or gas. No models for predicting this is at this point available.

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