Thermal analysis of HTGR helical tube once through steam generators using 1D and 2D methods

Nuclear Engineering and Design 355 (2019) 110352

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Thermal analysis of HTGR helical tube once through steam generators using 1D and 2D methods

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Xiaowei Li , Weikai Gao, Yang Su, Xinxin Wu Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China

A R T I C LE I N FO

A B S T R A C T

Keywords: High temperature gas-cooled reactor Once through steam generator Thermal deviation analysis Helical tube Two-dimensional method

The working temperature of High Temperature Gas-cooled Reactor (HTGR) steam generators is much higher than that of Pressurized Water Reactor (PWR) steam generators. Large temperature non-uniformities will make the HTGR steam generator deviate from the designed working temperatures, which should be avoided. The temperature distribution and thermal deviation should be analyzed. One-dimensional (1D) and two-dimensional (2D) methods for analyzing the thermal hydraulic performance of an HTGR steam generator were developed. In the 1D method, the tube and shell side governing equations were solved. They were coupled through the energy source terms. In the 2D method, simulation of the shell side helium cross flow over tube bundles was based on the two-dimensional CFD code Ansys Fluent. The tube side water flow was coupled to the 2D simulation of the cross flow over tube bundles through the heat transfer between the primary and secondary sides. The 1D and 2D methods were validated using the measured data of the Thorium High Temperature Reactor (THTR) steam generators and the High Temperature Gas-cooled Reactor Pebble-bed Module (HTR-PM) test steam generator. The 1D method can give the overall thermal performance of the steam generators. The 2D method can predict temperature non-uniformities in the tube bundles. The overall thermal performance of the steam generator was simulated. The thermal non-uniformities or thermal deviations caused by geometrical and thermal hydraulic deviations were analyzed. Radiation heat transfer contributes to 0.9% of the total thermal power. A one percent change of the secondary side water flow rate of a certain layer of tubes will result in a 5–6 °C variation of the outlet steam temperature of this layer of tubes. A helical diameter deviation of 1 mm will result in a maximum outlet steam temperature variance of 6–7 °C. An inlet helium temperature profile with a 10 °C variance will only result in a variation of 2 °C in the outlet steam temperature.

1. Introduction High Temperature Gas-cooled Reactors (HTGRs) are characterized by excellent safety features (Zhang et al., 2009; Wu and Zhang, 2000; Kunitomi et al., 2004). By modular construction, it may be economically comparable with pressurized water reactors (Zhang and Sun, 2007). Due to previous experiences gained during the construction and operation of gas cooled reactors, HTGRs are some of the most studied reactors among the Generation IV reactors. The steam generators (SGs) are the key heat transfer components of an HTGR. In order to reduce the size of its pressure vessels, the SG should be very compact. The heat transfer tubes are the pressure boundary of the coolant of HTGRs. So they require high reliability and the SG is classified as a class I component. The working temperature of an HTGR SG is much higher than that of a PWR SG, which makes its design and safe operation more

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difficult. Due to the critical working conditions of an HTGR steam generator (750 °C or even higher), its thermal hydraulic analysis and thermal deviation analysis become very important. Large temperature differences will damage the structure of the component or shorten its inservice time. Temperature non-uniformities in the steam generator will increase the working temperature of certain heat transfer tubes. High temperatures will make the tube material creep and lower its allowable stresses significantly. At a temperature around 675 °C, the allowable stresses of Incoloy 800H will decrease by 10% when its temperature increases by 15 °C (ASME, 2013). Heysham #1 and #2 Advanced Gascooled Reactor (AGR) were constructed in the early 1980s in UK. A large non-uniform temperature distribution in the AGR SG was observed during the commissioning period. In order to guarantee the maximum temperature of the heat transfer tubes below the design

Corresponding author. E-mail address: [email protected] (X. Li).

https://doi.org/10.1016/j.nucengdes.2019.110352 Received 11 April 2019; Received in revised form 12 September 2019; Accepted 16 September 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.

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Nomenclature a Cp d D F G h k K p Pr R Re t T ux ur z

Greek symbols

thermal diffusivity [m2s−1] specific heat [kJkg−1K−1] tube diameter, [m] helical diameter, [m] momentum source term, [Nm−3] mass flux, [kgm−2s−1] enthalpy, [Jkg−1] turbulence kinetic energy, [m2s−2] total heat transfer coefficient based on the outer tube diameter, [Wm−2K−1] pressure, [Pa] Prandtl number, dimensionless tube wall conduction thermal resistance based on the outer tube diameter, [W−1m2K] Reynolds number, dimensionless time [s] temperature, [°C or K] axial velocity, [m/s] radial velocity, [m/s] tube length in 1D method, [m]

λeff ε μ ρ Φ σt

effective thermal conductivity [Wm−1K−1] turbulence dissipation rate, [m2s−3] dynamic viscosity [kgm−1s−1] density, [kgm−3] energy source term, [Wm−3] turbulent Prandtl number, dimensionless

Subscript eff He i n o t x r w

effective helium inside the number of the node outside turbulent axial direction radial direction wall

tube, overall heat transfer rate and the pressure drop of the primary and secondary side fluids. A 2D method can also predict the temperature distribution in the radial direction in the SGs. It can take into account the influences of helical diameter, tube plugging, inlet velocity and temperature distributions on the temperature non-uniformity in the shell side. It can consider the thermal mixing of cross flow over tube bundles in the primary side of the SG, which is important for analyzing the temperature non-uniformity. Due to the large temperature non-uniformities encountered in the AGR helical tube once through steam generators, some researchers (Gane et al., 1985; Khan, 1988; Achenbach, 1987; Gulich, 1972) investigated this using numerical methods based on the concept of porous media. Simulation of the temperature distributions in High Temperature Gas-cooled Reactor Pebble-bed Module (HTR-PM) steam generators based on the porous media method were also carried out (Olson et al., 2014). The porous media method has the advantage of requiring a small amount of computational resources. However, its model needs empirical parameters and experiments to support, especially the mixing coefficients. It cannot account for some exact analyses, for example the wall effect (Li et al., 2014). The capabilities of computers have increased significantly in recent decades. Cross flow over tube bundles

values, the reactor power only reached 60% design power (Mathews, 1987). So more detailed and precise analysis of the temperature nonuniformity in the HTGR SG is required. New analysis methods should be developed accordingly. Many factors, including geometrical deviations and thermal-hydraulic deviations, have influences on the temperature non-uniformities in a SG. Geometrical deviations are mainly determined during the manufacturing process. They include the tube diameter uncertainty, wall clearance uncertainty, tube length uncertainty, tube helical diameter uncertainty, tube thermal expansion effect, etc. Thermal-hydraulic deviations may be caused by the geometrical deviations of the SG or by velocity or temperature non-uniformities at the inlet boundaries for the SGs. They include the flow rate and temperature nonuniformities in the primary and secondary sides among different units of helical tube bundles, inlet velocity non-uniformity and inlet temperature non-uniformity of one SG unit (one helical tube bundle), tube plugging, etc. One-dimensional (1D) and two-dimensional (2D) codes can be used to analyze the influences of these deviations on temperature distributions. A 1D code requires less computing resources, but it can only analyze the overall characteristics of the SG (or one helical tube bundle unit), including the temperature distribution along the

Fig. 1a. Geometrical model of HTR-PM steam generator. Cross section of steam generator with 19 units (Zhang et al., 2009). 2

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differences among the 19 units. So simulation of one helical tube bundle unit can represent the overall performance of the whole SG. The 1D method uses empirical correlations to calculate the pressure drop and heat transfer of the flow in the helical tubes and over the tube bundles of HTGR steam generators. In the 1D method, only one helical tube with an average helical diameter was simulated. The tube and shell side flow areas were simplified to one-dimensional single channel models. The shell side was simplified, and the characteristic diameter and velocity were set identical to the real model to make the flow resistance and heat transfer coefficient identical. The required input parameters are, the inlet temperature, pressure, mass flow rate of helium in the shell side and those of water in the tube side. The governing equations are shown below.

can be predicted with good accuracy using the CFD method by directly solving the Reynolds Averaged Navier Stokes (RANS) equations together with turbulence models (Barsamian and Hassan, 1997; Paul et al., 2008; Salpeter and Hassan, 2012; Hassan and Barsamian, 2004; Carasik et al., 2017; Lee et al., 2018; Feng et al., 2019; Blackall et al., 2019; Iacovides et al., 2014). And the mixing effect can be predicted correctly (Li and Wu, 2013). This provides alternative investigation methods (Li, 2012; Li et al., 2012) for analyzing the temperature nonuniformities in HTGR steam generators. 1D and 2D codes were developed for the thermal analysis of the HTR-PM SG. The one-dimensional governing equations for the shell and tube sides were solved in the 1D code. The 2D code was based on the commercial CFD code ANSYS Fluent. The governing equations including the 2D unsteady Reynolds Averaged Navier-Stokes equations and energy equation were solved for the shell side, and a 1D method was used for the tube side. The thermal mixing in the primary side caused by the tube bundles can be taken into account.

2.2.1. Governing equations for shell side The working fluid in the shell side of the SG is single-phase helium. The governing equations for the shell side are the steady-state mass, momentum and energy equations. Mass conservation equation:

2. Numerical method

dG =0 dz

The 1D method is helpful for understanding the overall performances of HTGR steam generators, and it is the basis for the 2D method. The 1D method will be introduced first, and the 2D method with thermal mixing capability will be introduced then.

where G = ρu, is the mass flux. Momentum conservation equation:

2.1. Geometrical model of HTR-PM SG

G

The steam generator of HTR-PM consists of 19 separate helical tube bundles (19 units). The cross section of steam generator with 19 units are shown in Fig. 1(a). The geometrical model of one unit is shown in Fig. 1(b). Each unit has 5 layers of helical tubes. There are 5, 6, 7, 8, 9 tubes for the No. 1, 2, 3, 4, 5 layers, and in total 35 helical tubes for one unit. There are connection tubes between the helical tube bundles and the outlet steam tube plate, and also connection tubes between the helical tube bundles and the inlet water tube plate.

where θ is the tube inclination angle, p is pressure, F is the momentum source term. Energy conservation equation:

G

dp du =− − ρg sin θ − F dz dz

dh =Φ dz

(1)

(2)

(3)

where Φ is the energy source term. The source terms in the momentum and energy equations are calculated using the empirical correlations for the pressure drop and heat transfer coefficient of helium cross flow over tube bundles (Idelchik, 1986; Zukauskas and Ulinskas, 1988).

2.2. One-dimensional method The 19 helical tube bundle units of the HTR-PM steam generator are identical. There only exist slight geometrical differences caused by fabrication tolerances and also slight thermal hydraulic differences induced by geometrical differences and inlet hot helium temperature differences. Thermal hydraulic differences include primary and secondary side mass flow rate differences and inlet hot helium temperature

2.2.2. Governing equations for tube side The flow and heat transfer in the tube side includes two single-phase flow regions and also two-phase flow regions. The single-phase flow regions include the subcooled water convection region near the inlet of the tube and the superheat steam convection region near the outlet of

Fig. 1b. Geometrical model of HTR-PM steam generator. Geometrical model of one unit of helical tube bundle. 3

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the tube. The two-phase flow regions were divided into two regions, which are the saturated nucleate boiling region and liquid deficient region. The working pressure in the secondary side is high, so the 1D homogeneous model is used for the two-phase flow regions. The steady-state governing equations for the single-phase water and single-phase steam flow regions are the same as that for the helium flow in the shell side. However, the expressions for the source terms in the momentum and energy equations are different. The governing equations for the two-phase flow regions are shown as follows: Mass conservation equation:

dG =0 dz

2.2.3. Discretization and numerical solution The governing equations (Eqs. (1) to (6)) were discretized using the finite difference method. The axial heat conduction of the helical tube was neglected. The grid is illustrated in Fig. 2. The local heat flux at the No. n node was calculated as,

q = Kn (THe, n − TH2 O, n )

where Kn is the local total heat transfer coefficient at No. n node based on the outer tube diameter do. THe, n and TH2 O, n are the local helium and water temperatures at No. n node. The heat flux can also be calculated using Eq. and Eq. was also used for calculating the tube wall temperatures.

(4)

q = hHe, n (THe, n − Two, n ) =

where G = ρhuh is the mass flux, ρh is the homogeneous fluid density, uh is the homogeneous fluid velocity. Momentum conservation equation:

G

duh dP =− − ρh g sin θ − FTP dz dz

h H2 o, n di (Two, n − Twi, n ) (Twi, n − TH2 o, n ) = Rw do (8)

The local total heat transfer coefficient at No. n node was calculated as,

(5)

Kn =

1 1 hHe, n

where FTP is the two-phase momentum term. Energy conservation equation:

dh G =Φ dz

(7)

+ Rw +

do h H2 O, n di

(9)

where hHe, n is the local heat transfer coefficient of helium cross flow over helical tube bundles at the No. n node, h H2 O, n is the local water heat transfer coefficient in the helical tubes, Rw is the tube wall conduction thermal resistance based on the outer tube diameter.

(6)

where Φ is the energy source term. The pressure drop in the single-phase region was calculated using the Ito correlation (Ito, 1959). The heat transfer coefficient in the single-phase region was calculated using the empirical correlation proposed by Yang and Tao (1998), which is based on the straight tube correlation and adopts a correction factor. The heat transfer and pressure drop in the two-phase flow region were calculated using the empirical correlations for straight tubes proposed by Collier and Thome (1994) and the Chinese standard method. In order to take the helical effect into consideration, a correction factor was also added. For the details of the correlations used in this paper, please refer to Olson et al., 2014 and Ma et al., 2014.

Rw =

ln(do/ di ) do 2λ

(10)

where λ is the thermal conductivity of the tube wall. The heat transfer rate used for calculating the source term of the shell side energy equation was equal to that for calculating the source term of the tube side energy equation. Thus, the tube side and shell sides are coupled. The pressures and the velocities for the 1D problem are decoupled automatically and can be calculated directly from the node at the inlet when the inlet conditions are given. The flow chart of the calculation procedure for the 1D method is shown in Fig. 3. The coupling between

Fig. 2. 1D grid distribution. 4

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Fig. 3. Flow chart of calculation procedure of the 1D method.

(Reynolds Averaged Navier-Stokes) governing equations. The flow on the shell side was approximated as being two-dimensional and axisymmetric. The governing equations for the tube side were the same as those in the 1D method. The governing equations for the shell side were solved by the commercial CFD code Fluent. Due to the axisymmetric assumption, the enthalpy increment of the water in one circle of a helical tube was obtained by multiplying the heat transfer rate by πD, where D was the helical diameter of the particular tube. Thus, the temperature and enthalpy of the tube side water along the tube can be predicted. The flow and heat transfer of the tubes in the different layers were calculated individually. These were all realized and incorporated into the Fluent software using User Defined Functions (UDFs).

the water and helium side was very strong. The solution of the two sets of discretized equations needs iteration. The two-phase flow heat transfer calculation also need iteration. A thousand of increments were enough for the accurate calculation of the temperature and pressure distributions. The convergence criteria for the tube side was that the residual of the heat flux was below 1 × 10−6 W/m2, and the residual of the pressure was below 1 × 10−9 Pa. The convergence criteria for the shell side was that the relative residual of the temperature was below 1 × 10−6 . 2.3. Two-dimensional method The advantage of the 2D method is that it can calculate the temperature distribution in the steam generator, which makes the thermal deviation analysis possible. The flow and heat transfer in the tube side was assumed to be one-dimensional. It was represented by uniform bulk values at the axial positions of each helical tube. The flow and heat transfer in the shell side was calculated directly by solving the RANS

2.3.1. Governing equations for shell side The governing equations for the shell side are the time-dependent two-dimensional axisymmetric compressible Reynolds Averaged Navier-Stokes equations and the energy equation. Mass conservation equation: 5

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ρur ∂ (ρur ) ∂ (ρu x ) ∂ρ + =0 + + r ∂r ∂x ∂t

2.4. Validation of the methods (11) 2.4.1. Validation of one-dimensional method The 1D code was validated by calculating the tube and shell side temperatures of the steam generator of the Thorium High Temperature Nuclear Reactor (THTR) and also that of the test steam generator of High Temperature Gas-cooled Reactor Pebble-bed Module (HTR-PM). The results are shown in Figs. 6(a) and (b). It can be seen that the calculated results are in good agreement with the THTR measurements at full power level (Henry and Elter, 1987). The calculated results are also in good agreement with the measurements from the test SG of HTRPM at 75% power level (Li et al., 2019). The temperature distributions in the HTGR steam generator have the characteristics of a large temperature difference in the superheat steam region and low temperature

Momentum conservation equations:

1 ∂ ∂ ∂ (ρu x ) + (ρu x u x ) + (ρru x ur ) r ∂r ∂t ∂x 2 1 ∂ (rur ) 1 ∂ ∂u x ⎞ ⎞ ⎤ ∂u ∂P ∂ ⎡ μeff ⎛2 x − ⎛ + + + 3 ⎝ r ∂r r ∂r ∂x ⎠ ⎠ ⎥ ∂x ∂x ⎢ ⎝ ∂x ⎦ ⎣ ⎡rμ ⎛ ∂u x + ∂ur ⎞ ⎤ ⎢ eff ⎝ ∂r ⎥ ∂x ⎠ ⎦ ⎣

=−

⎜

⎟

(12)

1 ∂ ∂ ∂ (ρur ) + (ρu x ur ) + (rρur ur ) r ∂r ∂t ∂x 1 ∂ ∂P ∂ ⎡ ∂u ∂u x ⎞ ⎤ μeff ⎛ r + =− + + r ∂r ∂r ∂x ⎢ ∂r ⎠ ⎥ ⎝ ∂x ⎣ ⎦ ⎡rμ ⎛2 ∂ur − 2 ⎛ 1 ∂ (rur ) + ∂u x ⎞ ⎞ ⎤ − 2μ ur + 2 μeff eff 2 ⎢ eff r ∂x ⎠ ⎠ ⎥ 3 ⎝ r ∂r 3 r ⎝ ∂x ⎦ ⎣ ⎛ ⎛ 1 ∂ (rur ) + ∂u x ⎞ ⎞ ∂x ⎠ ⎠ ⎝ ⎝ r ∂r ⎜

⎟

⎜

⎟

(13)

where μeff is the effective viscosity, μeff = μ + μt , μt is the turbulent k2

viscosity, μt = ρCμ ε . Boussinesq hypothesis and k-ε turbulence model was used for the modeling of the Reynolds stress term. Energy conservation equation:

∂ (ρT ) ∂ (ρT ) ∂ (ρT ) + ux + ur ∂x ∂r ∂t μt ⎞ ∂T ⎤ μ ∂T μ ∂T ∂ ⎡⎛ μ 1 μ ∂ ⎡⎛ μ + + t⎞ ⎤ + ⎛ + t⎞ = + ∂x ⎢ σt ⎠ ∂r ⎥ r ⎝ Pr σt ⎠ ∂x σt ⎠ ∂x ⎥ ∂r ⎢ ⎣ ⎝ Pr ⎦ ⎣ ⎝ Pr ⎦ ⎜

⎟

⎜

⎟

⎜

⎟

(14) where σt = 0.85 is the turbulent Prandtl number. 2.3.2. Geometrical model and grid distribution The 2D geometrical model of the shell side is shown in Fig. 4. It is one helical tube bundle unit of the HTR-PM steam generator. The tube bundles were assumed to be axisymmetric. The flow and heat transfer in the tube side were connected using the UDFs provided by Fluent. Fig. 5(a) shows the overall grid distribution of the model. The grid was refined in the near wall region to ensure that y+ was below 1. The local grid distribution is shown in Fig. 5(b). The total number of elements was about 1, 200, 000. 2.3.3. Boundary condition and solution strategy The mass flow rate in the tube bundle was specified at the entrance. A pressure boundary condition was prescribed at the outlet of the tube bundle. Non-slip and adiabatic wall boundary condition was assumed for the inner wall and outer wall. The operating pressure was 7 MPa. Convective and non-slip wall boundary conditions were assumed for the tube walls. The governing equations for the shell side were solved using ANSYS Fluent. The 1D code was used for calculating the tube side heat transfer coefficient, heat transfer in one loop of the helical tube and the water temperature increment in the helical tubes. Then the water temperature, tube side heat transfer coefficient were set as the convection boundary conditions in Fluent using UDFs. In some cases, averaged heat transfer coefficients were used for the subcooled water, boiling and superheat steam regions for easier convergence of the cases. The two-layer model was used for the near wall treatment. The coupling of the velocity and pressure was treated employing the SIMPLEC algorithm. The QUICK scheme was used for the discretization of the convection terms. The time step was 0.0001 s. The working fluid in the shell side was helium which properties varied with the working temperature.

Fig. 4. 2D axisymmetric geometric model of the steam generator. 6

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Fig. 5b. Grid distribution of the 2D model. Local grid distribution in the near wall region.

Fig. 6a. Validation of the 1D code.Tube and shell side temperature distribution of THTR SG at 100% power level.

Fig. 5a. Grid distribution of the 2D model. Overall grid distribution of the SG.

difference in the subcooled water region. This is determined by the design of the overall parameter of the nuclear power system (Zhang et al., 2016). The hot helium temperature (750 °C) of an HTGR is usually very high, while the steam temperature (570 °C) is determined by the operating parameters of the turbines. So the temperature difference will be large. For the THTR steam generator, the contribution of the subcooled water, two-phase boiling, and superheat steam regions to the tube length and thermal power are shown in Table 1. The subcooled water convection region accounts for more than 65% of total tube length, but it only contributes 40% of the total thermal power.

2.4.2. Validation of two-dimensional method The main objective of the 2D method is to investigate the temperature non-uniformities and thermal deviation of the HTGR steam

Fig. 6b. Validation of the 1D code. Tube and shell side temperature distribution of HTR-PM SG at 75% power level. 7

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Table 1 Tube length and thermal power distribution of THTR steam generator. Items

Subcooled water region

Two phase boiling region

Superheat steam region

Percentage of thermal power Percentage of tube length

39.6% 66.7%

19.4% 13.4%

41.0% 20.0%

generators. However, the 2D method should first predict the overall thermal parameters correctly. In order to test the thermal calculation capability of the 2D method, the temperature distribution along the tube in the HTR-PM SG with 100% power level was predicted using the 1D and 2D methods. The results are shown in Fig. 7(a). It is shown that the results of the 2D method agree well with the results predicted by the 1D method. The temperatures in the tube and shell side calculated by the 2D method are averaged over the cross section. The length of the subcooled water region accounts for more than 50% of total tube length, but it contributes less than 30% of the total thermal power. It is similar to that of the THTR SG. The shell side cross flow over tube bundles was simulated using the CFD code Fluent in the 2D method. The correctness of the calculation of the shell side flow and heat transfer was assured by the Fluent software. The work of Li and Wu (2013) showed that Fluent can predict the cross flow over tube bundles with sufficient accuracy. Fig. 7(b) further compares the calculated and measured averaged steam temperatures at the outlet of the 5 layers of helical tubes of the test HTR-PM SG. It can be seen that the calculated steam temperature distribution compares well with the measurements. Both of the results show a lower steam temperature for the first layer and a higher steam temperature for the fifth layer. We will further discuss this steam temperature differences among the 5 layers of tubes.

tube plate of the steam generator. Fig. 7(b) shows the outlet steam temperature of the five layers (lines) of tubes. It is shown that the steam temperature difference among the five layers of tubes is below 15 °C. The tube lengths of the 35 tubes in a helical tube bundle unit in the steam generator are identical, so ideally the outlet steam temperature should be identical. This temperature difference is mainly determined by the non-uniformities of the shell side cross flow and heat transfer (Li et al., 2014). The outlet steam temperature difference caused by geometrical and thermal hydraulic deviations will be discussed in Section 4. 4. Temperature non-uniformity analysis 4.1. Thermal hydraulic deviation analysis The temperature and flow rate deviations in the tube side and shell side will contribute to the temperature non-uniformity of the SG. The primary side helium temperature deviations are caused by the nonuniform temperature of the helium that comes out of the reactor core. The secondary side water temperature will usually be uniform, while the secondary side flow rate of each helical tube may have small differences due to the geometrical deviations. These include the inherent differences among the helical diameters of the 5 layers of tubes and also manufacturing differences of the tube inside diameters. The primary side helium flow rate for the different SG units (or helical tube bundles) may have small differences due to the fabrication tolerances of the helical tube bundles.

3. Overall thermal performance 3.1. Temperature distribution along the tube Fig. 8 shows the distributions of the shell side helium temperature, tube side water temperature, tube inner wall, outer wall and average wall temperatures along a tube of the HTR-PM steam generator using the 1D method. The tube wall temperature was relatively low in the subcooled water region (preheater) and boiling regions, while it increased obviously in the superheat steam region. There was a jump in the inner and outer wall surfaces temperature at approximately 46 m. This was because of the dry out of the convective boiling in the tube. This calculation results can provide temperature data for the determination of the material and wall thickness of the tubes. Fig. 9 shows the radiation heat flux from the inner and outer bounding walls (cylindrical) to the tube bundle surfaces. The results of the 1D and 2D methods are in good agreement. The 1D radiation heat flux in Fig. 9 was calculated based on the temperature shown in Fig. 8. The 2D radiation heat flux in Fig. 9 was calculated by Fluent using the P1 model. The emissivity was set as 1. The radiation heat flux is much lower in the single-phase water region. It then increases slowly along the tube length in the two-phase flow region. The radiation heat flux in the superheat steam region reaches a relatively high value. This is because the temperature in the superheat steam region is relatively high and the temperature difference between the helium and outside wall of the tube is very large. Integration of the radiation heat flux along the tube, yields a radiation heat transfer power of about 120 kW, which is approximately 0.9% of the total thermal power of the SG unit.

4.1.1. Helium flow rate deviation There are 19 helical tube bundle units in the HTR-PM steam generator. The helium flow rate in each steam generator unit may not be uniform due to the entrance effect and the different flow resistances of the 19 units caused by fabrication tolerances. The 1D method was used to simulate the influence of the helium flow rate on the outlet steam temperature non-uniformity. The helium flow rate deviation among the steam generator units will not vary much (about 1%). Fig. 10 shows the influence of a 1 percent helium flow rate variance on the steam

3.2. Outlet steam temperature distribution The uniformity of the outlet steam temperature is an important factor for evaluating the temperature uniformity in the steam generators. A large steam temperature difference is also not good for the

Fig. 7a. Validation of the 2D code. Tube and shell side temperature distribution along a tube in HTR-PM SG. 8

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Fig. 10. Influence of helium flow rate deviation on steam temperature. Fig. 7b. Validation of the 2D code. Comparison of calculated and measured outlet steam temperature distribution.

4.1.2. Helium temperature profile The helium that flows out of the reactor core will have large temperature differences (Zhou et al., 2014) due to the non-uniform nuclear power distribution in the reactor core and also the bypass flows through the gaps between the graphite bricks (Sun et al., 2013). Though the flow mixing structure at the bottom of the reactor is efficient, there may also be temperature non-uniformities when the helium flows into the steam generator units. The 2D method could be used to investigate the influence of the inlet helium temperature profile on the steam temperature non-uniformities. As shown in Fig. 11, a Gauss type temperature profile was assumed as the inlet helium temperature profile. The peak temperature variation was 10 °C. High temperature helium will transfer more heat to the tubes over which it flows, thus the steam temperature in these tubes will become higher. Figs. 12 shows the calculated outlet steam temperatures and the temperature deviations for one SG unit. It can be seen that the 10 °C temperature peak has a very small influence on the outlet steam temperature. The outlet steam temperature increase is below 2 °C. This is due to the thermal mixing effect of the tube bundle, which will make the helium temperature profile to become uniform after several rows of tubes.

Fig. 8. Temperature distributions along a tube of HTR-PM SG.

4.1.3. Water side flow rate deviation The water flow rate in the 35 tubes of one steam generator unit may also be not uniform due to the deviations of the tube inner diameter, orifice diameter, tube length, helical diameter, etc. The 2D method could also be used for the water side flow rate deviation analysis. Fig. 13 shows the outlet steam temperature variance induced by a

Fig. 9. Radiation heat flux distribution in the steam generator.

temperature. The influence of the helium flow rate on the steam temperature is almost linear. A 1 percent helium flow rate variation will make the steam outlet temperature change by 8.5 °C. Fig. 11. Helium inlet temperature profile. 9

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secondary side flow rate decrease of 1% in each layer of tubes. The results mean that when the secondary side water flow rate of a certain layer tubes decreases by 1%, then the outlet steam temperature of this layer will increase about 5–6 °C, while the outlet steam temperatures of the other 4 layers of tubes will increase about 1–2 °C. This clearly shows the thermal mixing effect of the tube bundle. The reduction in the secondary side flow rate will not only increase the outlet steam temperature of this layer of tubes but also increase that of the other layers of tubes. 4.2. Geometrical deviation analysis Geometrical deviations are caused mainly by manufacturing or fabrication tolerances. Helical diameter deviations will be carefully analyzed in the following section. Some geometrical deviations will result in thermal hydraulics deviations, which will also be discussed. 4.2.1. Helical diameter deviations The helical diameter of the heat transfer tubes will influence the transverse pitch of the tube bundle. Pitch variations will have influences on the helium flow channels in the primary side. Fabrication errors and thermal expansion will introduce helical diameter deviations. The extreme case is that the helical diameters of all the 310 rows of helical tubes in a layer shift in the same direction, which means the helical diameters of all the 310 rows of helical tubes of this layer increase or decrease the same amount of distance. The helical diameters of the five layers of tubes will be increased or decreases 1 mm to investigate the influence of the helical diameter deviations on the outlet steam temperature non-uniformity of the HTGR steam generators. Fig. 14(a) is the outlet steam temperature distribution with the helical diameter of the 1st layer of tubes increasing or decreasing by 1 mm. When the helical diameter of the 1st layer of tubes decreases, the helium flow channel area between the 1st and 2nd layer of tubes will increase, while the clearance between the inner cylinder wall and the 1st layer of tubes will decrease. This will make the helium flow rate for heating the 1st layer of tubes to decrease, while that for heating of the 2nd layer of tubes to increase. It is the opposite when the helical diameter of the 1st layer of tubes increases. Fig. 14(b) is the outlet steam temperature variance compared with the original ideal case. It can be seen from Fig. 14(b) that when the helical diameters of the 1st layer of tubes increase by 1 mm, the outlet steam temperatures of the 1st layer of tubes will increase by about 4 °C, while that of the 2nd layer of tubes will decrease by about 6 °C. The outlet steam temperatures of the 3rd, 4th, and 5th layers of tubes will not change much. The influence on the 5th layer of tubes are the lowest, which is due to the temperature

Fig. 12a. Influences of helium inlet temperature profile on outlet steam temperature. Outlet steam temperature.

Fig. 12b. Influences of helium inlet temperature profile on outlet steam temperature. Outlet steam temperature variance.

Fig. 13. Outlet steam temperature variance induced by a secondary side flow rate decrease of 1% Fig. 14a. Temperature change caused by helical diameter deviation of the 1st layer of tubes. Outlet steam temperature. 10

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4th layer of tubes will increase by about 2.5 °C, while that of the 5th layer of tubes will increase by 2 °C. Fig. 17(a) is the outlet steam temperature distribution with the helical diameter of the 4th layer of tubes increasing or decreasing by 1 mm. When the helical diameter of the 4th layer of tubes decreases, the helium flow channel area between the 3rd and 4th layers of tubes will decrease, while the helium flow channel area between the 4th and 5th layers of tubes will increase. This will result in the helium flow rate for heating the 3rd layer of tubes to decrease, while the helium flow rate for heating the 5th layer of tubes to increase, while those for heating the 1st, 2nd and 4th layers of tubes are almost unchanged. Fig. 17(b) is the temperature variance compared with the original ideal case. It can be seen from Figs. 17 that when the helical diameters of the 4th layer of tubes increase by 1 mm, the outlet steam temperature of the 1st and 2nd layers of tubes will increase by about 4 °C, while that of the 3rd layer of tubes will increase by about 5 °C. The outlet steam temperature of the 5th layer of tubes will decrease by about 4.5 °C, while that of the 4th layer remains almost unchanged. When the helical diameter of the 4th layer of tubes decreases by 1 mm, the outlet steam temperatures of the 1st and 2nd layers of tubes will decrease by about 3 °C, while that of the 3rd layer of tubes will decrease by about 4 °C. The outlet steam temperature of the 5th layer of tubes will increase by about 5 °C, while that of the 4th layer of tubes remains almost unchanged. Fig. 18(a) is the outlet steam temperature distribution with the helical diameter of the 5th layer of tubes increasing or decreasing by 1 mm. When the helical diameter of the 5th layer of tubes decreases, the helium flow channel area between the 4th and 5th layers of tubes will decrease, while the helium flow channel area between the 5th layers of tubes and the outer cylinder wall will increase. This will result in the helium flow rate for heating the 4th layer of tubes to decrease, while the helium flow rate for heating the 5th layer of tubes increases, while those for heating the 1st, 2nd and 3rd layers of tubes are almost unchanged. Fig 18(b) is the temperature variance compared with the original ideal case. It can be seen from Figs. 18 that when the helical diameters of the 5th layer of tubes increase by 1 mm, the outlet steam temperature of the 4th, 3rd, 2nd and 1st layers will increase by about 5 °C, 3 °C, 2.5 °C and 2.5 °C respectively, while that of the 5th layers of tubes will decrease by about 6 °C. When the helical diameter of the 5th layer of tubes decreases by 1 mm, the outlet steam temperature of the 4th, 3rd, 2nd and 1st layers of tubes will decrease by about 3.5 °C, 2.5 °C, 2 °C and 0.5 °C respectively, while that of the 5th layer of tubes will increase by about 7 °C. From the above analysis, the helical diameter deviation of 1 mm will result in a maximum outlet steam temperature change of 6–7 °C.

Fig. 14b. Temperature change caused by helical diameter deviation of the 1st layer of tubes. Outlet steam temperature variation.

diffusion or thermal mixing of the tube bundle. When the helical diameter of the 1st layer of tubes decreases by 1 mm, the outlet steam temperature of the 1st layer of tubes will decrease by about 7.5 °C, while that of the 2nd layer of tubes will increase by about 2.5 °C. The outlet steam temperatures of the other 3 layers of tubes change little. Fig. 15(a) is the outlet steam temperature distribution with the helical diameter of the 2nd layer of tubes increasing or decreasing by 1 mm. When the helical diameter of the 2nd layer of tubes decreases, the helium flow channel area between the 1st and 2nd layers of tubes will decrease, while the helium flow channel area between the 2nd and 3rd layers of tubes will increase. This will make the helium flow rate for heating the 1st layer of tubes to decrease, while the helium flow rate for heating the 3rd layer of tubes to increase, while those for heating the 2nd, 4th and 5th layers of tubes are almost unchanged. Fig. 15(b) is the temperature variance compared with the original ideal case. It can be seen from Fig. 15(b) that when the helical diameters of the 2nd layer of tubes increase by 1 mm, the outlet steam temperature of the 1st layer of tubes will increase by about 7 °C, while that of the 3rd layer of tubes will decrease by about 4.5 °C. The outlet steam temperatures of the 2nd, 4th and 5th layers of tubes will not change much. When the helical diameters of the 2nd layer of tubes decrease by 1 mm, the outlet steam temperature of the 1st layer of tubes will decrease by about 5.5 °C, while that of the 3rd layer of tubes will increase by about 4 °C. The outlet steam temperatures of the other 2nd, 4th and 5th layers of tubes change little. Fig. 16(a) is the outlet steam temperature distribution with the helical diameter of the 3rd layer of tubes increasing or decreasing by 1 mm. When the helical diameter of the 3rd layer of tubes decreases, the helium flow channel area between the 2nd and 3rd layers of tubes will decrease, while the helium flow channel area between the 3rd and 4th layers of tubes will increase. This will make the helium flow rate for heating the 2nd layer of tubes to decrease, while the helium flow rate for heating the 4th layer of tubes to increase, while those for heating the 1st, 3rd and 5th layers of tubes are almost unchanged. Fig. 16(b) is the temperature variance compared with the original ideal case. It can be seen from Fig. 16(b) that when the helical diameters of the 3rd layer of tubes increase by 1 mm, the outlet steam temperatures of the 1st layer of tubes will increase by about 4 °C, while that of the 2nd layer of tubes will increase by about 5 °C. The outlet steam temperatures of the 4th layer of tubes will decrease by about 3 °C, while that of the 5th layer of tubes will decrease a little. When the helical diameter of the 3rd layer of tubes decreases by 1 mm, the outlet steam temperatures of the 1st layer of tubes will decrease by about 1.5 °C, while that of the 2nd layer of tubes will decrease by about 4 °C. The outlet steam temperatures of the

Fig. 15a. Temperature change caused by helical diameter deviation of the 2nd layer of tubes. Outlet steam temperature. 11

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Fig. 15b. Temperature change caused by helical diameter deviation of the 2nd layer of tubes. Outlet steam temperature variation.

Fig. 17a. Temperature change caused by helical diameter deviation of the 4th layer of tubes. Outlet steam temperature.

Fig. 16a. Temperature change caused by helical diameter deviation of the 3rd layer of tubes. Outlet steam temperature.

Fig. 17b. Temperature change caused by helical diameter deviation of the 4th layer of tubes. Outlet steam temperature variation.

Fig. 16b. Temperature change caused by helical diameter deviation of the 3rd layer of tubes. Outlet steam temperature variation.

Fig. 18a. Temperature change caused by helical diameter deviation of the 5th layer of tubes. Outlet steam temperature.

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(Grant No. ZX06901). References Achenbach, E.J., 1987. Dispersion of hot spots in steam generators. In: Proceeding of specialists’ meeting on technology of steam generators for gas-cooled reactors, Switzerland. ASME, 2013. Boiler and Pressure Vessel Code, II Materials Part. D Properties (Metric). Barsamian, H.R., Hassan, Y.A., 1997. Large eddy simulation of turbulent crossflow in tube bundles. Nucl. Eng. Des. 172, 103–122. Blackall, J.L., Iacovides, H., Uribe, J.C., 2019. Modeling of in-line tube banks inside advanced gas-cooled reactor boilers. Heat Transfer Eng. https://doi.org/10.1080/ 01457632.2019.1640486. Carasik, L.B., Shaver, D.R., Haefner, J.B., Hassan, Y.A., 2017. Steady RANS methodology for calculating pressure drop in an in-line molten salt compact crossflow heat exchanger. Prog. Nucl. Energy 101, 209–223. Collier, J.G., Thome, J.R., 1994. Convective Boiling and Condensation. Clarendon Press, Oxford, New York. Feng, J. Y., Acton, M., Baglietto, E., Kraus A. 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Fig. 18b. Temperature change caused by helical diameter deviation of the 5th layer of tubes. Outlet steam temperature variation.

4.2.2. Tube inside diameter and tube length deviations The tube inside diameter influences the friction factors of the helical tubes. The lengths of the heat transfer tubes are almost identical, however the connection tube many vary a little. The deviations of the tube inside diameter and tube length will result in a deviation of the secondary side water flow rate. Thus the influences of tube inside diameter and tube length deviations on the outlet steam temperature can be analyzed based on the variation of secondary side water flow rate. 5. Conclusions The working temperatures of HTGR steam generators are much higher than that of PWR steam generators. Large thermal non-uniformities in HTGR steam generators should be avoided during operation. Geometrical and thermal hydraulic deviations in a HTGR steam generator will result in temperature deviations from the ideal design cases. Except for the basic thermal hydraulic design, thermal deviation analyses should be performed for HTGR steam generators. 1D and 2D thermal analysis methods were developed and validated. Both the two methods can predict the temperature distribution in the primary and secondary sides along the tube length. The 2D method can also include the thermal mixing of the cross flow over tube bundles and predict the outlet steam temperature distribution of the 5 layers of tubes. The thermal non-uniformities caused by geometrical and thermal hydraulic deviations were analyzed based on the developed methods. The main conclusions are: 1) The radiation heat transfer in a HTGR steam generator only contributes 0.9% of the total thermal power. 2) A one percent helium flow rate change will cause the outlet steam temperature to change by 8.5 °C. A one percent change of the secondary side water flow rate in the tubes of a certain layer, will result in a 5–6 °C variation of the outlet steam temperature. 3) Due to the thermal mixing effect of the tube bundles, an inlet helium temperature profile with a peak variation of 10 °C will only result in a variation of 2 °C of the outlet steam temperature. 4) A helical diameter deviation of 1 mm will result in a maximum outlet steam temperature change of 6–7 °C. Acknowledgments This work was financially supported by the National Natural Science Foundation of China (51576103) and the National S&T Major Project 13

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Zhou, Y.P., Li, F., Hao, P.F., He, F., Shi, L., 2014. Thermal hydraulic analysis for hot gas mixing structure of HTR-PM. Nucl. Eng. Design 271 (S1), 510–514. Zukauskas, A., Ulinskas, R., 1988. Heat Transfer in Tube Banks in Crossflow. Hemisphere Publishing Corporation, New York.

plant. Nucl. Eng. Des. 239 (7), 1212–1219. Zhang, Z.Y., Dong, Y.J., Li, F., Zhang, Z.M., Wang, H.T., Huang, X.J., Li, H., Liu, B., Wu, X.X., Wang, H., Diao, X.Z., Zhang, H.Q., Wang, J.H., 2016. The Shandong Shidao Bay 200 MWe high-temperature gas-cooled reactor pebble-bed module (HTR-PM) demonstration power plant: an engineering and technological innovation. Engineering 2, 112–118.

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