# Coolant Channel Model

This document describes the thermal hydraulics condition surrounding a single fuel rod to provide thermal boundary condition in the analysis of nuclear fuel behavior.

Energy conservation is used to derive the coolant enthalpy rise. Applicable heat transfer correlations are used to model the boiling curve prior to departure from nucleate boiling.

There are no applicable standards on the coolant channel model used in a fuel performance code. It should be recognized that one should refer to a thermal-hydraulics code for detailed modeling of a coolant channel. For the application in a fuel performance code, much emphasis should be placed upon the energy deposition and heat transfer characteristics and the flow distribution in axial or in radial direction can be ignored and certain assumptions can be made to reduce the computation in the coolant channel model.

Figure 1 shows the schematic of the coolant channel model. No finite element mesh was assigned to the coolant channel. The meshing of the one dimensional flow channel consists of several control volume with constant flow area A. Inlet pressure, coolant temperature (or enthalpy), and mass flux as a function of time are required input and they can provide the boundary conditions in solving one-dimentional momentum and energy equations. Heat input into the coolant consists of cladding outer surface heat flux and energy deposits due to interactions of neutrons and gamma rays with coolant water.

Figure 1: Schematic of One-Dimensional Coolant Channel with Upward Flow

Figure 2 shows a typical sub-channel at assembly interior for a square lattice in thermal-hydraulics analysis. As can be seen, each channel is bounded by four fuel rods and each fuel rod shares a quarter of its cladding outer surface with the coolant channel. For a fuel performance code, the coolant channel takes the same geometry shown in Figure 2, however, it should be noted that the heat flux from the bounding fuel rod surfaces for the coolant sub-channel all come from the fuel pin in the analysis to compute the coolant enthalpy for the sub-channel.

Figure 2: Geometry of an interior sub-channel channel

A more advanced approach could be averaging the coolant enthalpy in all the coolant channels surrounding a fuel pin of interest; for a fuel pin inside an assembly, there are four sub-channels surrounding the fuel pin and nine fuel rods provide heat input for the four sub-channels. Thus, besides the fuel pin of interest, power of eight neighboring rods is also needed to perform enthalpy calculation. For current development work, the coolant channel use heat generated from the same fuel rod and then the calculated heat transfer coefficients are applied to the fuel rod in the thermal solution.

### Governing equations

This session starts with equations for one-component, one-dimensional, and two-phase compressible flow and by making assumptions and integrating those equations over a control volume, a set of working equations were derived with the emphasis on the heat transfer characteristics between the cladding and coolant water.

Mass conservation: (1) where, is the density of the mixture of the steam and liquid and is the void fraction of the steam. is the mass flux of the mixture of the steam and liquid in the unit of .

Momentum equation is given by: (2)

(3) where, x is the vapor quality, A is the flow area, is the friction factor, and is the perimeter of the flow channel.

The energy is equation is given by: (4)

(5)

(6) A few assumptions are applied:

1. Coolant channel has a constant flow area

2. Pressure drop analysis is neglected and coolant pressure at any axial location is inlet pressure

3. Flow is homogeneous and no slip between the liquid phase and the steam, then the void fraction is related to vapor quality by: (7) Plugging Eq. 7 into Eq. 3 and Eq. 5, one can get and

4. Pressure is allowed to have variation in time; however the work done by the pressure on the coolant is neglected.

With above assumptions, the momentum equation is not needed in the analysis and the governing equations can be reduced to: (8)

(9)

In the operating conditions of Light Water Reactors, fuel rods are surrounded by flowing water coolant; the flowing coolant carries the thermal energy generated from nuclear fission reaction and transfers the heat into a steam generator or drives a turbine directly. To predict the thermal response of a fuel rod, thermal hydraulic condition of the surrounding coolant needs to be determined. Such condition in modeling the energy transport aspect of the coolant in Bison code is described by a single coolant channel model. This single channel is used mathematically to describe the thermal boundary condition for modeling the fuel rod behavior. This model covers two theoretical aspects, i.e., the local heat transfer from cladding wall into the coolant and the thermal energy deposition in the coolant in steady state and slow operating transient conditions.

Assumptions and limitations of the coolant channel model are summarized below:

• Closed channel: The lateral energy, mass, and momentum transfer in the coolant channel within a fuel assembly is neglected. Therefore, the momentum, mass continuity, and the energy equations are only considered in one-dimension, i.e., the axial direction.

• Homogeneous and equilibrium flow: For the flow involving both the vapor and liquid phases, the thermal energy transport and relative motions between the two phases are neglected. This essentially assumes the two-phase flow is in a form of one pseudo fluid.

• Fully developed flow: In the application of most heat transfer correlations, the entrance effects are neglected. The heat transfer is assumed to happen in a condition that the boundary layer has grown to occupy the entire flow area, and the radial velocity and temperature profiles are well established.

• Pressure drop neglected: The pressure drop due to flow induced resistance is not accounted for in the coolant channel model. Instead, coolant pressure as a function of time and axial location can be an input provided by user through a hand calculation or using a computer code.

## Coolant Enthalpy Model

In steady state operation, the enthalpy rise in a coolant channel with incompressible fluid can be derived using energy conservation equation: (10) where is the coolant enthalpy at inlet in (J/kg), is the coolant enthalpy at axial location z in (J/kg), is axial location (m), is fuel rod surface heat flux (W/m), is fuel rod linear heat generation rate (W/m), is the fraction of heat generated in the coolant by neutron and gamma rays (dimensionless), is heated diameter (m), is coolant mass flux (kg/sec-m), is flow area of the coolant channel (m).

The mass flux, pressure, and coolant temperature at the inlet of coolant channel are provided as input for calculating coolant enthalpy rise. With calculated enthalpy and input coolant pressure, the corresponding thermodynamic condition can be determined using a steam table. The coolant temperature can be obtained and would be used in the convective boundary condition to compute the clad temperature. The thermal-physical properties of water and steam are evaluated at the corresponding bulk coolant temperature and/or at the cladding wall temperature for the use of calculating heat transfer coefficients between the cladding wall and the coolant.

The inlet mass flux, pressure, and coolant temperature can be provided as functions of time in the code input. Allowing the variation of inlet thermal-hydraulic conditions can be used to model a quasi-steady state when the velocity and thermal energy of coolant at a given location are assumed to achieve the equilibrium condition instantaneously.

## Pre-CHF Heat Transfer Correlations

Depending on the flow rate, flow pattern, and cladding wall surface heat flux, the heat transfer from cladding wall outer surface to coolant can be characterized into different heat transfer regimes.

A set of heat transfer correlations to describe the heat transfer condition prior to the point of Critical Heat Flux (CHF) is described follows:

### Dittus-Boelter correlation

Under forced flow condition and when the coolant is still in the liquid phase, the heat transfer from the cladding wall to the coolant is in the regime of single phase forced convection, and the heat transfer can be described by Dittus-Boelter equation. (11) The equation is applicable for 0.7 Pr 100, Re 10,000, and $L/D > 60$. Fluid properties are evaluated at the arithmetic mean bulk temperature (Todreas and Kazimi, 1990).

#### Jens-Lottes correlation

(12) where is the cladding wall super heat = - in (K). is the cladding wall surface heat flux (W/m-K)), and P is the coolant pressure (Pa). This correlation is developed based on data at a pressure between 500 psi (3.45 MPa) and 2000 psi (13.79 MPa) in sub-cooled boiling regime. The heat transfer coefficient is given as: (13)

#### Thom correlation

A similar correlation is given as follows: (14) The heat transfer coefficient is: (15) This correlation is for water at a pressure between 750 psi (5.17 MPa) and 2000 psi (13.79 MPa); but much of Thom's data were obtained at relatively low heat fluxes according to Tong and Weisman (1996).

#### Shrock-Grossman correlation

Shrock-Grossman heat transfer correlation is used in the regime of saturated boiling. The heat transfer coefficient is given as: (16)

(17) where is the steam quality, is the latent heat of vaporization (J/kg), is the heat transfer coefficient in the liquid phase at the same mass flux (J/kg), is the mass flux (kg/m-sec), , , and are constants as follows with the values:

#### Chen's correlation

An alternative correlation that is used in the saturated boiling regime is Chen's correlation. Chen's correlation consists of a convective term () and a nucleation term (): (18) is the modified Dittus-Boelter correlation: (19) F is a factor to account for the enhanced heat transfer due to the turbulence caused by vapor. (20)

(21) The nucleation term is the Forster-Zuber equation: (22)

(23)

(24) S is a suppression factor: (25) Where ; is the Reynold number for liquid phase only.

#### Rohsenow correlation

The Rohsenow correlation (Liu and Kazimi, 2006) is used to represent the heat transfer during a very short period of vapor bubbles nucleation, growth, and departure follows natural convection that occurs during the fast heat up of the fuel rod. The heat flux is given as: (26) where is the heat flux (W/m), is the latent heat of vaporization (J/kg), is the liquid viscosity at saturated temperature (kg/m-sec), is the density of vapor at saturated temperature (kg/m), is the density of liquid at saturated temperature (kg/m), is the surface tension energy at saturated temperature (N/m), is the acceleration due to gravity (m/s), is specific heat of liquid at saturated temperature (kJ/kg-K), is the wall temperature (K), is the saturated temperature (K), and is the Prandtl number.

## Critical Heat Flux Correlations

The sub-cooled and saturated boiling can enhance the heat transfer; however at a critical condition when the cladding outer surface is enclosed by vapor film, the heat transfer can deteriorate significantly, the corresponding heat flux is the Critical Heat Flux (CHF). The following correlations are implemented in Bison to calculate CHF, which can be used to estimate the thermal margin in a coolant channel.

#### EPRI-Columbia correlation

(27) where is the critical pressure ratio=system pressure/critical pressure, is the local mass velocity (Mlbm/hr-ft), is the inlet quality, and ^2$). The following parameters in the table below are used in the EPRI-Columbia correlation. Table 1: Parameters used in the EPRI Columbia correlation Model ParameterParameter Value G^{0.1}$
for cold wall, both and are set equal to 1.0

is the non-uniform axial heat flux distribution parameter: (28) Y is Bowring's non-uniform parameter defined as: (29)

#### GE correlation

(30) (31) The correlation is applicable for mass fluxes less than lb/ft-hr.

#### Zuber correlation

Taken from Tong and Tang (1997), the Zuber correlation is (32) where is the critical heat flux (W/m), is the latent heat of vaporization (kJ/kg), is the acceleration due to gravity = 9.8 (m/s), is the density of vapor at saturation temperature (kg/m), is the density of liquid at saturation temperature (kg/m), and is the surface tension energy at saturation temperature (N/m).

#### Modified Zuber correlation

The modified Zuber correlation (Liu and Kazimi, 2006) is included in the critical heat flux correlation selection option in Bison, and can be selected by user input. This correlation is based on pool boiling critical heat flux hydrodynamics and is applicable to very low flow conditions. It was developed for critical heat flux calculations in LWRs in severe accident conditions. (33) where is the critical heat flux (W/m), is the correction factor for bulk subcooled fluid conditions, is the latent heat of vaporization (kJ/kg), is the acceleration due to gravity (here set as 9.8 (m/s)), is the density of vapor at saturation temperature (kg/m), is the density of liquid at saturation temperature (kg/m), and is the surface tension energy at saturation temperature (N/m).

The correction factor for bulk subcooled condition is (34) where is the specific heat of saaturated liquid (kJ/kg-K), is the saturation temperature (K), and is the bulk fluid temperature (K).

#### BIASI correlation

BIASI correlation is a function of pressure, mass flux, flow quality, and tube diameters. The correlations are provided in following equations. For kg/m-s, the Eq. 35 is used; for higher mass flux, the Eq. 35 or Eq. 36 whichever higher is used. (35) (36) where (37) (38) The parameter ranges for the correlation are given in the table below.

Table 2: Parameters ranges for which the BIASI correlation is defined

Model ParameterParameter Value
0.003-0.0375 m
0.2-6.0 m
0.27-14 MPa
100-600 kg/m-s

#### MacBeth correlation

The MacBeth correlation (Geelhood, 2014) was developed using based on compilation of a large amount of CHF data from a wide variety of sources. The database consists entirely of burnout tests for vertical upflow in round tubes. This correlation can be extrapolated to CHF in annuli and rod bundles at low pressure. The MacBeth CHF correlation is separated for low flow conditions and high flow conditions.

At low flow conditions, the correlation defines the critical heat flux as: (39) where is the critical heat flux (MBtu/hr-ft), is the hydraulic diameter, based on wetted perimeter (in), is the latent heat of vaporization (Btu/lbm), is the mass velocity (lbm/hr-ft), and is the equilibrium quality.

At high flow conditions, the correlation defines the critical heat flux as: (40) where A and C are the empirical parameters that were defined using statistical optimization for two overlapping sets of data.

The parameters A and C for the MacBeth's 12-coefficient model are formulated as: (41)

(42) where are the empirical coefficients (See Table 3)

Table 3: Coefficients for MacBeth's 12-coefficient model for various reference pressures

Model Coefficient560 psia Reference Pressure1000 psia Reference Pressure1550 psia Reference Pressure2000 psia Reference Pressure
2371143665.5
1.20.8110.5091.19
0.4250.221-0.1090.376
-0.94-0.128-0.19-0.577
-0.03240.02740.0240.22
-0.111-0.06670.463-0.373
19.312741.717.1
0.9591.320.9531.18
0.8310.4110.0191-0.456
2.61-0.2740.231-1.53
-0.0578-0.03970.07672.75
0.124-0.02210.1172.24

The acceptable parameter ranges for the correlation are given in the table below.

Table 4: Parameters ranges for which the MacBeth 12-correlation model is defined

Model ParameterParameter Value
Pressure15-2700 psia
Mass Velocity0.0073-13.7 Mlbm/hr-ft
Hydraulic diameter0.04-1.475 in
Heated length1.0-144 in
Axial power profileuniform

The EPRI correlation is used as the correlation for a Pressurized Water Reactor (PWR) environment. The GE correlation is used as the correlation for a Boiling Water Reactor (BWR) environment. Alternatively, an input temperature at critical heat flux is allowed, which would use the selected heat transfer in the nucleate boiling regime and the input temperature to compute the critical heat flux.

## Post-CHF Heat Transfer Correlation

The post-CHF heat transfer regime is divided into transition boiling and film boiling. The transition boiling heat transfer regime occurs when the cladding wall temperature exceeds the Critical Heat Flux (CHF) temperature, but remains below the minimum film boiling temperature. The heat flux decreases significantly with increasing temperature in this regime. Three heat transfer correlations are implemented for the transition boiling regime. The three correlations are McDonough-Milich-King, modified Condie-Bengtson and Henry correlations. The film boiling heat transfer regime occurs when the wall temperature reaches the minimum film boiling temperature. Four correlations are provided for the film boiling region. The correlations are Dougall- Rohsenow, Groenveld, Frederking and Bishop-Sandberg-Tong correlations. The heat transfer correlations at CHF and in the post-CHF regimes implemented in the Bison code is described as follows:

#### Transition Boiling

McDonough-Milich-King correlation and modified Condie-Bengtson correlation are implemented for the transition boiling regime.

### McDonough-Milich-King correlation

The McDonough-Milich-King correlation (Todreas and Kazimi, 1990; Rashid et al., 2004) for forced convection transition boiling is given as (43) The heat transfer coefficient is: (44) where is the critical heat flux (kW/m), is the transition region heat flux (kW/m), is the wall temperature at critical heat flux (K), is the bulk temperature of coolant (K), is the wall temperature in the transition region (K), is the system pressure (MPa), and is the transition boiling heat transfer coefficient (kW/m-K).

Table 5: Parameters ranges for which the McDonough-Milich-King correlation may be applied

Model ParameterParameter Value
Pressure5.5 - 13.8 MPa
Mass flux271.246 - 1898.722 kg/m-sec
Channel geometrytube
Diameter0.00386 m
Length0.3048m
Fluidwater

### Modified Condie-Bengtson correlation

The modified Condie-Bengtson correlation (Rashid et al., 2004) for high flow rate transition boiling is given as follows: (45) The heat transfer coefficient is: (46)

(47) where is the critical heat flux (Btu/hr-ft), is the transition heat flux (Btu/hr-ft), is the film boiling heat flux at (Btu/hr-ft), is the wall temperature at critical heat flux (F), is the saturation temperature (F), is the cladding wall temperature (F), and is the transition boiling heat transfer coefficient (Btu/hr-ft-F).

At the CHF point, = , and (48)

At , the critical heat flux is equal to the sum of the film boiling component and the transition boiling component to ensure the predicted boiling curve is continuous.

### Henry correlation

The Henry correlation (Liu and Kazimi, 2006) for transition boiling has been developed to address the heat transfer at cold zero power condition and at high subcooling conditions. The heat flux in the transition boiling regime determined by an interpolation of the critical heat flux and minimum heat flux is given as follow: (49) where (50)

The minimal temperature for the Henry correlation is strongly affected by the surface condition as well as the subcooling of the coolant and is given as: (51) where (52)

(53)

• is the minimum heat flux (W/m)

• is the critical heat flux (W/m)

• is the temperature at critical heat flux (K)

• is the minimal stable film boiling temperature(K)

• is the homogeneous nucleation temperature (K)

• is the bulk temperature (K)

• is the cladding wall temperature (K)

• is the pressure (Pa)

• is the thermal conductivity of subcooled liquid (W/m-K)

• is the density of subcooled liquid (W/m)

• is the specific heat of subcooled liquid (kJ/kg-K)

• is the thermal conductivity of cladding wall (W/m-K)

• is the density of cladding wall (W/m)

• is the specific heat of cladding wall (kJ/kg-K)

• is an empirical parameter taken as 3.3

## Film Boiling

Four correlations, Dougall-Rohsenow correlation, Groenveld correlation, Frederking correlation and Bishop-Sandberg-Tong correlation, are provided for modeling the heat transfer in the film boiling region. In the transition from the transition boiling regime to the film boiling regime, the intercept of the selected film boiling correlation and transition boiling correlation was used to determine the minimum film boiling temperature and minimum film boiling heat flux.

### Dougall-Rohsenow correlation

The Dougall-Rohsenow correlation (Dougall and Rohsenow, 1961; Rashid et al., 2004) for forced convection stable film boiling was developed for high flow rate and low quality (x 0.3) flow. The heat transfer coefficient is given as: (54) where is the mass flux (kg/m-sec), is the hydraulic diameter (m), is the thermal conductivity of vapor (W/m-K), is the viscosity of vapor (kg/m-sec), is the density of vapor (kg/m), is the density of liquid (kg/m), is the specific heat of vapor (J/kg-K), and is the local quality.

The vapor properties of the Prandtl number are evaluated at the saturation temperature. The data range for this correlation is given below.

Table 6: Parameters ranges for the Dougall-Rohsenow correlation

Model ParameterParameter Value
Pressure0.1154 - 0.1634 MPa
Mass flux450.268 - 1109.396 kg/m-sec
Heat flux45.426 - 131.862 kW/m
Exit qualityup to 0.4
Channel geometrytubes
Diameter, inner0.004572 m, 0.01036 m
Length0.381 m
Fluidfreon

### Groenveld correlation

The Groenveld correlation (Todreas and Kazimi, 1990; Rashid et al., 2004) for forced convection stable film boiling heat transfer coefficient is: (55) where the parameter Y is given as (56) whichever is larger, and where is the mass flux (kg/m-sec), is the hydraulic diameter (m), is the thermal conductivity of vapor (W/m-K), is the viscosity of vapor (kg/m-sec), is the density of vapor (kg/m), is the density of liquid (kg/m), and is the local quality.

The coefficients a, b, c and d are given in Table 7 below. The Prandtl number of the film is given by (57) where is the specific heat of vapor at film temperature (J/kg-K), is the viscosity of vapor at film temperature (kg/m-sec), and is the thermal conductivity of vapor at film temperature (W/m-K)

The vapor properties of the Prandtl number should be evaluated at the film temperature. (58) where is the saturation Temperature (K) and is the cladding wall temperature (K) Prandtl number is currently evaluated at the saturation temperature in the code.

Table 7: Groenveld correlation coefficients a, b, c, d

ParameterValue
a0.0522
b0.688
c1.26
d-1.06

The applicable range of data for annuli geometry is shown in the Table 8 below.

Table 8: Range of data for Groenveld correlation

ParameterData Range for Annuli Geometry
Hydraulic Diameter (mm)1.5 - 6.3
Pressure (MPa)3.4 - 10
Mass Flux (kg/m-sec)800 - 4100
Heat Flux (kW/m)450 - 2250
Quality0.1 - 0.9

### Frederking correlation

The Frederking correlation (Liu and Kazimi, 2006) for turbulent film boiling heat transfer coefficient during RIA is: (59) where is the turbulent film boiling heat transfer coefficient (W/m-K), is the thermal conductivity of vapor (W/m-K), is the modified latent heat of vaporization (kJ/kg), is the acceleration due to gravity = 9.8 (m/s), is the density of vapor at saturation temperature (kg/m), is the density of liquid at saturation temperature (kg/m), is the saturation temperature (K), and is the cladding wall temperature (K).

The modified latent heat is given as (60) where is the latent heat of vaporization (kJ/kg), is the specific heat of vapor at saturated temperature (kJ/kg-K), is the saturation temperature (K), and is the cladding wall temperature (K).

### Bishop-Sandberg-Tong correlation

The Bishop-Sandberg-Tong correlation (Geelhood, 2014) for film boiling heat transfer coefficient is: (61) where is the coolant thermal conductivity at the film temperature (W/m-K), is the hydraulic diameter (m), is the Reynolds number with fluid properties evaluated at the film temperature, is the Prandtl number with fluid properties evaluated at the film temperature, is the density of vapor at saturation temperature (kg/m), is the density of liquid at saturation temperature (kg/m), and is the bulk fluid density (kg/m). This correlation is defined by the properties of the vapor film at the wall and the film temperature. The film temperature is defined as (62) The film boiling heat flux for this correlation is: (63) The bulk fluid density is defined as (64) The equilibrium void fraction is defined as (65) where, is the equilibrium quality (dimensionless).

## Logic to Determine Heat Transfer Regime

The boiling curve in the Bison code depends on the selected pre-CHF, CHF, and post-CHF correlations. The diagrams in Figure 3 shows the criteria used in the selection of different heat transfer regimes.

Figure 3: Schematic of heat transfer regimes selection criteria}

Dittus-Boelter correlation is used for the single phase liquid forced convection and for the single phase vapor forced convection. Thom or Jens-Lottes correlation is used for the sub-cooled boiling regime. Thom, Jens-Lottes, or Chen correlation is used for the forced boiling convection regime. Shrock-Grossman correlation is used for the forced boiling convection and vaporization regime. In the transition boiling regime, either the MCDonough-Milich-King ocrrelation or the modified Condie-Bengtson correlation is ued. In the film boiling regime, Dougall-Rohsenow or Groenveld correlation is used. is the temperate at the onset of nucleate boiling. is the temperature at the critical heat flux. The selection of different types heat transfer correlations is described in the users manual. The logic described above is not applicable for radiation heat trasnfer and reflood heat transfer modes. They can be activated by using input heat transfer mode.

## FLECHT Reflood Heat Transfer Correlations

An empirical approach for modeling the reflooding phase of a LOCA is using the correlations derived from Full Length Emergency Cooling Heat Transfer (FLECHT) tests (Cunningham et al., 2001; Cadek et al., 1972).

Two reflood heat transfer correlations are implemented in Bison code. The first correlation is provided in Cunningham et al. (2001), and the second one is described in Cadek et al. (1972).

The heat transfer correlations compute heat transfer coefficients during reflooding phase of LOCA as a function of flooding rate, cladding temperature at the start of flooding, fuel rod power at the start of flooding, flooding water temperature, pressure, rod elevation and time. The applicable ranges of these variables are shown in Table 9 and Table 10 for the heat transfer correlations given in Cunningham et al. (2001) and Cadek et al. (1972) respectively. The variables are defined as follow:

• = flooding rate (in/s)

• = Peak cladding temperature at start of flooding (F)

• = fuel rod power at axial peak at start of flooding (kW/ft)

• = reactor vessel pressure (psia)

• = equivalent FLECHT elevation (ft)

• = flood water subcooling at inlet (F)

• = time after start of flooding as adjusted for variable flooding rate (s)

• = heat transfer coefficient (Btu/(hr-ft-F))

• = radial power shape factor = 1.0 for a nuclear fuel rod = 1.1 for electrical rod with radially uniform power

• = flow blockage ()

### Generalized FLECHT correlation

The generalized FLECHT correlation from Cunningham et al. (2001) divides the reflood heat transfer into four time periods: period of radiation only, period I, period II, and period III.

The heat transfer due to radiation is modeled during the time range of t 0 and t $\leq t_{1}$. The heat transfer coefficient expression is given as (66) where (67)

(68)

(69)

(70)

(71)

(72)

#### Period I

During Period I, the flow develops from the radiation dominated pre-reflood condition to heat transfer conditions in reflooding phase. (73) where is defined as (74)

(75) is defined as (76)

The heat transfer coefficient during Period I is calculated as follow (77) where (78)

(79)

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

#### Period II

During this period, the heat transfer coefficient reaches a plateau with a rather slow increase. The time range for Period II is (90) where (91) The heat transfer coefficient during Period II is computed by the equation (92)

(93)

(94)

(95)

(96)

(97)

(98)

(99)

(100)

#### Period III

During this period, the flow pattern might have changed to film boiling regime ,and the heat transfer coefficient increases rapidly as the quench front approaches. The time range of Period III is (101) is the time of quenching.

The heat transfer coefficient during Period III is calculated as follow (102) where (103)

(104)

(105)

#### Modification for Low Flooding Rates

The heat transfer coefficients for Periods I, II, and III is multiplied by a factor f to best match the test data performed at low flooding rates. The factor f is calculated as follow (106) where (107)

(108)

(109)

(110) The above correlations are valid over the following ranges of parameters (Cunningham et al., 2001):

Table 9: Range of applicability of generalized FLECHT correlation

VariableApplicable range of variable in British unitApplicable range of variable in SI unit
Flooding rate0.4 - 10 in/s0.0102 - 0.254 m/s
Reactor vessel pressure15 - 90 psia0.103 - 0.62 MPa
Inlet coolant subcooling16 - 189 F264.3 - 360.4 K
Initial cladding temperature300 - 2200 F420 - 1478 K
Flow blockage ratio0 - 75 %0 - 75 %
Equivalent elevation in FLECHT facility2 - 10 ft0.6096 - 3.048 m

### WCAP-7931 FLECHT correlation

The WCAP-7931 correlation (Cadek et al., 1972) divides the reflood heat transfer into three time periods which are designated as Period I, Period II, and Period III.

#### Period I

The time range of Period I is (111) where is defined as (112)

The quench time is defined as (113) is defined as (114)

The heat transfer coefficient during Period I is calculated as follow (115) where (116)

(117)

(118)

(119)

(120)

#### Period II

The time range of Period I is (121) where (122)

The heat transfer coefficient during Period II is computed by the equation (123) where (124)

(125)

(126)

(127)

(128)

(129)

#### Period III

The time range of Period III is (130)

The heat transfer coefficient during Period III is calculated as follow (131) where (132)

(133)

(134) The above correlations are valid over the following ranges of parameters:

Table 10: Range of applicability of FLECHT correlation from WCAP-7931 report

VariableApplicable range of variable in British unitApplicable range of variable in SI unit
Flooding rate0.4 - 10 in/s0.0102 - 0.254 m/s
Reactor vessel pressure15 - 90 psia0.103 - 0.62 MPa
Inlet coolant subcooling16 - 189 F264.3 - 360.4 K
Initial cladding temperature1200 - 2200 F922 - 1479 K
Flow blockage ratio0 - 75 %0 - 75 %
Equivalent elevation in FLECHT facility4 - 8 ft1.219 - 2.438 m

## Properties for Water and Steam

Properties for water and steam consist of thermodynamic properties, transport properties, and other physical properties used in the heat transfer correlations. They are implemented based on a few standards specified by the International Association or Properties for Water and Steam (IAPWS). The thermodynamic properties, or the steam tables, are implemented in the IAPWS95 library, included as a submodule in Bison.

## Sodium Coolant

Sodium coolant for fast reactors can also be simulated in Bison. The model uses the same framework as the above calculations for water/steam, but with appropriate correlations for liquid sodium. The model uses the modified Schad correlation (Waltar et al., 2011) by default for triangular subchannels (135) where is the Nusselt number, is the Peclet number, and is the pitch-to-diameter ratio and is applicable for . For , the term is set to 1.0.

The Lyon's Law correlation (Lyon, 1951) is generally used for heat transfer from a rod to flow within a surrounding circular tube for liquid metals (136) applicable for and .

The Seban and Shimazaki correlation (Subbotin et al., 1963) is specific to liquid sodium heat transfer from a rod to fluid within a surrounding circular tube with constant rod wall temperature (137) and is applicable for , , and .

Sodium properties are taken from the ANL/RE-92/2 report (Fink and Leibowitz, 1995): (138) where is thermal conductivity, is enthalpy, and units are SI.

At high tempeature, radiation heat transfer can occur from cladding outer surface to surrounding core structure components. In simulated LOCA tests at Halden, heat can be transferred to the heating element as well. Radiation heat transfer is described by following equation: (139)

(140) where and are the surface emissivities of the cladding and heater, respectively,f and and are the radii of the two surfaces, is the Stefan-Boltzmann constant ().

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