**3. Numerical analysis and performance results of the RSG**

The mathematical simulation of the SG normal and transient operation and behaviour was performed with system computer code RELAP5 [5] on the example of a two-loop PWR power plant Krško [6].

Correlations for the heat transfer calculation in steam generators are given first. Then, SG modelling requirements and mathematical models of the plant are presented, followed by analytical results of SG performance in accident conditions during the loss of the NPP heat sink.

### **3.1. Physical models of thermal hydraulic and heat transfer phenomena**

#### *3.1.1. Thermal hydraulic conditions*

The most important SG parameter subjected to regulation is the SG level. If the level is too low, the insufficient heat removal by the secondary side may cause evaporation of the reactor coolant, thus overheating of the reactor core. On the other hand, if the level is too high, the steam exiting the steam generator would carry water droplets (the void fraction would be higher than zero) which can be damaging to the turbine. The SG level is maintained by the feedwater flow by means of controller which continuously compares measured feedwater flow with steam flow and a compensated steam generator downcomer water level signal with a water level set point. A functional diagram of the steam generator level control system is shown in **Figure 3**.

tracted from a desired reference SG level. That signal is then corrected by a proportional-integral

and K2

According to **Figure 3**, in the case the reference level signal is larger than the measured level or the steam flow is larger than the feedwater flow, the feedwater flow will be increased by increasing the feedwater control valve area. In the opposite case, the control valve flow area

Two types of water levels are measured inside the steam generator: the narrow range (NR) level and the wide range (WR) level. The term "water level" should not be taken literally since no free water surface in the SG secondary side can be established. The fluid is in a state of boiling, and in the area above the tube bundle, steam quality steadily rises from the top of the U-tubes to the inlet in the steam separators. Thus, the level is deduced from the pressure difference, pressures being measured at two different heights. The level is affected by variations

In general, the SG level is a measure of a pressure difference inside the steam generator compared to a pressure difference between the liquid and gas phases. It is calculated by the expression:

The height in the expressions for the pressure difference is the distance between the measurement taps. For both the narrow and wide range measurements, the upper tap is in the

*Δ* 100% − *Δ* 0%

s)) and added to a difference between steam flow and feedwater flow

are scaling factors and τ<sup>1</sup>

(1+1/τ<sup>3</sup>

, τ<sup>2</sup> and τ<sup>3</sup>

(1)

s)) and sub-

time con-

s)) before being used

The measured steam generator level is compensated by a lag controller (1/(1+τ<sup>1</sup>

signals. The resulting signal goes through a final PI correction (K<sup>2</sup>

in the fluid density as well as residual pressure drops.

**Figure 3.** Functional diagram of the SG level control system.

gh and Δp100% = ρ<sup>l</sup>

[%] <sup>=</sup> <sup>100</sup><sup>⋅</sup> *<sup>Δ</sup>* <sup>−</sup> *<sup>Δ</sup>* \_\_\_\_\_\_\_\_\_\_ 0%

gh.

stants depending on the design of nuclear power plant control system.

(PI) controller (K1

will be decreased.

where Δp0% = ρ<sup>g</sup>

(1+1/τ<sup>2</sup>

172 Heat Exchangers– Advanced Features and Applications

for feedwater flow control. Parameters K<sup>1</sup>

The RELAP5 code is a six-equation one-dimensional, nonhomogeneous, nonequilibrium transient system code. It solves mass, momentum and energy conservation differential equations for the two phases, gas and liquid, hence, the six conservation equations. The equations will not be presented here because they are standard fluid conservation equations, although they include many fine transport mechanisms in order to realistically simulate thermal hydraulic system behaviour [5]. For example, momentum equations take into account wall friction, momentum transfer due to interface mass transfer, interface frictional drag and force due to virtual mass. The interface mass and heat transfer terms are also treated by the mass and energy conservation equations.

Efficient and reliable SG operation requires efficient steam separation. Separators must be capable of achieving very low moisture carryover. High carryover will result in turbine efficiency losses as well as the potential for turbine blade erosion. Efficient steam separator design also requires that the primary separation stage has the low pressure drop and low steam carryunder in the downcomer flow in order to support efficient recirculation through the tube bundle. Furthermore, to allow flexibility in water level operation, the separators must be able to operate over a wide range of water levels.

The computational separator model consists of a special hydraulic volume component with junction flows (**Figure 4**). A steam-water inflowing mixture is separated by defining the quality of the outflow streams using empirical functions. The void fraction of the vapour at the separator outlet junction J1 depends on the void fraction in the separator thermal hydraulic volume according to the left curve in **Figure 5**. If the vapour void fraction in the separator volume is larger than the input parameter, labelled as VOVER, the outlet void fraction will be 1.0, and pure gas vapour will be released out of the separator. If the separator vapour void fraction is less than the value of VOVER, then the outflow is going to be a two-phase mixture of gas and liquid. Thus, changing the VOVER parameter, the code user can control the state of fluid leaving the separator. In the case of VOVER parameter being small, the separator is going to act as an ideal separating device, discharging pure vapour, and in the case of VOVER having a high value, close to 1.0, the separator component will perform as a standard junction releasing fluid in the same state as is entering the separator.

**Figure 4.** Typical separator volume and junctions.

**Figure 5.** Outlet junctions void fractions.

The flow of the separator liquid drain junction is modelled in a manner similar to the steam outlet except that pure liquid outflow is assumed when the volume liquid void fraction is greater than the value of VUNDER (**Figure 5**). Typical values of VOVER and VUNDER parameters are 0.001 and 0.1, respectively.

#### *3.1.2. Correlations for the heat transfer calculation*

Steam generator operation depends on the wall-to-fluid heat transfer on the secondary side. During the steady-state operation, the steam generator water level is constant, but during the transient, it can vary in the large range; thus, both convective and boiling heat transfers occur across the U-tubes. The total wall heat flux is composed of convective heat transfer to gas and liquid phases, boiling and condensation heat fluxes. During boiling, the saturation temperature based on the total pressure is the reference temperature, and during condensation, the saturation temperature based on the partial pressure is the reference value. The expression for a heat flux is given as

$$q\_{\text{wall\\_total}} = h\_{g,\text{g}}(T\_w - T\_g) + h\_{g,\text{spf}}(T\_w - T\_{\text{spf}}) + h\_{g,\text{spp}}(T\_w - T\_{\text{spp}}) + h\_{l,l}(T\_w - T)\_l + h\_{l,\text{spf}}(T\_w - T\_{\text{spf}})\_l \tag{2}$$

where

The computational separator model consists of a special hydraulic volume component with junction flows (**Figure 4**). A steam-water inflowing mixture is separated by defining the quality of the outflow streams using empirical functions. The void fraction of the

thermal hydraulic volume according to the left curve in **Figure 5**. If the vapour void fraction in the separator volume is larger than the input parameter, labelled as VOVER, the outlet void fraction will be 1.0, and pure gas vapour will be released out of the separator. If the separator vapour void fraction is less than the value of VOVER, then the outflow is going to be a two-phase mixture of gas and liquid. Thus, changing the VOVER parameter, the code user can control the state of fluid leaving the separator. In the case of VOVER parameter being small, the separator is going to act as an ideal separating device, discharging pure vapour, and in the case of VOVER having a high value, close to 1.0, the separator component will perform as a standard junction releasing fluid in the same state as is enter-

depends on the void fraction in the separator

vapour at the separator outlet junction J1

174 Heat Exchangers– Advanced Features and Applications

**Figure 4.** Typical separator volume and junctions.

**Figure 5.** Outlet junctions void fractions.

ing the separator.


Correlations used to calculate heat transfer in the steam generators, depending on the heat transfer regimes, are given in **Table 2**.

For the convective heat transfer, the correlation is given by Churchill and Chu [7]:

 = ( 0.825 + 0.387 <sup>1</sup>/<sup>6</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_ (1 + ( \_\_\_\_\_ 0.492 Pr ) 9 /16) 8 /27 ) 2 , (3)

where Nu, Ra and Pr are Nusselt, Rayleigh and Prandtl numbers, respectively.


**Table 2.** Wall heat transfer correlations.

The Forster-Zuber correlation [8] for the nucleate boiling heat transfer coefficient is given as

$$h = 0.00122 \frac{\mu\_{\!\!\!}^{0.9} \rho\_{\!\!\!\!}^{0.8} \rho\_{\!\!\!\!}^{0.8} \rho\_{\!\!\!\!\!}^{0.48} \mathfrak{g}^{0.28}}{\sigma^{0.5} \mu\_{\!\!\!\!\!\!}^{0.29} \theta\_{\!\!\!\!\!}^{0.24} \rho\_{\!\!\!\!\!}^{0.24}} \,\Delta \,\mathcal{T}\_{\text{w}}^{0.24} \,\Delta \,\!\!\!\!\!\!\!\/^{0.75} \,\tag{4}$$

where k is the thermal conductivity, cp the specific heat, ρ the density, g the gravity acceleration, σ the surface tension, μ the viscosity, hlg the enthalpy of boiling, ΔTw the difference between the wall and fluid temperatures and Δp the difference between the saturation and total pressures.

The Chen transition boiling model [9] considers the total transition boiling heat transfer to be the sum of individual components, one describing wall heat transfer to the liquid and a second describing the wall heat transfer to the vapour. The model is expressed as

$$\mathbf{q} = \mathbf{q}\_{\rangle} \mathbf{A}\_{\rangle} + 0.0185 \,\mathrm{Re}^{0.83} \,\mathrm{Pr}^{13} \left(\boldsymbol{\mathcal{T}}\_{\mathbf{v}} - \boldsymbol{\mathcal{T}}\_{\boldsymbol{\mathcal{S}}}\right) \mathbf{(1} - \boldsymbol{\mathcal{A}}\_{\boldsymbol{\vartheta}}) \tag{5}$$

where q is the transition boiling heat flux and A<sup>l</sup> is the fractional wall-wetted area.

The Bromley [10] correlation for the heat transfer coefficient during film boiling is given as

 <sup>=</sup> 0.62*<sup>α</sup>* ( *ρ* 2 (*ρ* − *ρ*) lg \_\_\_\_\_\_\_\_\_\_\_\_\_\_ ( <sup>−</sup> ) Pr ) 0.25 , (6)

where α is the void fraction and L the characteristic length.

Finally, the Nusselt [11] correlation for the condensation heat transfer coefficient is given as

$$h = 1.1006 \left( \frac{\text{g } \rho\_l \Lambda \rho}{\mu\_l^2 \text{Re}\_l} \right)^{13} . \tag{7}$$

#### **3.2. Numerical simulation: the NPP model and results**

#### *3.2.1. The RELAP5 computational model of the power plant*

The nodalization scheme (mathematical model) of the two-loop PWR nuclear power plant, analyzed herein, is shown in **Figure 6**. The boxes represent hydraulic control volumes (CVs) connected by junctions. The reactor is in the middle of the figure, connected by pipes with two steam generators (SG1 and SG2). The scheme also includes other important NPP components and systems: the pressurizer; the safety injection system; the main feedwater; and auxiliary feedwater systems, steam lines, etc.

**Figure 6.** RELAP5 nodalization of the NPP Krško.

The Forster-Zuber correlation [8] for the nucleate boiling heat transfer coefficient is given as

where k is the thermal conductivity, cp the specific heat, ρ the density, g the gravity acceleration, σ the surface tension, μ the viscosity, hlg the enthalpy of boiling, ΔTw the difference between the wall and fluid temperatures and Δp the difference between the

The Chen transition boiling model [9] considers the total transition boiling heat transfer to be the sum of individual components, one describing wall heat transfer to the liquid and a

0.24  Δ 

Churchill and Chu [7]

( − )(1 −

 ) 0.25

is the fractional wall-wetted area.

0.24 Δ 0.75 , (4)

), (5)

, (6)

. (7)

 0.79 0.45 *ρ* 0.49 0.25 \_\_\_\_\_\_\_\_\_\_\_\_ *σ*0.5 *<sup>μ</sup>* 0.29 lg 0.24 *ρ*

second describing the wall heat transfer to the vapour. The model is expressed as

+ 0.0185 Re 0.83 Pr 1 /<sup>3</sup>

The Bromley [10] correlation for the heat transfer coefficient during film boiling is given as

 *ρ* 2

Finally, the Nusselt [11] correlation for the condensation heat transfer coefficient is given as

The nodalization scheme (mathematical model) of the two-loop PWR nuclear power plant, analyzed herein, is shown in **Figure 6**. The boxes represent hydraulic control volumes (CVs)

 *ρ* <sup>Δ</sup>*<sup>ρ</sup>* \_\_\_\_\_ *μ* 2 Re ) 1 /3

(*ρ* − *ρ*) lg \_\_\_\_\_\_\_\_\_\_\_\_\_\_ ( <sup>−</sup> ) Pr

= 0.00122

**Heat transfer phenomena Correlation**

Subcooled or saturated transition boiling Chen et al. [9] Subcooled or saturated film boiling Bromley [10] Condensation heat flow Nusselt [11]

Subcooled or saturated nucleate boiling Forster and Zuber [8]

Convection to noncondensable steam-water mixture, supercritical, single-phase liquid or gas flows

176 Heat Exchangers– Advanced Features and Applications

saturation and total pressures.

**Table 2.** Wall heat transfer correlations.

=

where q is the transition boiling heat flux and A<sup>l</sup>

= 1.1006 (

**3.2. Numerical simulation: the NPP model and results** *3.2.1. The RELAP5 computational model of the power plant*

where α is the void fraction and L the characteristic length.

<sup>=</sup> 0.62*<sup>α</sup>* (

Regarding the steam generator model, on the primary side, the inlet plenum is represented with two control volumes (215 and 217), the tube sheet inlet with one CV (219), the U-tube section with 20 control volumes (upward part of U-tubes—CVs 223, 22501–22508, 227, and downward part of U-tubes—CVs 233, 23501–23508, 237), the tube sheet outlet with one CV (241) and the outlet plenum with two CVs (243 and 245).

On the secondary side, the downcomer is represented with 11 control volumes (411, 41301– 41310), the riser section also with 11 CVs (415, 41701–41709+-, 419), the separator with one CV (421), the steam plenum with dryer with one CV (423), the separator bypass region with two CVs (425, 427) and, finally, the steam generator dome with one control volume – 429. This is the model of the SG1. The SG2 has the same model with the different numbering.

#### *3.2.2. Steady-state calculation*

The first step in the NPP and the SG model development is the qualification of the code nodalization. This is done by comparing plant's main operating parameters with computer steady-state simulation at full power. Parameters of interest are primary and secondary system pressures, reactor coolant, feedwater and steam mass flow rates, reactor coolant temperatures, pressurizer and steam generator liquid levels, primary and secondary system fluid masses, heat transfer inside SGs, circulation ratio, etc. If the calculated values differ less than approximately 1% than the real plant data, we can say that nodalization is qualified for the plant safety analyses.

The steam generator qualification process of the RELAP5 model includes calculation of pressures, temperatures, fluid flows and liquid levels inside the SG. Additionally, geometrical representation of the computational model and calculation of SG conditions at partial loads need also to correspond to manufacturer data, as shown in **Figures 7** and **8**, respectively. The SG pressure at a lower load is higher than the pressure at a higher load. As the pressure increases, water evaporates at a slower rate, and a total SG secondary fluid mass is 50% higher at a 10% load than at a 100% load. The steam flow at a full load is 540 kg/s, while at a 10% load, the flow is only 40 kg/s, achieved by increase of pressure and decrease in feedwater temperature. The pressure difference of 1 MPa, as observed in **Figure 8**, results in steam temperature change of only 10 K. The highest impact of load change is on the circulation ratio which decreases from the value of 41 at a 10% load to a value of 3.7 at a full load.

**Figure 7.** SG secondary side volume versus height.

**Figure 8.** SG steam pressure versus load.

#### *3.2.3. Transient calculation*

The steam generator qualification process of the RELAP5 model includes calculation of pressures, temperatures, fluid flows and liquid levels inside the SG. Additionally, geometrical representation of the computational model and calculation of SG conditions at partial loads need also to correspond to manufacturer data, as shown in **Figures 7** and **8**, respectively. The SG pressure at a lower load is higher than the pressure at a higher load. As the pressure increases, water evaporates at a slower rate, and a total SG secondary fluid mass is 50% higher at a 10% load than at a 100% load. The steam flow at a full load is 540 kg/s, while at a 10% load, the flow is only 40 kg/s, achieved by increase of pressure and decrease in feedwater temperature. The pressure difference of 1 MPa, as observed in **Figure 8**, results in steam temperature change of only 10 K. The highest impact of load change is on the circulation ratio

which decreases from the value of 41 at a 10% load to a value of 3.7 at a full load.

**Figure 7.** SG secondary side volume versus height.

178 Heat Exchangers– Advanced Features and Applications

In order to illustrate SG dynamic behaviour, a representative accident of the loss of all electrical power was selected. The unavailability of AC power supply to NPP systems means that important operational and safety components, such as big pumps which provide driving force to primary and secondary system coolant flows, will not work. There will not be feedwater flow into the SG, and the water level will decrease. On the primary side, the reactor core will heat up and subsequently, if in the meantime, no power restoration occurs, melt. Plant conditions will be further complicated by the loss of reactor coolant through damaged reactor coolant pump seals. The seals are normally cooled by the high-pressure water flow provided by the charging pump which, without electrical motor drive, does not operate.

There are two ways of mitigating accident consequences. First, there is a passive steamdriven auxiliary feedwater (AFW) pump that can provide water for the cooling of steam generators. The steam is taken directly from the SG outlet. In addition, the plant operator can reduce pressure by controlling SG relief valves and prolong the time to the core damage because of increased cooling of the reactor coolant system (RCS). Power supply needed for those actions is provided by the accumulator batteries installed at the plant. The three scenarios ((1) no AFW flow, no SG depressurization; (2) AFW flow, no SG depressurization; (3) AFW flow, SG depressurization) were analyzed according to the aforementioned mitigating options.

The AFW system water injection provided secondary-side heat sink. The SG pressure was maintained at 8 MPa by the operation of SG safety valves (**Figure 9**). Natural circulation was established in the primary system after the stoppage of coolant pumps, heat source being the core decay heat and heat sink of the steam generators. Primary system water evaporated in the core, and steam condensed in the SG U-tubes. The heat (**Figure 10**) was transferred to the secondary-side boiling water which level (**Figure 11**) was maintained by injection from the AFW system. Oscillatory behaviour was due to operation of safety valves, and continuous short-term steam releases to keep pressure at 8 MPa. Large condensate storage tank (CST) water inventory (860 m3 ) provided AFW flow for almost 70 h. Depletion of CST inventory led to dry out of the SGs. The CST tank could be filled up at any time during the accident, but in this conservative analysis, no provision of maintaining the CST water inventory was taken into account. Soon after the CST depletion, the RCS was heated up, water boiling in the core accelerated and the core started to uncover. If there is no immediate action to inject water in the core, the core will melt.

**Figure 9.** SG pressure.

**Figure 10.** Heat transfer in steam generators.

scenarios ((1) no AFW flow, no SG depressurization; (2) AFW flow, no SG depressurization; (3) AFW flow, SG depressurization) were analyzed according to the aforementioned

The AFW system water injection provided secondary-side heat sink. The SG pressure was maintained at 8 MPa by the operation of SG safety valves (**Figure 9**). Natural circulation was established in the primary system after the stoppage of coolant pumps, heat source being the core decay heat and heat sink of the steam generators. Primary system water evaporated in the core, and steam condensed in the SG U-tubes. The heat (**Figure 10**) was transferred to the secondary-side boiling water which level (**Figure 11**) was maintained by injection from the AFW system. Oscillatory behaviour was due to operation of safety valves, and continuous short-term steam releases to keep pressure at 8 MPa. Large condensate storage tank (CST)

to dry out of the SGs. The CST tank could be filled up at any time during the accident, but in this conservative analysis, no provision of maintaining the CST water inventory was taken into account. Soon after the CST depletion, the RCS was heated up, water boiling in the core accelerated and the core started to uncover. If there is no immediate action to inject water in

) provided AFW flow for almost 70 h. Depletion of CST inventory led

mitigating options.

180 Heat Exchangers– Advanced Features and Applications

water inventory (860 m3

the core, the core will melt.

**Figure 9.** SG pressure.

**Figure 11.** SG narrow range water level.

Operator action to rapidly depressurize secondary side to 2 MPa using SG relief valves led to fast primary-side cooldown and depressurization. The primary and secondary systems were closely coupled with little difference in pressures and temperatures. Both systems were in saturated conditions, temperature depending on the saturation pressure. Since the core decay heat was only a couple of percents of the total power and the heat transfer area in the SG was very large, the temperature difference was only few kelvins, and, thus, pressures were almost the same. With the operator action, the CST was emptied at ~202,500 s. That was 30,000 s earlier than in the case without any operator actions due to higher AFW flow required during SG steam extraction to satisfy prescribed RCS cooldown rate. Reducing the primary pressure to 2 MPa enabled the actuation of accumulators' water injection which quenched the core. Therefore, although secondary-side heat sink was lost earlier, the larger RCS water inventory increased the time to core damage. In the case with no AFW flow, the core heat-up started after 2.2 h. In the case with the AFW availability, the core temperatures started to increase after 65 h for the case without SG depressurization, and after 70 h when the operator controlled the SG pressure.
