[1] Plant dynamics calculation code

The plant dynamics calculation treats plant control, stability, and response at abnormal transients and accidents. A simple model for heat transfer calculation in a plant is the node junction model as shown in Fig. 2.36 [27]. In the node junction model, reactor components are represented as 1D nodes and connected; the connected items include core, upper and lower plena, downcomer (inlet coolant flows down the downcomer region which is placed between the core and reactor pressure vessel), main feedwater and main steam lines, valves, etc. The node junction calculation begins from an upstream node based on mass and energy conservation laws. The momentum conservation law is also considered for pressure drop and flow rate distribution calculations. In the case of Fig. 2.36, for example, the core is described with two single channel models at average and maximum power. The core design calculation is performed at steady state as mentioned before, but a transient single channel model is used in the plant dynamics calculation.

A transient radial heat conduction in fuel can be expressed as

Fig. 2.36 Node junction model

where,

Cp: specific heat of pellet (J/kg-K) kf. thermal conductivity of pellet (W/m-K) qm: power density (W/m3) r: radial distance (m)

Tf. pellet temperature (K) pf pellet density (kg/m3).

The heat transfer from cladding outer surface to coolant is evaluated by Newton’s law of cooling. A proper heat transfer coefficient should be used considering flow regime and temperature, pressure, etc.

Plant dynamics analysis codes are constructed with the node junction model and the following reactor kinetics model. The simplest kinetics model is the point reactor kinetics model (point reactor approximation) as

Щг = А£Т±пЮ+ІліСіЮ

dt А і=і

dCiit) pi (Л.. л

——= — n(t)—AiCi(.t) u = l~6.) dt A

End Time Step?

Fig. 2.37 Plant dynamics analysis model and calculation flow chart

where,

n(t): number of neutrons

Ci(t) number of delayed neutron precursor for group i t: time

Pi. fraction of delayed neutron group i

6

p — Ipi

i = 1

Ap: reactivity

Л: prompt neutron generation time (s)

decay constant of precursor of delayed neutron group i (s_1)

Tave (t): average fuel temperature (K)

Pmod(t): moderator density (g/cm3).

The reactivity feedback of fuel temperature (Doppler effect) and moderator density is applied to the point reactor kinetics equations. Even though large power reactors such as LWRs show space-dependent kinetic characteristics, the reactivity feedback effect can be expressed with the point reactor approxima­tion using a space-dependent weight function (adjoint neutron flux).

Figure 2.37 shows an example of a node junction model in a plant dynamics analysis code and its calculation flow chart. This model describes coolant flow which is fed to the water moderation rods through the top dome of the reactor vessel for a supercritical water-cooled thermal reactor (Super LWR) [28, 29].

The mass, energy, and momentum conservation equations used in the node junction model are a time-dependent form of the single-channel
thermal-hydraulic model in the core design. In the case of single channel, single phase, and one dimension, the mass conservation equation is given by Eq. (2.116)

(2.116)

dt dz

and the energy conservation equation is given by Eq. (2.117).

diph) dipuh) Pe „

dt ~

The momentum conservation equation is expressed as

and the state equation is

P=p(P, h)

where,

t: time (s)

z: position (m)

p: coolant density (kg/m3)

u: fluid velocity (m/s)

h: specific enthalpy (J/kg)

q": heat flux at fuel rod surface (W/m2)

A: flow area of fuel channel (m2)

Pe: wetted perimeter of fuel rod (m)

P: pressure (Pa) g: gravitational acceleration

Dh: hydraulic equivalent diameter of fuel channel (m)

Re: Reynolds number 6: vertical angle of fuel channel f: frictional coefficient

for example, f = 0.0791 x Re-0 25 (Blasius equation).

In the case of a water rod, the energy transferred to the water rod should be considered in the energy conservation equation of the fuel rod cooling channel. In the case of light water, the steam tables of the Japan Society of Mechanical Engineers (JSME) or the American Society of Mechanical Engineers (ASME) can be used in the state equation of the coolant.

The following introduce heat transfer coefficients for water-cooled reactors. In a single-phase turbulent flow, for example, the heat transfer coefficient can be obtained from the Dittus-Boelter correlation

(2.120)

where,

De: hydraulic equivalent diameter (m)

G: mass flux (kg/m2-s)

h: heat transfer coefficient (W/m2-K)

k: thermal conductivity (W/m-K)

Pr: Prandtl number p: viscosity coefficient (Pa-s).

If nucleate boiling occurs, the heat transfer for PWR can be described by the Thom correlation

1 (ATsate^ У dT V 0.022 )

where,

h: heat transfer coefficient (W/m2-K)

P: pressure (MPa)

dT: temperature difference between wall surface and fluid (K)

ATsat: wall temperature elevation above saturation temperature (K)

and it is applied for the range of

pressure: 5.2-14.0 MPa mass flux: 1,000-3,800 kg/m2-s heat flux: 0-1600 kW/m2.

The Jens-Lottes correlation can be used for BWRs.

If there is a thin liquid film in annular flow with a high steam quality, for example, the heat transfer to a vertical upward water flow can be represented by the Schrock-Grossman correlation

(2.122)

і _/ x у-9ЛеЛ0-5(ж)0’1 Xtt l-x) pgJ pf)

where,

De: hydraulic equivalent diameter (m) G: mass flux (kg/m2-s)

h: heat transfer coefficient (W/m2-K) kf. thermal conductivity of water (W/m-K)

Pr^: Prandtl number of water X: steam quality pf. water density (kg/m3) pg: steam density (kg/m3)

Pf. viscosity coefficient of water (Pa-s)

Pg: viscosity coefficient of steam (Pa-s)

and it is applied for the range of

pressure: 0.3-3.5 MPa mass flux: 240-4,500 kg/m2-s heat flux: 190-4,600 kW/m2.

In the post-dryout region of BWRs, namely, the steady-state film boiling state of mist flow where steam flow is accompanied with liquid droplets, for example, the heat transfer can be described by the Groeneveld correlation.

where,

h: heat transfer coefficient (W/m2-K)

kg: thermal conductivity of saturated steam (W/m-K)

Prv, w: Prandtl number of steam at wall surface De: hydraulic equivalent diameter (m)

G: mass flux (kg/m2-s)

Pg: viscosity coefficient of steam (Pa-s)

X: steam quality

pf. water density (kg/m3)

pg: steam density (kg/m3)