Simulation modelling for dynamic performance studies in a channel model of a supercritical-pressure, direct cycle light water reactor

Pergamon

PII:

SO306-4549(96)00008-4

Ann. Nucl. Energy Vol. 24, No. 4. pp. 325-338, 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0306-4549/97 $I 7.00 + 0.00

TECHNICAL NOTE SIMULATION MODELLING FOR DYNAMIC PERFORMANCE STUDIES IN A CHANNEL MODEL OF A SUPERCRITICAL-PRESSURE, DIRECT CYCLE LIGHT WATER REACTOR M. SULTAN, M. SOBHY, E. ELSHERBINY

and F. DIMITRI

lnshass Research Center, Cairo, Egypt (Received 21 November 1995)

Abstract-Dynamic performance studies in a channel model of a supercriticalpressure, direct-cycle light water reactor is carried out in this paper. In this supercritical-pressure system where some parameters are temperature dependent within the channel, neutronic and thermal-hydraulic calculations are carried out to define the distributions of heat transfer coefficient and power generated along the supercritical channel model. A linearized mathematical nodal model representing the heat transfer system is given for the supercritical channel. A digital-analogue simulation technique is used to study the dynamic performance of the channel model from the transient response of the fuel and coolant temperatures at different nodes in the supercritical channel model. Transient responses are displayed for different modes of perturbation of the system. Copyright 0 1996 Elsevier Science Ltd

INTRODUCTION The performance of light water reactors is upgraded in supercritical-pressure, direct-cycle light water reactors (SCLWRs). The reactor and plant system is simplified by eliminating the recirculation system, steam separators and dryers. The reactor vessel is smaller than that of a normal Pressurized Water Reactor (PWR). The required core flow rate is about one-eighth that of a PWR, and the steam turbines are smaller. The vessel wall is not very thick despite the supercritical pressure. The coolant density in SCLWRs decreases continuously in the core under a system pressure of 250 bars, and the coolant is directly fed to the turbine (Koshizuka et al., 1994). The void reactivity coefficient will be reduced due to the absence of recirculating flow. The thermal efficiency is higher and the capital cost is lower because of the system simplification in SCLWRs. Several types of SCLWR were recently conceived by selecting the cooling cycle, cladding material and fuel (Oka and Kataoka, 1992). In this paper dynamic performance studies are carried out for an SCLWR channel. This channel consists of a single fuel rod with surrounding supercritical coolant and cell outer radius 325

Technical

326

Note

denoted by R,. Some parameters which are considered constant in conventional LWRs are temperature-dependent parameters in the physical and thermal-hydraulic calculations of SCLWRs. Therefore, a nodal model for the temperature field is adopted to determine the generated power and flux distributions in the axial direction of the SCLWR channel. Some parameters from a conceptual design (Yoshiaki and Koshizuka, 1993) are adopted in this paper for the present studies. DISTRIBUTION

OF GENERATED POWER AND TEMPERATURE CHANNEL MODEL OF FUEL AND COOLANT

IN SINGLE-

Dynamic performance studies require neutronic and heat transfer calculations to define the distributions of generated power, flux and temperature in the axial direction of the SCLWR channel. Figure 1 illustrates the nodal model for the temperature field in terms of the two-dimensional cylindrical coordinates of the reactor channel (Yoshiaki et al., 1995). The ‘i’ and 7 in Fig. 1 refer to the node numbers in the axial and radial directions of the core, respectively. In this paper the nodal model consists of 40 fuel nodes equally spaced in the axial direction. With regard to the core radial direction, 11 nodes under mirror boundary condition are considered. Cell calculations were first carried out using the collision probability method to define the effective multiplication factor KeB for different pitch-to-diameter ratios (P/D), and for the following parameteric values: fuel enrichment = 5.6% fuel diameter = 0.886 cm thickness of stainless steel cladding material = 0.057 cm outer diameter of fuel rod = 1 cm Figure 2 depicting the relationship between Keff and (P/D) shows an optimum value of P/D = 1.68. However, for thermal-hydraulic considerations and also to avoid a large core size, a value of P/D = 1.4 is taken in the calculations of the generated power and the temperature distribution in an SCLWR channel. Therefore, the following values of the channel parameters are defined: pitch P = 1.4 X 0.886 = 1.2404 cm volume of coolant/node I/ = 10.733 cm3 heat transfer area/node Ar = 44.77 cm2 fuel mass in channel/node Mr = 93.124 g

*i, j Fig. 1. A nodal model temperature

field in terms of 2-D cylindrical

coordinates.

Technical

321

Note

The flow rate and generated power per channel can be define from the definition of total reactor power, inlet temperature, total number of channels and supercritical pressure in the reactor. Therefore, additional channel parameters are given as follows: supercritical pressure = 250 bars inlet coolant temperature = 3 10°C flow rate in nodes Fc = 69 gls power generated/channel = 78 kW A numerical method of solution based on neutron flux diffusion equations expressed in finite differences is used to calculate the nodal temperature and power distributions in the axial direction of the single-channel model, as follows. An initial guess for the axial temperature distribution is made using previous similar studies (Yoshiaki and Koshizuka, 1993). Then the values of water density and specific heat are defined for each node from the curves in Figure 3. The nuclear macroscopic cross-section can then be calculated for each node. The deduced cross-sections are introduced in a two-dimensional diffusion code for calculating the neutron flux in the axial direction. The generated power can then be defined in the axial direction for each node. By using the heat transfer equations in the single-channel model as explained in the next section, a new axial temperature distribution of the coolant can be defined. This axial temperature distribution 1.1

L . -=\. n

1.0 /

r:

y” 0.9

1

0.81

1.2

1.

I I.6

I I .4

I 1.8

I 2.0

P/D

Fig. 2. Change in K,, with P/D ratio.

0.8 -

--

780

Density

- 60

- 20 /

-------------* 0

I 0.2

I 0.4

I 0.6

\ I 0.8

‘-__ I .o

Height

Fig. 3. Change

in specific heat and water density with height.

Technical Note

328

is used instead of the initial guess in the next iteration process. This iteration process is carried out until a convergence with respect to temperature is attained for the axial temperature distribution and consequently for the generated power distribution in the channel model. The convergence rate of this iteration process can be increased by using an overrelaxation technique for the axial coolant temperature distribution after each iteration process, as expressed by the following equation: N Tij=(l-

N-l ,N l+‘,)TiJ+ WTij

(1)

where W = overrelaxation factor N = iteration index N Tij = overrelaxated temperature for the Nth iteration ,N Tjj = temperature calculated in iteration N before overrelaxation The convergence criteria can be expressed by the following accuracy equation: accuracy = (

$ i=

$,NGj’-E)/(ii

3,

gj

(2)

in which ‘I’ and ‘J’ denote the total number of nodes in the axial and radial directions, respectively. The iteration process leads to an actual solution for the flux and temperature distributions in the SCLWR channel. HEAT TRANSFER

In this section we investigate (1) the heat transfer equations to determine the temperature distribution in the iteration process mentioned in the previous section and (2) the heat transfer coefficient from cladding to coolant in different nodes. With regard to (l), the coolant temperature distribution is established by calculating the increase in temperature in each of the 40 nodes, considering the interaction with adjacent 480-

360

-

320

-

240

0

I 0.2

I 0.4

I 0.6

I 0.8

I 1.0

Height

Fig. 4. Coolant temperature distributions with height.

Technical Note

329

nodes. Figure 4 depicts the results of these calculations, relating the coolant temperature to the normalized height of the SCLWR channel. From the steady-state heat transfer equations (Isachenko et al., 1987) the following balance equation between the power generated in the fuel and the power gained by the coolant in the axial direction can be obtained.

(7;+ I

-

Tj) =

RF’G 2~; 4, 1, (R,*- &?.A v,C,,i

R,

= radius of fuel rod = cell radius

G .& +1 C,,, 4 “, J;

= energy released per fission (~200 Mev) = fission cross-section = neutron flux for node i = coolant specific heat at constant pressure for node i = length in vertical direction of node i = coolant velocity in node i = coolant density in node i

where R,

(3)

The product f; vi can be considered as constant (Isachenko et al., 1987). With regard to (2) above, i.e. the determination of the heat transfer coefficient from cladding to coolant in each node, both the Reynolds number Re,, (Krasnoshchekov et al., 1977) and the Nusselt number NU,, of the water turbulent flow are calculated. d

v. f

ReFi =Y

C,

where de = equivalent diameter of channel = 4 X cross-section of stream/wetted perimeter = 0.959 cm Pi, = dynamic viscosity of water (g/s cm). The values pci for different nodes were calculated as a function of temperature (Krasnoshchekov et al., 1977). In order to determine the cladding surface temperature, the heat transfer coefficient from the cladding to the supercritical pressurized water must be calculated through the Nusselt number. In the case of an incompressible liquid in turbulent flow through a channel which has Prandtl numbers Pr > 0.7, the Nusselt number NuFi and the heat transfer coefficient (Y,are determined by the following correlations (Krasnoshchekov et al., 1977): Nu FI =0021 .

Re F, O*PrF,“.43 (Pr,,/Pr,,)O.” k

a, = Nu,; 2

4

(4) (5)

The subscripts ‘u’ and ‘a’ indicate that the physical properties of the liquid involved are selected according to fluid temperature t, or wall temperature t,. Using the above-mentioned numerical method, the fraction of generated power in the axial direction, QF, and the heat transfer coefficient are calculated for 40 lumped nodes. The data for these 40 fuel nodes were lumped again in 10 fuel nodes representing the average of each 4 nodes. Table 1 gives the values of QF,‘, cut,fC,,’ andf;’ for these 10 lumped nodes.

330

Technical Note MATHEMATICAL

MODEL FOR THERMO-HYDRODYNAMIC SCLWR CHANNEL

OF

In this section a linearized mathematical model representing the SCLWR channel heat transfer based on a nodal approximation is given. There are 10 heat transfer nodes for fuel in this modelling approach. Figure 5 shows the schematic of the fuelkoolant heat transfer model in which two coolant nodes are used for each fuel node. The inlet coolant node is the average coolant temperature used to establish the temperature difference for the heat flux distribution. Therefore, a linearized mathematical model representing the heat transfer system in the SCLWR channel can be given by the following equations, in which 6 denotes deviation from the steady state: d 6 T,, dt

= ar S P - b, (6 T,, - 6 Tci9)

d 6Tc,r _ - dp (S T,? dt

d 6 Tc,, -_ dt

- 6 Tc,,,) - e, (6 T,,, - 6 To,,)

(7)

dr (6 TFi’ - 6 T,,,,) - eiq (6 T,,;, - S Tc,,,)

Table 1. Values of parameters for 10 nodes Node no. i’ 1 2 3 4 5 6 7 8 9 10

QFi’

ff;

0.00386 0.00925 0.01962 0.04046 0.07289 0.11879 0.19174 0.29953 0.18947 0.11396

0.6739 0.6702 0.6606 0.647 1 0.5788 0.4845 0.4389 0.4099 0.3887 0.3797

Tcli’

WD)ci

f;

2.64 3.25 4.20 8.50 12.50 4.40 3.24 2.04 1.40 1.10

0.72 0.69 0.63 0.53 0.42 0.32 0.22 0.17 0.16 0.14

Tc*i’

Coolant

Coolant

outlet

inlet

Fig. 5. Schematic of fuel-coolant heat transfer model.

’

Technical

331

Note

where

= volume of coolant/node in lumped system = 42.932 cm3 = specific heat of fuel/node = 0.36 Jig “C c,, (M C,), = thermal capacity of fuel/node = 134.1036 JPC = average fuel temperature in node i’ ST,,, = average coolant temperature in fuel node i’ aT,w = outlet coolant temperature in fuel node i’ ST,,,, = channel inlet temperature SK,, = change of power generated in channel SP From the values of the parameters given previously, the coefficients a;, b,,, d; and e; can be evaluated for the IO-node channel as given in Table 2. V’

DIGITAL-ANALOGUE SIMULATION FOR DYNAMIC STUDIES IN AN SCLWR CHANNEL

PERFORMANCE

In this paper dynamic performance studies on the SCLWR channel are carried out using the Digital-Analogue (D-A) simulation technique described below. An analogue simulation of the SCLWR channel is performed as shown in Figure 6, based on the mathematical model given by equations (6H8). From Figure 6 it can be seen that for the 10 fuel nodes scheme, there are 122 operation blocks in the analogue simulation. With such a large number of operation blocks the D-A simulation technique is the most practical means of carrying out dynamic performance studies in an SCLWR channel. Computer programs such as TUTSIM (Klee, 1984), Mat-lab or Vis-Sim can be used to perform the required D-A simulation. From the analogue simulation of Figure 6 a model and parameter listing can be initiated and the data fed to any of the above-mentioned D-A simulation computer programs. The TUTSIM D-A simulation program is used in this paper to study the dynamic perfor-mance of the heat transfer system in the (SCLWR) channel for the following perturbations: Table 2. Values of parameters for 10 nodes Node no. i’ 1 2 3 4 5 6 7 8 9 10

a,* 0.000029 0.000062 0.000146 0.000302 0.000544 0.000886 0.001430 0.002234 0.001400 0.000850

h;

C$

PC

0.89990 0.89490 0.88214 0.86410 0.77290 0.6469 0.5861 0.5474 0.5191 0.5070

1.065 0.860 0.656 0.318 0.1931 0.459 0.5650 0.8381 1.1580

4.464 4.658 5.102 6.064 1.652 10.044 14.609 18.906 20.088 22.957

1.4397

Technical Note

332

(a) a step increase in the channel generated power, (b) a step increase in the inlet channel temperature, (c) a combination of the perturbations mentioned in (a) and (b).

RESULTS

Figures 7-12 give the results of the D-A simulation. Figure 7 (a)-(c) shows the transient response of the fuel cladding temperature at different nodes due to a 10% step increase in the channel generated power (6P = 7800 W). Figure 7 (d) shows the steady-state deviation values of the fuel cladding temperature at different nodes.

I SP

, *

LEGEND

Fig. 6. Analogue

simulation

I and 2)

@

Constant (blocks no.

0

Gain (blocks no. 3 10 92)

D

integrator (blocks no. 93 to 122)

of a (SCRDCLWR) channel, with 10 nodes of lumped parameters. Values in gain blocks are given in Table 2.

Technical Note

333

(a) 5

I

TUTSIM

Node IV

0

20 (b)

50 TUTSIM

Node VII

VI V

I 20

0

:) TUTSIM

Node VIII

Cd)

Fig. 7. Transient

Node number (I ) responses (a,b,c); steady state deviation values of fuel clad temperature 10% step increase in channel generated power.

(d), for

334

Technical Note

TUTSIM

TUTSIM

0

20

Node

Fig. 8. Transient

number

(1)

responses (a,b,c); steady state deviation values of average coolant temperature (d), for 10% step increase in channel generated power.

Technical Note

335

Figure 8 (a)--(c) shows the transient response of the average coolant temperature of nodes for the same peturbation. Figure 8 (d) shows the steady-state deviation values of the average coolant temperature at different nodes. (a) 20rm

TUTSlM

(b)

20

TUTSIM

I 20 PC, zo-

TUTSIM

Sec.

Fig. 9. Transient responses (a,b,c) of fuel clad temperature at different nodes due to 5% step increase in channel inlet temperature ST,“.

336

Technical Note

Figure 9 (aHc) shows the transient response of the fuel cladding temperature at different nodes due to a 5% step increase in the channel inlet temperature (W,, = 15.5”C). (a) 20-

TUTSIM

Node I

1) TUTSIM

) TUTSIM

Sec.

Fig. 10. Transient

responses (a,b,c) of average coolant temperature at different nodes due to 5% step increase in channel inlet temperature ST%,,

331

Technical Note TUTSIM

Sec.

Fig. 11.

1

J

-50

20

Sec.

Fig. 12. Figure 10 (a)-(c) shows the transient response of the average coolant temperature of nodes for the same perturbation. Figure 11 shows the transient response of the fuel cladding temperature 6T, and average coolant temperature 6T,, at node 8, due to both a 10% step increase in the channel generated power and a 5% increase in the channel coolant inlet temperature. Figure 12 shows the transient response of the fuel cladding temperature ST, and the average coolant temperature ST,, at node 8, due to both a loo/o step increase in the channel generated power and a 5% reduction in the channel coolant inlet temperature. REFERENCES

Isachenko, V. et al. (1987) Heat Transfer. Mir Publishers, Moscow. Klee, H. (1984) Digital-Analogue (D-A) simulation program (TUTSIM), Florida, College of Engineering.

U. 0 Central

338

Technical Note

Koshizuka, S. et al. (1994) Annals of Nuclear Energy 21, 177. Krasnoshchekov, E. et al. (1977) Problems in Heat Transfer. MIR Publishers, Moscow. Oka, Y. and Kataoka, K. (1992) Annals of Nuclear Energy 19, 243. Yoshiaki, 0. and Koshizuka, S., (1993) Nuclear Technology 103, 295. Yoshiaki, 0. et al. (1995). Fission Reactors, Nuclear Technology 109, 1.