^

d C * . * • •

- "H- - . : yif^s.'. -' . " .-_;•.

BLANK PAGE

J . I

?' ru-'ij -"i * h - U " red S ' J V ^ • ••

U S Dep-:'' i* Cor*-—--;"-'

5285 P •• R JVJ. fsVv.ci Sc< -g* •• a '. — • .: 2 : v Pnce: Pr.nted Copy S5.0C: V c- • ,•-.-• S? :b

Th s report was prepared as ar. account of work sponsored by the Unted States Governrren! Neither the United States nor the Energy Research and Development Administration United State; Nuclear Regulatory Commiss.on. nor any of fe:r ems oyees. nor my of their contrac'ors. subcontractors, or ther empioveei makes jnv wa rranty. expres? or implied, or assumes any legal liability or responsibility for the accuracy completeness or usefi,.ness of any information, apparatus, product or process disclo ed. or epresents that IS use would not infringe privately owned rights

I

0RNL/NUREG/TM-I6 NRC-8

Contract No. W-7405-eng-26 t

HTGR STEAM GENERATOR MODELING

T. H. Ker l in , Director

This work was performed under subcontract 4122 by the UNIVERSITY OF TENNESSEE Nuclear Eng.neering Department KnoxviHe, Tennessee 37916

Manuscripc Cospleted — 6-2-76 Da*e Published - Ju ly . 1976

Prepared for the U.S. Nuclear Regulatory Commission Office c»" Nuclear Reactor Regulation

Under Interagency Agreement ERDA 40-545

NOTICE: This document contains information of a preliminary nature and was prepared primarily for internal use at the Oak Ridge National Laboratory. I t is subject to revision or correction and therefore does not represent a f ina l report.

OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37830

operated by UNION CARBIDE CORPORATION

for the ENERGY RESEARCH AND DEVELOPMENT ADMINISTRATION

A

tr

i i i

CONTENTS

Paje

Abstract v

Introduction v * i

I . Fort St . Vrain Steam Generator Modeling

by S. I . Chang 1

1.1 A Description of the System y 1.2 Lumped Parameter Analysis 6 1.3 Model Formulation 7 1.4 Mathematical Model 8 1.5 Linearization of Equations jg 1.6 Results and Discussion 32 References 49

I I . Nonlinear Modeling of a HTGft Steam Generator

by Ming Huei Lee 5 1

References 5 6

I I I . Gas-Cooled Steam Gener2tor Linear Model by David G. Renfro 5 8

IV. Nonlinear. Nodal System Model

by J. G. Thakkar 72

IV.1 Steam Properties 72 IV.2 Reactor Tc IV.3 Reheater 73 IV.4 Helium Circulator 74 IV .C Future Work 78 References 79

1

V

ABSTRACT

Research activities at The University of Tennessee on gas cooled

reactor dynamics are described. The main activity is on steam generator

modeling, using approaches ranging from a relatively simple linear

representation to a detailed nonlinear representation. Model compari

sons will involve simulations of the Fort St. Vrain reactor steam

generator, with emphasis on the evaluation of accuracy vs computation

costs. A smaller effort is also in progress for modeling the reactor

core, the main turbine and the blower turbines. Preparations are de

scribed for using te«t data from Fort St. Vrain for validating the

dynamic models and identifying important design parameters in the

plant.

Vll

INTRODUCTION

The research activities at The University of Tennessee on gas cooled

•eactor dynamics during 1975 are discussed in this progress report.

The main purposes of this work are:

1. component modeling. The emphasis is

CHAPTER I

FORT ST. VRAIN STEAM GENERATOR MODELING

S. I . Chang

1.1 A Description of the Systeir.

The Fort St. Vrain plant has two identical once-through type

steam generator loops arranged in paral lel , each of which consists of

six individual steam generator modules, located within the cavity cf the

prestressed concrete reactor vessel (PCRV) and beneath the reactor co»-e

(see Figure 1.1). Each steam-generator module contains a heat-transfer

section with a main steam production region and an integral reheater,

as ».?11 as a PCRV penetration section with suitable inlet and outlet

connections and helium-pressure barriers.

The steam generator is used to transfer the heat frcm the heliuir

coolant to the water/steam. The helium flow that cools the reactor core

enters the steam generator at high temperature and gives up its heat,

f i rs t to the reheat steam section, then to the once-through main steam/

water section. The sections produce main steam and reheat steam as

demanded by the plant load and main turbine requirements.

The main design parameters of the system are shown in Table 1.1.

The modules (Figure 1.2) are divided into three bundles (heating

surface sectiois) arranged one above the other, made up of helically

would tuoes supported by perforated plates attached to the central

support structure.

o

ORV. OWG ?6 7 7S9

WTTQMiOO PCNETWmoW

r i g . 1 .1 . rort S t . Vrain 330-MW(e) HTGR.

3

TABLE 1.1

FORT ST. VRAIN NUCLEAR POWER PLANT STEAM GENERATOR DESIGN CONDITIONS

100% Steam Flow 252 Steam Flow Load Generator Output (MM) Station Output (MM)

Helium Flow (lb/hr) Outlet Pressure (psia) Inlet Temperature (F) Outlet Temperature (F) Pressure Drop (psi)

Feedwater/Hain Steam Flow (lb/hr) Outlet Pressure (psia) Inlet Temperature (F) Outlet Temperature (F) Pressure Drop (psi)

Reheat Steam Flow (lb/hr) Outlet Pressure (psia) Inlet Temperature (F) Outlet Temperature (F) Pressure Drop (psi)

342.0 81.2 330.2 67.4

3,452,400 1,026,000 686 589

1263 1035 632 501

3.47 0.36

2,305,326 574,776 2512 2419 403 299

1005 1000 590 45

2,245,366 556,928 600 151 673 570

1002 1000 42 11

A

ORM. 3WIC -6 ''60 - *T » • Q I

(M !• Ill a

big. £.2. ic-t St. Vrain Nuclear Power Plant Steam Generator Module.

5

1.1.1 Helium Circui t ry

Helium leaves the core and travels downward to the steam generator. The gas then passes over the reheater, finishing superheater, pre-superheoter, evaporator, and economizer surfaces consecutively. Cold helium exits at the bottom of the steam generator modules and flows into the four suction chambers of the helium circulate-*.

1.1.2 Main Steam Circuitry Feedwater is distributed to the 12 modules through flow-control

valves and enters each module through a ringheader. The heat-transfer surfaces of the economizer-evaporator-presuperheater (EES) and finishing superheater (SHI1) bundles are composed of helically wound tubes originating at the feedwater subheaders. In the botvMi EES bundle, the water is preheated, evaporated, and presuperheated. The steam flow is counter current to the helium flow passing over these tubes. Then the crossover tubes lead the presuperheated steam to the finishing superheats (SHII) which is arranged for co-current flow above the EES bundle. This arrangement is required to minimize tube metal temperatures and enhances stability in this section. Superheated steam leaves the SHII bundle via tubes and is collected in the main steam ringheader, which is connected to the main steam piping.

1.1.3 Integral Reheater f. > -cuitrv Cold reheat steam enters each steam-generator module through

an annul us. The steam is discharged from the annulus to the reheater (RH) bundle. The RH bundle, also composed of helically wound tubes, is

6

the uppermost heating surface in the steam generator, and is arranged

for counter current flow. Hot reheat steam is carried via the

central reheat pipe through the primary and secondary closures ano

returns to the intermediate pressure (IP) turbine.

1.2 Lumped Parameter Aralysis

The dynamic analysis of a subsystem such as a once through steam

generator (OTSG) is a distributed parameter problem. However, a lumped

parameter approach is commonly used for i t s simplicity and ease of

solution by either analog or digital techniques. The lumped parameter

modeling technique generally results in the physical system being

represented in the following fom::

G(x,x,t) = F^x.t) + F 2(u,t) 1.2.1

Y(t) = F 3(x,u,t) . 1.2.2

For dynamic analysis, i t is often des i r^ le to have a linear set of

coupled differential equations. Equation (1.2.1) can be linearized by

expanding each term in a Taylor Series about in operating point. If

the terms above first order are neglected, vhe result is valid as

long as the variables stay close to the operating point.

The form of the linearized set of equations i s :

A

7

Branch of the Reactor Engineering Department of General Atomic in

the mathematical model LAP (Linear Analysis Program).

1.3 Model Formulation

The linear model of LAP includes the primary coola.it system,

secondary coolant system, and reactor system. Each of these sub

system models was derived from first principles (conservation of mass

momentum, energy, and neutrons) except for an apprj imation used

for the main steam section in the OTSG in early studies. An empirical

or transfer function approach was adopted for the main steam section.

This empirical model was obtained by tuning arbitrary constants in

LAP's' ' transfer function ty^_ model to produce agreement between

LAP and T A P . ^

The development and implementation of a linear steam generator

model derived from first principles was undertaken at GA (work

performed by W. 0. Leech* "̂ ) because it was believed that an OTSG

linear model based on first principles would be superior to the original

model for two reasons. First, a model based on first principles

yields a better physical understanding. Second, the tuning of the

arbitrary constants which appear in the transfer function involved a

large degree of qualitative judgement as to when satisfactory

agreement was reached between TAP and LAP.

In order to account for movement of the beginning and end of

the boiling zone, the main steam section was divided into three regions

with moving boundaries corresponding to the economizer, evaporator,

and superheater. The first node upper boundary was defined as tie

http://coola.it

8

length (from the inlet) required for the water to be heated to

saturation. The second node upper boundary, was defined by the

additional length required to heat the water from saturated liqi.id

to saturated vapor. The third node was defined as the remaining length

by the steam generator. This formulation allows movement of the points

where boiling begins (OX quality) and ends (100% quality) and yields

a low order representation of the physical system that retains all of

the physical processes usually considered important.

1.4. Mathematical Model

1.4.1 Assumptions

In this model ng, the assumptions can be grouped into two sets:

those relating to equation developments for each system component and

those made in obtaining a simplified conceptual system from the real

system. More details of system assumptions or approximations are made

in the development following this section.

Assumptions listed in the following five points refer specifically

to equation development for the m,in steam section bundle.

1. Fluid properties a-e uniform at any gi/fn cross section.

2. Fluid properties for each node are weighted by the properties

at boundaries.

3. No heat conduction occurs in the flow direction.

4. Dynamic effects of gas-pressure changes are negligible, so

the gas flow rate is assumed to be time dependent only.

5. Balanced flow and uniform heat flux exist, at any node in the

main steam sections.

9

1.4.2 Basic Equations

Figure 1.3 shows a counter flow heat exchanger. The locations X ? and X, are free to move but always remain at fixed quality. For the

steam generator model, X~ is the location of saturated liquid (0% quality)

and X., is the location of saturated vapor (100* quality),

a. Conservation of Mass

The basic relationship is:

Flow in - Flow out = rate of change of mass stored.

ORNL-OWG 76-7763

TS,

I WG,

i

TG, • TG, t

WG, > # T G Z G A S

W G 3 » • TG3 WG4

1 TG 4

i

WG, i

TG, • TG, t

TG 2 — l b 3 » • TG3

WG4 1

TG 4

• TM, ( >TM2 • TM 2 « TM 3 • TM 3

1 Pi Pi P 2 97 STEAM P 3 • HS3

P,

1

• »HS, • H S , < >ws 2 • ^ 1 » 3 • HS3 HS4'

1

WS, WS, ws2 ws3 ws3 ws4 MST |

TUBE FROM ECONOMIZE R EVAPORATOR SUPERHEATER TUBE TO FEEOWATER TURBINE PUMP

TS, FEEDWATtR TEMPERATURE

MST : MAIN STEAM TEMPERATURE

Fig. 1.3. Schematic Representation of a Moving Boundary Model for a Counter Flow Heat Exchanger.

in

Gas side WG. » W G 1 n 0=1,2,3) L 4 J

where WG = gas flow rate (lb/sec) WG ; = inlet gas flow rate. Here we assume the gas flow rate is a function of time only and

the same in every section. Water/steam side

(MS. - L. . A • V -

11

Conservation of Energy The basic relationship i.: Energy flow in - Energy flow out - Rate of energy transferred

out = rate of ̂ ergy stored, a. Gas

< W Gi +l ' H G i + l + Ag * pi +l * K Gi+l • W " < H Gi * H 6 i +

A g • P f - HG, • *,) - Q g m - A^ • C y • {L B i • TG, (X. + 1 - X.)] 1.4.4

where HG = enthalpy of gas (Btu/lb) p. = gas density (lb/ft )

Q = heat transfer rate from gas to metal (Btu/sec) C = constant volume specific heat for the gas (Btu/lb°F) TG » gas temperature (°F)

We assume that the helium is an ideal gas under the conditions that exist in the steam generator. Then

HG = C (TG • 459.6) I.4.5

where C = constant pressure specific heat for the gas (Btu/lb°F). P

The gas density can also be related to the gas pressure and gas

temperature.

P i « PGi • 0.373343/(TG1 • 459.6) (i - 1 , 2 , 3, 4) . 1.4.6

12

Equation (1-4.4) can be rearranged to yield

WG i + 1 • HG i + 1 - HG. • m. - Q g m , A g • Cy • ^ . TG\

( x i + 1 - x . ) ]

- A g * * i + l ' H G i + l * V l + A g • p i * H G i * * i 1.4.7

b. Metal

The basic relationship i s :

Heat flow in - Heat flow out = rate of heat storea in the rcetal

where TM = average metal temperature for the node (°F) C p m = metal heat capacity (Btu/lb-°F) Q = heat transfer rate from metal to steam (Btu/lb)

M = metal mass per unit length (lb/ft). We ass'ime the metal boundary temperature to be given by the arithmetic average of adjacent node temperatures

TM i = \ • (7M i = 1 + TTl) for i = 2, 3. 1.4.9

Substitute Equation (1.4.9) into (1.4.8) and rearrange to yield

* , • , - £ • o v m . , ) . * . ] . 1-4.10

13

The heat transfer rates are evaluated as follows:

-C .

Q = [Heat transfer area • (TG^ - Tff.)]/[KG. • (WĜ ) g i +

R- • 0.5]

_ -C . Q~ = [Heat transfer area - (TM. - TS-)]/[K . • (VJS-) S 1 +

no 1 1 SI 1 ^ • 0.5]

1.4.12

where

K = constant for gas to metal heat transfer 9

C = flow exponent for gas to metal heat transfer

K = constant for metal to water heat transfer

C = flow exponent for metal to water heat transfer

R = thermal resistivity cf the metal times the tube thickness

TS = water/steam teirperature.

C. Water/Steam

{ H S . . HS, - i - • HS, • A • X,) - (HS 1 t, • HS, + ) - 1 — • HS j t )

* - *i*i» * t i - * - Sr % * •» - -'Vi ~ *«" '• 4-' 3

where

HS = enthalpy of water/steam (Btu/lb)

u = internal energy of water/steam (Btu/lb).

Rearranging Equation (1.4.13) yields:

us, • HS, - wsi+, • HS1+, • H- • » • 3E i j j • i, •

14

Conservation of Momentum

Since i t is assumed the flow on the gas side of the steam

generator is time dependent only, there is no need for a momentum

equation in that region.

In the water/steam, the basic principle i s :

][ forces = Rate of change of momentum.

P - P - K K T 2 - ( ] ) • PITCH . ( X - + 1 - X.) 1 1 ' T 1 144 • v-

«

15

ws1 -. / ( P 0 • - P1>" are functions of pressure with

fixed quality.

16

3. No heat conduction in the flow direction.

4. Balanced flow and uniform heat flux exists at any node.

5. In the economizer, the flow in equals flow out.

6. The average water/steam flow rate used in the equation is

assumed to be a linear average of the flow in and out of

each node.

7. The water/steam flow rate is proportional to the square root

of pressure difference.

1.4.3 Summary of Equations

Now for the purpose of review, the following equations are a

summary of those that must be solved for each node of a moving

boundary heat exchanger.

The twelve differential equations are:

W S i " W S i + l = A { ^ C ^ - ( X i + 1 - X.) - 1 — - X i + 1

+ h' V ( i = u ly 3 ) 1.4.20

Note that X] = X 4 = 0

WG 1 + 1 • HG. + 1 - WG, • HG, - Q g m - Ag • Cv . ^ •

[p\ • TG. • ( X 1 + , - X,) ] - A g • P . + 1 • HG. + 1 • X . + 1

+ A • P i • HGi • X. ( i = 1 , 2, 3) 1.4.21

17

V" V = M • CP* • t ( X i * l - V • TH 1

4 < ™ I - T W • * 1 + i - \ - (™i - ™i-i } * * i ]

(i = J,

18

5. Three gas flews (WGj, WG 2> WG 3) described by algebraic Equation (1.4.24).

6. Three water/steam flow (WS 2 > WS 3, WSJ described by

algebraic Equation (1.4.25).

The assumed knowns, or inputs to the problem are:

1. Inlet water pressure (P,)

2. Inlet water flow and enthalpy (WS,, HS.).

3. Fixed locations (X,, X.)

4. Inlet gas temperature and flow (TG 4, WG. ).

1.5 Linearization of Equations

The previously derived equations are linearized in the following

sections.

1.5.1 Equation (1.4.20) Linearization

Equation (1-4.20) is repe-ted as follows:

H S I - « « • A 4 % • iv, - v - ^ • *,•!+ J? • v '-5-1

We assume V- is a function of HS. and P.. The average

quantities HS. and F. are obtained by weighting the end point values

for each node as follows:

HS. = W 1 • HS i + (1 - H.) • H S i + 1 1.5.2

P = u. • p • (l - w,.) • P i + 1 . 1.5.3

19

The linearized form of Equation (1-4.20) is

6WS 1 - «WS- + 1 = A l sl 6 X i + 1 • A 2 j i 6H S i • ^ 6H$.^

**f i b»i i 6 , i ^

where

A i i = A • ( r - y—* 1 , 1 M v i + l

(Note: All the values for each variable in the coefficients represent

the steady state in the system.)

*2.1 "* ' Ll • i » p / l 2 i = I 1 1

= 0 i = 2, 3

(V is only a function of pressure when i = 2, 3).

3V, i = 3 A3,i - - * • h • ^ s + A 2

= 0 i = 1 , 2

A 4 , i - - A * L i * Hs/Vi

A 3 ^ i 5,1 -" " A ' L i ' ( 3 V > i + 1 / V

A6,i =~A ' (v7 - V 6HS. can be related to the temperature TS. as follows:

3HS, 3HS. 6 H S i " (-5P7 >TS 1 6 p i + «*T57)P1 6 T S i '

20

Since enthalpy is almost independent of pressure for subcooled water, i.e.,

3HS,

For i = I, we have

ails, 6 H S i = 5 T S r

1 K i

The f ina l form of the l inear ized Equation (1.4.20) i s :

fo. i = l .

3HS, 3 W S 1 - 6 W S 2 = A l , l 6 *2 + A2'l * ^ \ ' 6 f S l l ' S A

m

i * 2

6WS 2 - 6WS3 = A 6 ? 2 5 X 2 + ( A 4 > 2 + A 5 > 2 ) 5P 2 + A 1 ̂ 1-5.5

l = 3.

The term HS4 can be related to the main steam temperature (TS4) and

pressure ( P j as fo l lows:

3HS. 3HS.

3HS. 3HS.

21

Af ter 6HS. is replaced by «P 3 and sTS^, we have

3HS, «WS3 - 6WS4 - A 3 j 3 ( *) 6 fS 4 + 6 1 6P 3

4 v1

+ A 6 j 3 6 X 3 1.5.6

where

3HS 6 1 = < A4,3 + A5,3> + A 3,3 *

22

The l inear ized form of Equation (1.4.21) i s as follows

B, • 5HG.., + B 0 . 5WG. + B, • «TG-., + C. - 5TG-1,1 l+ l 2,1 l 3,1 i+ l 4,1 l

+ B. . 5TG. + B c . 5X.^, + Bc - *X- + B, . 6TM. = 4,1 l 5,1 i+ l 6,1 i 7,1 l

where

A 7 , i * X i + l + A 8 , i * X i + A 9 , i 6 T G i

— — " ( C n i + ^ E l , 1 = * - O D - L . • ( T G . - T t y . K g i - C g | • WG ^

-C . ? (K • • WG 9 + 0.5R.) gi I

E 2 - i = * * 0 D * ( K g i * W G ° * 5 R i }

B l , i = H G i + l - E l , i - ( 1 - F i }

B 2 , i = - H G i - E l , i * F i

B, - - WG. , • C 3 , i i+ l pg

/

4,1 " h,i • L i

3. • = - WG. • C 4 , i i pg

Bs.i = - E2,i • fre,-1™,)

B 6 , i = " 8 5 , i

6 7 , i ' E 2 , i • L i 1.5 .9 ( c o n t . j

23

h,i - A 9 " ( C v g • h • T G i - '1*1 - H B , t l )

AQ . = - A • (C • p. • TG. - p. • Mb) 8, i g vg i i i 7'

A- , = A • C • L. • p. 9,1 g vg 1 M i

For i = 1

B, 1 6WG2 + B 2 1 6WG, ^ B., 1 6TG 2 + C. , 5TG, + 6. 1 5TG

+ B 5 , l 6 X 2 + B 7 , l 6 ™ 1

A 7 , 1 6 X 2 + A 9 , 1 6 T G 1 1.5.9

1 = 2

Bl,2 6 W G 3 t B2,2 6 W G 2 + P3,2 6 T G 3 + C4,2 6 T G 2 + B4,2 6 T G 2

+ B c 5 5X_ + B, 0 6X- + B. 0 6TM, = 5,£ 3 6,Z 2 /,2 2

A7,2 6 X 3 + A8,2 **2 + A9,2 < T G2 1.5.10

i = 3 Bl,3 6 W G 4 + B2,3 6 W G 3 + B3,3 6 T G 4 + C4,3 6 T G 3 * B4,3 6 T G 3

+ B, , 6X 0 + 8., ., 6TH-

A 8 , 3 5 X 3 + A 9 , 3 6 T G 3 1.5.11

24

Note that e i ther the variable 5TG or the variable 6TG can be eliminated

by using Equation (1 .5 .8 ) . In order to make changes of the weight factor

(F.) i n the system dynamic response study convenient, we keep both

variables in the equation.

1.5.3 Equation (1.4.22) Linearization

Equation (1.4.22) is repeated as fol lows:

JL_ i Q - 0 = M • C • [ ( X . A , - X.) • TM. + ± • (TM. - TM. A J • wgm ins pm l x i+l v i 2 v l i +V

* i + l " 2~ ' ( ™ i " ^ i - l * * X i ] ( 1 = K 2 * 3 ) ' U 5 ' 1 2

We assume the term TS\ in 0 is a function of P. and HS~.. l ins i i

And, l e t

P. = W. • P. + (1 - W.) • P.^. i l l ' r l+l

HSi = Wi • HS i + (1 - W.) • H S i + r

The l inearized form of Equation (1.4.22) as follows:

B 8, i 6 X i + l - b 8 , i 6 X i + B9,1 «™1 + V i 6 T G i

+ B, , . 6WG,., + B v . , 6WG. + (B . . . + B., .) 6P\ .2,1 i+ l U , i l 14,1 15, i l

+ B 1 6 , i 6 H \ + B 1 7 , i 6 H S i + l + B 1 8 , i m i * B 1 9 , i ' 5 W S i + l

= A l l , i 6 T M i + A 1 2 , i 6 X i + l + A 13 , i 5 X i

25

where -CS, E, t » T - 0 0 • (KS. • WS i • 0.5R ) _ 1

-(CS,*1) £S.i " **°° * L i * " i " C S i * H S * ( T"i " * V

" C S i -2 • (KS t • «S l • 0.5 Rj) '

* 8 . i " £ 2 . i • ^ I - T R i ) . £ 4 i . ( T ? f i . T s - 1 ) . 0 .

(For at the steady state, the heat transferred in equals the heat

transferred out.)

B 9 , i " - E 2 , i * L i * E 4 , i * L i

B 10, i * E 2 . i ' L i

B 13. i " E l . i ' F i

B 14. i ' E 4 . i ' ^ m " L i

«15.1 " E4.1 • ^ ^ * h

B i 6 . i ' E4.i • imf>p - L i « - '

» 0 1 » 2, 3

(For i * 2 , 3, TS. is function of P. only with fixed quality.)

26

3TS.

1 = 1 , 2

18,1 5,1 i

B 19.1 = " E5.1 * " ' V

A, , . » M • C . L. 11,1 pro i

A i 2 . i 4 M - c p B - ( ™ > - T , w

*13.i " " ? H * V (™i - ^ - l *

The f i n a l l inear ized equations fo r each node are:

For \ » 1

B 9 , l + 6 ™ 1 + B 1 C , 1 6 T B 1 *• B 12.1 d W G 2 + B 1 3 , l 6 W G 1

aHS, * < B14,1 + W 4P1 + B 16, l * ^ \ ' * T S 1 + B 1 8 , l 6 W S 1

* B i 9 j « W V V i 5 ^ i + ' V , 5 * 2 I * 5 ' 1 2

27

i = 2

B 9 ,2 fi™2 + B 1 0 , 2 6 T G 2 + B 1 2 , 2 6 W G 3 + B 13.2 6 W G 2

+ ( B 1 4 , 2 + B 1 5 , 2 ) i P 2 + B 18 ,2 6 W S 2 + B 19.2 d W S 3

= A n > 2 6 T K 2 + A ] 2 ) 2 « X 3 * A 1 3 i 2 oX2 1.5.13

i = 3 B 9,3 6 ™3 + B 10,3 6 T G 3 + B 12,3 6 W G 4 + B 13.3 5 H G 3

3HS4 + G 2 6P 3 + B 1 7 j 3 • ( a T ^ ) p 4 . 5 T S 4 + B 1 8 > 3 6HS3 + B ] g 3 « S 4

= A 1 U 3 6TM3 • A 1 3 > 3 6X 3 1.5.14

3HS4 where

6 z e t B 1 4 t 3 H 1 5 ( 3 + B 1 7 t 3 . ( 1 - W 3 ) - ( ^ T S ^

1.5.4 Equation (1.4.23) Linearization

Equation (1.4.23) is repeated as follows:

MS. • HS. - WS. + 1 - H S . + 1 • < U " A " i t % * °1 •

(X. + 1 - X,)] * A • [ 1 - • H S i . *. - 1 — • HS i + 1 • X i + 1 ]

( i = 1 , 2, 3). .1.5.15

Assume the 0. and V. are functions of P. and HS.. Let

3HS « H Si • (^'quality • i Pi < ^ 2 . 3 > -

28

Note: I t is 0* quality at boundary 2 and 100* quality

at boundary 3. The linearized form of Equation (1.4.23) is as follows:

B 2Q.i 6 H S i + B 2 1 , i 4 M S i * B 22 , i < H S i + l + B23,i « * i * l

- B 24,i 6 X i + l + B24,i 6 X i " B 25.i 6 ™i + B 2 6 . i fiPi

+ B 2 7 , i iPHI + B23,i * P M = A U , i ' 5 * S i + A 1 6 , i ' '» S i * l

* ( A 1 5 i + A 1 7 i ) 3 P . • A 1 8 i 1 « i + 1 • A 1 9 J 5 i ,

where sT5.

B 20 , i ' " l " E 4 , i • • S ' • ' = 0 i • 2. 3

B 2 1 , i = H W E 5 . i _ 3TS4

B22,i - " « M • E 4 , i • L i ' ' 3

= 0 i = 1 , 2

B 2 3 , i = - H S i + l + < 1 - , ' i » - E 5 , i

B 2 4 , 1 - - E 4 , i - < T H i - T J i '

B,r • = - E, , • L.

Z5,i 4,1 i

aTs. aTs". «, • - E4, • s • v * . * ( *#« , 4 1 !

I n B 2 6 , i ' B 27, i> and B ^ . , we le t

6 P i = 2 ^ 6 p i - l + 6 f V

29

for 1 = 1

1 = 1

1 = 3

1 3 H S 2 B26 1 = D l " 2 ' M S ' ( aP~^ Z 6 J ' z a P 2 quality

B27 1 * " I ' M S ' ( a F ^ < f / » l * 3 K 2 quality

B28.1 " °

1 aHS. oHS~ B 26,2 = D 2 " 2 ' W S * C (aPY" ,quality " ( ?P 2 ~ ) qual i ty 1

1 3 H S 3 B27,2 = " 2" * H S ' ( ^ q u a l i t y

1 3 H S 2 B 28,2 = 2 ' U S * (5P 2~ )quality

3HS B 2 6 . 3 s 0 3 + ? * M S - ^ q u a l i t y

B27,3 = °

1 3 H S 3 B28,3 " 2 * W S ' (1PJ )quality

A14,i " A * Li * r ' ' [V1 * ^ P , " °1 ' ^ V i ' 1 = 0 1 « 2 , 3

, 3U. 3V.

9 aO. 3V,

•jyj

i * 3

* 0 I » 1 , 2

D i H S i * l

» 8 » 1 Vi v i + 1

0, HS,

The linearized equations for each node are: 1 • 1

dHS B 20 .1 * ( 3 T ^ ) P 1 * 3 T S 1 * B 21.1 4 W S 1 * B 23.1 * S 2

~ B 2 4 , l aZ " B ? 5 , l fiTRl + B 26,1 * P 1 + B 27 ,1 &P2 3HS,

" A 1 4 , l ' ( n s ; V 6 t S l + ( A 15 ,1 + A 1 7 , l } * P 1 + A 13,1 *h ( 3 7 ) 1 M

B 21 ,2 6 M S 2 + B 23,2 6 W S 3 " B 24,2 j X 3 * B 24 ,2 « X2

- B 2 5 2 «TH2 • B 2 6 > 2 6P2 * B 2 7 2 «P3 • B 2 8 > 2 6P}

'

: i

where 3HS.

S ' ^ l ^ ' - y • B22.3 • < i ^ T S 4 aHS

G4 " A15.3 * A17.3 + ' A16.3 ' < w f \ '

The nine l inear ized algebraic equations are

6UGi - «WG1n (1 - 1, 2, 3) 1.5.18

ATG- = F 1 • 6TGi + (1 - F.) • «TG.+1 (i « 1, 2, 3) 1.5.19

«WS2 » KlCg • (5P] - 6P2) 1.5.20

where

KK2 = — 2(PrP2)

6WS3 = KK 3 • (6P 2 - 5P 3 ) 1.5.21

where

US KK3 =

2(P 2 -P 3 )

5WS4 = KK4 • (6? 3 - fiP4) 1.5.22

where

WS KK„ »

4 2M 4) 3 4' The linearized equations from (1.5.4) to (1.5.22) can then be

arranged in a matrix form.

32

1.6 Results and Discussion

The linearized equations presented in the above section were

used to calculate the frequency response and time response of the main

steam generator (no reheater) for selected inputs. In obtaining these

results, two different sets of equations were used. In the f i r s t set,

the feedwater pressure, P Q , (see Figure 1.3) is held constant; hence

the description "constant pressure model" (CPSG). In the second set,

the variable MS, (feidwater flow) is held constant; hence, the

description "constant flow model" (CFSG). The frequency response and

time responses for the constant pressure and constant flow models were

calculated for e ± input selected.

The code SFRMOD^5' was used to calculate the frequency resoonse.

The input to SFRMOD is of the form:

A , £ * A 2 ; = ? 1.6.1

where

A, = matrix of constant coefficients related to differential

te-rns

A 2 = matrix of constant coefficients related to algebraic

terms

x = vectors of system variables

? = forcing vector for the selected input.

Before calculating the time response, i t was necessary to

transfer the system of algebraic and differential equations (1.6.1)

33

describing the steam generator to the following form

dX. -£ = AX. + ? . 1.6.2 dt d d where X. = vector of state variables a A = matrix of constant coefficients f. = forcing terms for the system input.

(5) This was performed using the computer code PUREDIfF.v ' The input to PUREDIFF consists of the A, and A- matrices and 7 vector as in Equation (1.6.1). The A,, A-, and A matrices appear in Table 1.2 and Table 1.3.

TABLE 1.2

THE VARIABLES ASSOCIATED WITH CPSG FOR A } AND A 2 MATRIX

Number Variables

1 , 2 , 3 6 P r 6 P 2 , 6? 3

4 . 5 , « . 7 6TG,, 6TG 2 , 6TG 3 , 6TG4

8 , 9 , 10 6TB,, 6 T M 2 , a?M3

1 1 , 12 6Xn, 5XA

13 6MST

14, 15, 16 S T G . , fiTG"2, 6TG3

17, 18, 19, 20 6WS r 6WS2, 6WS3, «WS4

34

T^BLE 1.3

THE STATE VARIA8LES ASiOCIATFD WITH CPSG FOR REDUCED A MATRIX

Number State Variables

1 . 2 , 3, 4 s P r « P 2 . « P 3 , «TG4

5 , 6 , 7 «TK,. 6t i^ , sTilj

8, S SX ,̂ 6X3

10 «MST

11,12,13 6TG,. oTS^, 6TG3

A1(I ,J ) 1 1 2.61053E-01 2 11 2.28625E 01 3 i2 1.05238E 01 8 8 7.41049E 03 9 11-7.91655E 03

10 12-1.09647E 04 12 2 9.45497E 02 13 3 3.67353* 02 14 11-3.11099E 02 15 12-3.11099E 02 16 16 2.59237E 01

A2(I .J ) 1 17-1.00000E 00 2 19 l.GOGOOE 00 4 4-7.5OO00E-0J 5 5-6.70000E-01 6 6-3.00OO0E-O1 7 7 2.00000E 00 8 14-9.37232E 02 9 2-9.14281E 01 9 18 4.74777E 01

10 10 3.89902E 03 IC 19 3.52938E 01 11 2 2.59346E 01 l i 17-4.71644E 02 12 2 3.78754E 01 12 11 1.54129E 03 12 19 1.02896E 03 13 10-2.94904E 03 13 19-1.08763E 03 14 5-5.95539E 02 14 14 9.37232E 02 15 9-5.95840E 02 l i 15 5.95840E 02 16 10-S.49973E 02 17 1 6.18079E 00 18 2 7.24S00E 00 19 3 '..85951E 00 20 20 l.OOOOOE 00

1 11 3.38096E 01 2 12 1.47444E 0) 3 13-5.05814E-01 8 11-7.91655E 03 9 12-1.09647E 04 11 J. 4.14207E 02 12 11 1.57921E 04 13 12 9.52171E 03 14 14 3.67501E 01 15 15 1.97891E 01

1 18 l.OOOOOE 00 3 19-1.00000E 00 4 5-2.50000E-01 5 6-2.30000E-01 6 7-7.00OOOE-O1 8 1-8.77428E 01 8 17 8.58445E 01 9 9 2.35927E 03 9 19 2.33845E 01

10 13-1.83209E 03 10 20 8 . .3522E 01 11 8-2.61185E 03 11 18 7.48713E 02 12 3-2.76181E 01 12 12-1.54129E 03 13 2 2.76181E 01 13 12 1.8M34E 03 13 20 1.48584E 03 14 8-9.37232E 02 15 5 5.95539E 02 i5 l l -1 .61186t : 01 16 6 5.95539E 02 16 12-1.89635E 03 17 17 l.OOOOOE 00 18 18 l.OOOOOE 00 19 19 l.OOOOOE 00

2 2 1.01479E OJ 3 3 3.13884E-01 7 7 l.OOOOOE 00 9 9 4.71117E 03

10 10 7.51123E 03 11 11-6.98903E 02 12 12 8.68772E 03 13 13-4.25746E 02 15 11 3.11099E 02 16 12 3.11099E 02

2 18-1.00000E 00 3 20 l.OOOOOH 00

-4 14 l.OOOOOE 00 5 15 l.OOOOOE 00 6 16 l.OOOOOE 00 8 8 3.54908E 03 8 18 2.86148E 01 9 15-5.95840E 02

10 3 1.42560E 02 10 16-9.49973E 02 t l 1 1.13677E 02 11 11-1.39502E 03 12 1-2.59346E 01 12 9-1.76343K 03 12 18-8.24805E 02 13 3-1.25829E 02 13 13 2.04697E 03 14 4 5.95539E 02 14 11 1.45888E 03 15 6-5.95539E 02 15 12 1.611B6E O'i 16 7-5.95539E 02 !6 16 0.49973E 02 l o 1-7.24500E 00 19 2W./5951E 00 20 3-1.37705E 00

35

TABLE T . 3 (Continued)

THE REDUCED Al-MATREC IS 1 1-2.0806900E 01 1 2 1.3222581E 01 1 5 6.2250166E 00 1 8 3.3246164E 00 2 1 6.5625935E 00 2 2-8.12296"»0E 00 2 3 2.4341621E 00 2 5 3.2062525E-01 2 6 5.0771427E 00 2 8-4.2665825E 00 2 9 4.4378195E 00 3 1-2.629428SE 00 3 2 5.6758375E 00 3 3-1.1549730E 01 3 5-3.3963531E-01 3 6 2.2627077E 00 3 7 2.8572784E 01 3 8-2.1591740E 00 3 9-1.5588333E 01 3 I0-1.9833328E 01 4 4-2.0000000E 00 5 1-1.9712412E-01 5 2 1.4782137E-01 5 5-5.3030968E-01 5 8-2.7431257E-02 5 11 1.2647772E-01 6 1 4.7485876E-01 6 2-2.6879144E-01 6 3-8.7153971E-02 6 5 4.1298445E-02 6 6-1.3141069E 00 6 8 7.3300046E-01 6 9-7.1094406E-01 6 12 1.2646991E-01 7 1 5.9283441E-01 7 2-3.4774709E-01 7 :-o.:T97608E-02 7 5 7.6574564E-02 7 fi-5.1015300E-01 7 7-5.1910526E-01 7 8 4.8681015E-01 7 9-4.4591367E-01 7 10 2.4390888E-01 7 13 1.264G112E-01 8 1-2.364I872E-01 8 2 1.1217344E-01 8 5-4.8073113E-02 8 8-2.5674574E-02 9 1 4.0627545E-01 3 2-2.3232561E-01 9 3-4.1429687E-02 9 5 5.2477356E-02 9 6-3.4961307E-O1 9 8 3.3361596E-01 9 9-3.0558920E-01 10 1 6.8181896E 00 10 2-4.9869909E 00 10 3-1.6296978E 00 10 5 8.8068461E-01 10 6-5.8672705E 00 10 7 1.773230CE 01 10 8 5.5988035E 00 10 9-1.6029999E 01 10 10-1.2308591E 01 11 1-2.0013561E 00 11 2 9.4958210E-01 11 4 2.4830109E 01 11 5 2.5095078E 01 11 8-3.9918015E 01 11 11-4.7107468E 01 11 12 3.2246902E 01 il 13-3.5471603E 01 12 1 1.0103191E 01 12 2-5.4155455E 00 12 3-6.5127712E-01 12 4-1.0479436E C2 12 5 1.5806589E 00 12 6 2.4610168Z 01 12 8 8.7103363E 01 12 9-8.62591B6E 01 12 12-7.5017975E 01 12 13 1.4970625E 02 13 1-4.8762445E 00 13 2 2.7884445E 00 13 3 4.9725193E-01 13 4 7.6581772E 01 13 5-6.2984949E-01 13 6 4.1961641E 00 13 7 3.6651230E 01 13 8-4.0041618E 00 U 9 7.6815918E 01 13 13-1.1323302E 02

The time response calculations were performed using the computer

code MATEXP.

36

TABLE 1.4

DESCRIPTION OF EACH ANALYSIS RUN

Description of Perturbed Constant Flow Model or Run No. Input Variable Constant Pressure Model

1 Helium flow rate (1 lb/sec) CFSG

2 Helium flow rate (1 lb/sec) CPSG

3 Helium i n l e t temperature ( i°F) CFSG

4 Helium in le t temperature (1°F) CPSG

5 Feedwater f lew rate (1 lb/sec) CFSG

1.6.1 Discussion of Results fo r Run No. 1 (Figure 1.4 to 1.7).

Run No. 1 shows the CFSG model time response and frequency

response fo r a 1 lb/sec increase i n helium flow ra te . Increased helium

flow rate increases the heat t ransfer rate from the helium to the

secondary coolant through the tube w a l l . For constant out le t steam

header pressure and feedwater flow ra te , the steam pressure

i n i t i a l l y rises and then approaches the or ig inal value. This is

because there is no increase in feedwater flow to sustain the r i se .

The main steam temperature (TS4) increases s ign i f i can t l y (1.44°F),

whereas the temperature r ise for the helium out let node (average)

is re la t i ve ly low (0.63°F). The steady values of each output in time

response af ter 300 sec are close to the magnitude of the frequency

response at low frequency (0.004 rad/sec) jus t as they should be.

1

37

oo* 00 02- OO'Oll- 00 09 , 0 1 * (S33U030IHSWHd

e s: 0, «_# m c —* 3 z: Q

1/ • tr. y - \

o v; v. •^ a» N ^ ' ce

3 >> O o »—* c u. 41 3 E CT 3 01 - ^4 u •—1

ml* 5J z

, o m * J

. -~ oi 14 • 3 QC u

->-(

ORNL OWC It JJt* OHNI llWd lb nth

i v i i i i n n — J i i r m i l „•> i » 11 mi . i i i i m i l , \o* To^ "i or h of i o"

FREO. IRRO/SEC.) TC1B

FREO. (RAO/SEC.) TG1B

10' j f i i m i l J—i » i mil—;,i i i i i mi .-t—ri i M I I I TO 10' ICf 10' FREQ. (RflO/SEC.)

P3B 00

FREO. (RAO/SEC.) P3B

Fig. 1.6. Frequency Response of Helium Outlet Temperature to Helium Flow (CFSG).

Fig. 1.7. Frequency Response of Main Steam Pressure to Helium Flow (CFSfi).

39

The response curves of the outputs are almost monotonic and

settle down within 165 seconds (0.006 rad/sec in frequency domain).

The time lag is primarily due to the thermal capacitance of the metal,

water/steam mass and the thennal resistance from the primary to the

secondary coolant.

1.6.2 Discussion of Results for Run No. 2 (Figure 1.8 to 1.11)

Run No. 2 shows the CPSG model time response and frequency

response for a 1 lb/sec increase in helium flow rate. The system

OBM-DWG 76-7766

^ . 0 0

^ . 0 0

^ . 0 0

80.00 160.00 240.00 TIME (SEC)

80.00 160.00 TIME (SEC)

240.00

*Af 0.0004 320.00

A= 0.6453

320.00

e e o*= 1-5023

80.00 160.00 240.00 TIME (SEC)

320.CO

Fig. 1.8. CPSG Time Response to a 1 lb/sec Step Increase of Helium Flow.

I

H W 3 -o re i •»: < • • * - •

r-r OC C • 1 re i— rr \C

(rfc)

in rr K

AB^MK) 10° JO"* JO"' itf 101

i imiuT • •••••••r • i i n , i , r a i iiinuT

PHASE (DEGREES) -150.00 -100.00 -50.00 0.00

re -r. >— I H- re -n C XI TJ 3 c m ency

Flo

o

£ J) ?c D

y—v re o n x N t/2 O C 3 w X

SEC

- re "

*F

re 3 •o re

c re

X! ' C

10

X • 1

x re

JO"1

o -n N 3) m o o

OD JO

m n

o

Mb/sec' ABSIMR)

2 3 H S 6 7 6 9 J t f I I I I IQI I I I

PHASE(0EGREES1 -40.00 -20.00 0.00

CD JQ

20.00

Ot I

* l

ORNL-DWG 76-7769 O o

v 3 — i i 1 1 H I T — j i i I I I I I I n—i i i I I I I I . i—i i i I I I I I ,

Tcry ho-* \o'1 \

42

1.6.3 Discussion of Results for Run No. 3 (Figure 1.12 to 1.15)

Run No. 3 shows the CFSG model time response and frequency response

for a 1°F increase in helium in le t temperature. Increased temperature

of the primary helium coolant increases the rate of transfer of thermal

energy from the primary to the secondary coolant. "P3B" increases

i n i t i a l l y , ther, returns to the or iginal value due to the constant

O R M OWG 76 7770

^ . 0 0

^Too

'V.oo

80.00 TIME (SEC)

A = 0 . 0 0 6 0 160.00 210.00 320.00

A * - A = 0 . 3 7 1 1

80.00 160.00 240.00 320.00 TIME (SEC)

e e c*= 1 . 6 4 0 7

80.00 160.00 240.00 320.00 TIME (SEC)

Fig. 1.12. CFSG Time Response to a 1°F Step Increase of Helium In l e t Temperature.

OHNI own 76 >>n

10"" _y > i m n i — - t > i I ' I I I I I I — , , i i t " ' " t fjt' i i »i it T5 W icf1 ^ o 4

FREQ. (RflD/SEC.) TS4

fej < » • — • — i 'S

^ ^ •

44

i«"ti ;>» i . *

T J i l l mil—-a—1 • 1 mil—,1 i i i m«i . 1—1 1 1 mi l I f f* ™ 0 ^ \0'1 10* W

FREO. (RflO/SEC.) P3B

F i g . 1 .15. Tempera ture .

— 3 1 1 nu l l _ 1 • ' i •"'L ?—1 1 1 mil . 1—r 1 t m n , l(T \ f ™ 0 ' 10* I t f

FflEO.(RflD/SEC.) P38

Frequency Response of Main Steam Temperature to Helium I n l e t

feedwater f low. Al l the temperature nodes increased following the

increase of helium i n l e t temperature.

1.6.4 Discussion of Results fo r Run No. 4 (F ig j re 1.16 to 1.19).

Run No. 4 shows the CPSG model time response a.id frequency

response fo r a 1°F increase in helium i n l e t temperature. The system

transient responses for th is run are s imi lar to those of Run No. 3.

1

OHNL DWG 76 7 7 74

^ . 0 0

V. oo

en

^"oo

, , A = 0.0002 80.00 160.00 240.00 320.00

TIME (SEC)

A= 0.3798

80.00 160.00 i^O.OO 320.00 TIME (SEC)

e a » w * * *> A = 1 . 6 7 4 4

80.00 TIME (SEC)

160.00 240.00 320.00

Fig. 1.16. CPSG Time Response to a 1°F Step Increase of Helium Inlet Temperature.

OHM ()WO 'IS I'll

"\Q* "̂ Q-l1 \Q-X' ' \tf' '"""^ (), FREQ. (RAD/SEC.)

1S4

To' l o° FREQ. (RRD/SEC.)

TS4

W

Fig. 1.17. Frequency Response of Main Steam Temperature to Helium Inlet Temperature (CPSG).

OHM OW(i IA 1)16

10' 751—i i i i n i i — . , i i i 11 m i — 7 ^ — i i i T I I I I « i i t 11 m Yo- W ^ V

FREQ. IRflO/SEC.) 1 GIB

1 a—t i i mil—z?—t ' ' m i l 1'—i i r m i l — . r i i r inn .

Tor* "io \ o' T O 0 \ o FREO. (RflO/SEC.)

TG1R Fis. 1.18. Frequency Response of Helium Out

let Temperature to Helium Inlet Temperature.

(JKNI I)*(I It III)

W W Sd> FREQ. (RflD/SEC.)

P3B at

'io' itf FREO. (RflD/SEC.)

P3B

Fig. 1.19. Frequency Response of Main Steam Pressure to Helium Inlet Temperature (CPSC).

47

1.6.5 Discussion of Results for Run No. 5 (Figure 1.20 to 1.23).

Run No. 5 shows the CFSG time response and frequency response

fo r a 1 lb/sec increase in feedwater flow ra te . For constant helium

flow rate and i n l e t temperature, the temperature in the secondary

coolant c i r c u i t drops (-2.74°F) following the increased feedwater f low

ra te . The helium out le t temperature also drops (-0.68°F).

For constant ou t le t steam header pressure, the main steam pressure

increases by 0.71 psi in order to sustain the increase in the feedwater

flow ra te .

^ . 0 0

0.00

F3!

O.tO

OHNL DWG 76 777B

, , ,A= 0.7073

80.00 160.00 2>I0.00 TIME (SEC)

320.00

80.00 '60.00 240.00 TIME (SEC)

A= -0.6783 320.00

-2.7389 80.00 160.00 240.00

TIME (SEC) 320.00

i-ig. 1.20. CFSG Time Tvesponse to a 1 l b / s e c Step I n c r e a s e of Feedwater Flow.

'•nfc V: rt rt

1 rr?

H H -rJ 9 0 3 • • o ( I i—* n • a h j rt t—• C • •-! n

"1 rt ^ 0 fC

£ •n c (i) fD rt D a. O « *< fi: r t 73 rt n M •J.

~o I T J o I—" D 0 in 3T (D

^ o Ci i-n • T ; tn 2 n BJ

flBS(MR) ,io° ,io" JO"1 ,io° jo1

1 I IIHill • I fii.i.r 1 I mini • f l i m n r

PHRSE(DECREES) 70.00 110.00 150.00 190.00

(Tb7i« )

acr* >-= 00 r- • 3

•a I-I Q • M r^ C t*j IT • c fr. -r. rt (D C J3

C **1 C

P3B

m o

£ 3 5 ns r> •33 a. >< o £ V . B) PC W) r t re m rt tc o •1 T3

o «̂ •*: D l— 03 c re z o

q •4

RBS(HR) PHASE (DEGREES! J Or 1 Cr -120.00 -so.oo -uo.oo

• ~ — • • • • • p • • _ | • •

3

0.00

8fr

4?

U^^L iS**\* -

FREO. (RAD/l-EC.) 7G1B

——a—i i i mri 3—i i 111 m .1 i i 11 m i . i i i 11 m i . icr* "icr* Her' "ICT lo*

FREO. (RflO/SEC.) TG1B

Fig. 1.23. Frequency Response of Main Steam Pressure to Feedwater Flow (CFSG).

References

1. Richardson, D. C , "Linear Modeling and Control of High Temperature Gas-Cooled Reactor (HTGR) Power Plant," Gulf-GA-A12062, April 1972.

2. Savery, C. W., e t a l . , "TAP - A Fortran IV Program for tne Transient Analysis of the HTGR Power Plant Performance," Gulf General Atomic, 1966.

50

3. Leech, W. D., Personal communication, General Atomic, 1974.

4 . Ray, A. , "A Nonlinear Dynamic Model of a Once-Through Subcri t ical Steam Generator," M. S. Thesis, Northeastern Universi ty, 1974.

5. Harrington, R. M., Personal communication, The University of Tennessee, 1974.

6. B a l l , S. J . and R. K. Adams, "MATEXP: A General Purpose Dig i ta l Computer Program for Solving Ordinary D i f fe ren t ia l Equations by the Matrix Exponential Method," ORNL-TM-1933, Oak Ridge National Laboratory (August 1967).

I 51

CHAPTER I I

NONLINEAR MODELING OF A "TGR STEAM GENERATOR

Ming-Huei Lee

A nonlinear model is important wher large disturbances or various

operation levels are involved in the simulation of a steam generator.

I t is expected to be more accurate than the l inear model for these

conditions, The nonlinearity of the system parameters and the

spatial effects cause the problem to be quite complex, Moreover,

since the uncertainties in the system parameters are s ign i f i can t ,

the results w i l l have uncertainties. Therefore, i n the nonlinear

model, the spatial e f fec t , the nonlinear parameters and the

uncertainties in system parameters are three major areas of concern.

Practical methods for handling spatial effects require the use

of lumps to represent the spatial d is t r ibut ions. However, as the

number of lumps is increased, the computing time is also increased.

I t is important to know whether a few-lump model can provide the

necessary accuracy. In th is study, the number of lumps w i l l be

varied to f ind the number required to reach asymptotic performance as

the number of lumps increases.

The system parameters which are involved in major nonlinear

terms include the thermal conduct ivi t ies, heat capacit ies, and

viscosit ies of f l u i d (steam, water, helium) and/or tube metal,

surface tension of water, supression factor and some other factors

in the heat transfer correlat ions. These parameters may be evaluated

using available subroutine programs or by using emperical equations

which are obtained by least square methods.

52

In studying the uncertainties in system responses due to parameter

uncertainties, the heat transfer coefficients and two phase multiplier

factors are most important, since their inaccuracy could be up to 20%.

Other important parameters with significant uncertainties will also

be studied for their sensitivities.

The dynamic equations for the transient state were derived from

the fundamental differential equations. The conservation of energy,

the conservation of momentum, and the conservation of mass equations

were developed into a moving boundary model. The developed model was

compared to different models from a literature survey so that an

assessment of model assumptions and approximations could be made.

The literature was revi wed to find the data and correlations

required for the steam generator model. A short summary of the

information from the literature and other preparations for the model

development follow:

A. Survey on heat transfer correlations and/or friction factors,

1. For single phase

a. Seban and McLaughlin's correlation (2)

b. Mori and Nakayama'sx correlation c. Pressure Drop and Heat Transfer in Coils summarized

(3) by Srinivasan et a l . (4)

d. Srinivasan's ' correlation e. I to 's corre lat ion.

All of the above correlations are for helical coils -

one of the features of the HTGR steam generator. The

correction factor for helical tubes from (e) or (a) was

53

selected to use in this study, since the applicable

range of Reynold numbers and curvature i s adequate for

HT6R steam generator.

For two phase, steam/water

a. Borishanskii, et a l ' s . ' correlat ion (stra ight tube)

b. Owhadi, Be l l , and Crain's correlation (coi l tube) (8)

c. Crain and Bel l 's correlat ion (co i l tube) (9)

d. Chen's correlation (st ra ight tube)

e. Rohsenow's * correlat ion (stra ight tube)

f . Miropolski i 's ' correlat ion (st ra ight tube)

(a) through (e) are for nucleate bo i l ing and subcooled

bo i l i ng , ( f ) is for f i lm bo i l i ng .

Chen's correlat ion, Rosenhow's corre la t ion, and Borishanskii 's

corrections were selected for use in the nucleate boi l ing

and subcooled boi l ing region. The last two correlations

were evaluated at high pressure and high temperature.

The Miropolskiy's correlat ion was selected to use in

the dryout region for high pressure and high temperature.

The correction factor for hel ical coi ls is the same as for single phase, which is based on the discussion in (b)

(91 or the suggestion of Co l l ie r . '

For single phase, steam (13)

a. Miropolskii and Shitsmanv ' correlation b. Bishop et al's/ 1 4' correlation The correction factor for coil tube is the same as for single phase water.

54

4. For helium side heat transfer and pressure drop through banks

of helical tubes, the G i l l i ' s J ; correlations were selected

to use. Grimi^on's - straight tube bank data are necessary

for evaluating G i l l i ' s correlations.

Development of the two phase pressure gradient in homogeneous

models by employing Colebrook f r i c t i o n coef f ic ient , I to 's

curve pipe correction factor, and McAdams ' ' two phase

viscosi ty 's def in i t ion was made.

Obtaining the ercperical equations for the nonlinear parameters.

The thermal conductivit ies of the f l u id and tube metal, the

viscosit ies of f l u i d , the surface tension of water, suppression

factor and so on are necessary to obtain the i r emperical equations

for practical use. Some of these equations may be found in the

l i te ra ture survey; some of them were obtained by using l inear

regression methods to f i t the numerical tab le . .

Steady State Program.

The steady state program has been developed in the multilump case

so that the steady state distr ibut ions of the state variables can

be obtained. This was done for three purposes: (a) To study the

differences of the results by using di f ferent correlations and to

compare them to the HTGR steam generator design data; (b) to

evaluate the adequate average state variables for the transient

study in a few lump models; (c) to provide the i n i t i a l conditions

to the transient case.

55

ihe steady state prog rain contains the main program and the following

subroutines:

1) SUBROUTINE FRCTN 2) SUBROUTINE TWOFRC 3) SUBROUTINE SUBFRC 4) SUBROUTINE EQCKT 5) SUBROUTINE TW0EK1 6) SUBROUTINE TW0EK2 7) SUBROUTINE SUBEK 8) SUBROUTINE VSCST 9) SUBROUTINE VISFG 10) SUBROUTINE SUBVIS. 1) , 2), and 3) are used to evaluate the Colebrook friction factors by an i terat ion method for single phase steam, t»o phase steam/

water and single phase water respectively. 4 ) , f ; ) , 6 ) , and 7)

are used to evaluate the equivalent thermal conductances, tube

temperature and tube surface temperature by an i tera t ion method for

single phase steam, two phase steam/water dryout region, two phase

steam/water nucleate boi l ing region, and the single phase water

region respectively. 8 ) , 9 ) , and 10) are used to evaluate the

dynamic viscosit ies of the above three regions.

The steady state program is now being tested and debugged.

In the fu ture, the following are planned:

1. Comparison of steam generator models using d i f ferent assumptions,

approximations, and correlat ions.

2. Steam generator dynamic simulation.

56

3. Sensi t iv i ty to the system parameters.

4. Study of small and large disturbances.

5. Extension of the model from a few lumps to many lumps.

References

R. A. Seban and E. F. McLaughlin, "Heat Transfer in Tube Coils with Laminar and Turbulent How," In t . J . Heat Mass Transfer, Vol. 6, pp. 387-395, 1963.

Yasuo Mori and Wataru Makayama, "Study on Foroed Convective Heat Transfer in Curveo Pipes," In t . J . Heat Ma.~s Transfer, Vol. 10, pp. 681—695, 1967.

P. S. Srinivasan, S. S. Nandapurkar, and F. A. Holland, "Pressure Drop and Heat Transfer in Coi ls," The Chemical Engineer, London, 1968, Ho. 218, P.CE113.

P.S. Srinivasan, S. S. Nandapurkar, and F. A. Holland, "Fr ict ion Factors for Coi ls," Trans. Instn. Chem. Engrs.,Vol. 48, p. T156, 1970.

H. I t o , "Fr ic t ion Factors for Turbulent Flow in Curved Pipes," J . of Basic Engineering, p. 123, June 1959.

A. A. Andreevskii, V. M. Borishanskii, V. S. Fokin, V. fJ. Fromzel, G. P. Danilova, G. S. Bykov, and V. A. Chistyakov. "Recommendations for Computing the Heat Transfer Coefficient in a Two-Phase Steam-Water Flow in the Pipes of a Steam Generator," Heat Transfer-Soviet Research, Vol. 5, Mo. 3, p. 130, 1973.

Al i Owkadi, K. J . B e l l , and B. Crain, "Forced Convection Boil ing Inside He*Ically-Coiled Tubes," I n t . J . Heat Mass Transfer, Vol. 11, pp. 1779-1793, 1968.

B. Crain and K. J . B e l l , "Forced Convection Heat Transfer to a Two-Phase Mixture of Water and Steam in a Helical Co i l , " American Ins t i tu te of Chemical Engineers Symposium Series, No. 131, Vol. 69, p. 30, 1973.

J . G. Co l l i e r , Convective Boil ing and Condensation, McGraw-Hill Book Company, 1972.

1

57

10. W. H. Rohsenow, "A Method of Correlating Keat-Transfer Data for Surface Boi l ing of Liquids," Trans, of ASME, Vol. 74 3 pp. 968-976 (1952).

11. L. Duchatelle, L. de Nucheze, and M. G. Robin, "Theoretical and Experimental Study of Phcnix Steam Generator Prototype Modules," Nuclear Technology, Vol. 24, p. 12-", 1974.

12. Z. L. Miropolski i , "Heat Transfer in Film Boil ing of a Stean-Water Mixture in Steam-Generating Tubes," AEC-tr-6252, 1963 (Teploenergotika, Vol. 10, No. 5, p. 49, 1963).

13. L, Miropolskii and M. E. Shitsman, "Heat Transfer to Water and Steam at Variable Specific Heat ( in near c r i t i ca l region)," Soviet Physics-Technical Physics,Vol. 2 , No. 10, p. 2196, 1957.

14. A. A. Bishop, F. J . Krambeck, and R. 0. Sandberg, "High Temperature Supercrit ical Pressure Water Loop, Part I I I , Forced Correction Heat Transfer to '.uperheated Steam at High Pressure and High Prandtl Numbers," WCAP-2056 (p t . 3) 1964,

15. P. V. G i l l i , "Heat Transfer and Pressure Drop fcr Cross Flow Through Banks of Mul t is tar t Helical Tubes with Uniform Incl inat ions and Uniform Longitudinal Pitches/ ' Nuclear Science and Engineering, 22, pp. 298-314 (1965).

16. E. D. Grimison, "Correlation and Ut i l i za t ion of New Data on Flow Resistance and Heat Transfer for Cross Flow of Gases Over Tube Banks," Transactions of the American Society of Mechanical Engineers, Vol. 59, p. 583, 1937.

17. L. F. Moody, "Fr ic t ion Factors for Pipe Flow," Transactions of the ASME, Vol. 66, p. 671, 1944. ~~

18. W. M. McAdams, et a l . , "Vaporization Inside Horizontal Tubes-II-Benzene-Oil Mixtures," Transactions of the ASME, Vol. 64, p. 193 (1942).

58

CHAPTER I I I

GAS-COOLED STEAM GENERATOk LINEAR MODEL

David G. P^nfro

This phase of the project is concerned with the development and

investigation of a l inear , moving-boundary model for the Fort St.

Vrain steam generators. In this progress report, a br ie f history

of the work in th is area w i l l be reviewed before describing the current

project in greater de ta i l .

General Atomic uses a l inear model of the HTGR-NSS in part of the

design process of the plant control system. This mathematical model is

called LAP (Linear Analysis Program). I t was developed by combining

l inear models of the reactor, primary coolant, and secondary coolant

systems. Each of these models was derived from f i r s t principles using

the basic conservation equations except for the steam generator. GA

was unable to develop a satisfactory steam generator model by the time

the rest of the system models were ready, so an empirical or transfer

function model was substituted. However, a rodel based on the physical

system s t i l l was considered to be more desirable. The development

and evaluation of this model is the purpose of the current project.

General Atomic developed a lumped-pa-ameter, three-node, moving-

boundary mcdel for the steam generator. The lumped parameter approach

was chosen fo r i t s s impl ic i ty and ease of solut ion. The moving boundary

technique attempts to account for movement of the beginning and end of

the boi i ing zone as steam generator conditions change. This is accomplished

by dividing the main steam section into three regions with moving

boundaries between the economizer and evaporator zones and the evaporator

59

and superheater zones. This representation allows a f a i r l y simple model

that overcomes the d i f f i c u l t i e s encountered by f ixed boundary models.

The oroject at The University of Tennessee is based on the or iginal

GA model just described. A version of LAP and the iecessary data

including steady state parameters were supplied by GA. To improve the

physical va l id i ty of the time responses of the or ig inal GA steam generator

model, S. I . Chang at UT made several basic modifications. These included

the revision of equations to allow for replacement of node-boundary state

var ables by node-averaged state variables where applicable. A sketch

of the steom generator model with variables as modified by Chung is

shown in Figure I I I . l . As indicated by the f i gu re , both node-boundary state

TS,

WG,

TG,

HS, WS,

• TG.

TM,

• H S , WS.

WG,

TG,

•== „. , .WG, jTG 2 GAS TG,

oTM, >TM,

WS,

P? STEAM • • WS,

( T V ,

WS,

ORNL-OWG 76-7763

W G 4 • T G 3 <

T G ,

• T M ,

1

_^_ HS3 HS 4 t WS, WS,

MST

TUBE FROM ECONOMIZER EVAPORATOR SUPERHEATER TUEE TO FEEDWATER TURBINE PUMP

TS : : FEEDWATER TEMPERATURE

MST MAIN STEAM TEMPERATURE

Fig. III.l. Steam Generator Nodal Structure.

60

node-averaged variables remain in the model equations. As suggested by

GA, these are related by using a weighting factor, W., as fol lows.

7 i = Y i y i + Y i + i ° - V • ° - w i i 1 - I I I J

where Y. and Y. , are the boundary quantit ies and Y. is the node-averaqed

quantity. The weighting technique is very s igni f icant since i t is

used for temperatures on the primary side and for flow rates, pressures,

and enthalpies on the secondary side. In addit ion, internal energies,

specif ic volumes, and temperatures on the secondary side are calculated

from the previously-averaged pressures and enthalpies. GA suggested

specif ic values for each of the three nodal weight factors at which

they arrived by physical and in tu i t i ve reasoning. These suggested

weight factors influenced the calculation of the steady state

parameters they supplied. Chang used these weight factors and the rest

of the GA-supplied data to produce time responses that seemed reasonable

and va l i d . However, when he varied the weight in the helium entrance node,

(W.A to investigate i t s e f fec t , the time responses were s imi lar in character

to the previous ones but markedly oi f ferent in magnitude. This result

was a subject for some concern for at least two reasons. F i r s t , the

primary side infl.;ence was not thought to be so s ign i f icant . Also, GA

arrived at the suggested weight", f a i r l y a rb i t ra r i l y so there was l i t t l e

reason to believe they were the ideal ones to use.

This area of concern with weighting factors along with a basic

need to veri fy the model modifications independently were the reasons

behind th is phase of the project at UT. The i n i t i a l purposes were to

veri fy the unexpected s igni f icant dependence on the primary side weighting

61

factors, next to investigate these effects on both the primary and

secondary sides, and then to offer some explanation for the dependence

or to suggest changes to improve the weighting technique. Since the

beginning of this work, the investigation has broadened to include a

more general sensitivity study of the mode 1 and examination of model

assumptions made in its development.

The first task was to verify the work done on the model at UT by

S. I. Chang. An attempt was made to duplicate his development and compare

the results at each stage. First, the model equations were derived from

the basic conservation equations. Next, the equations were linearized

and set up in the model form with specified state variables and

perturbation variables. All discrepancies found were resolved by

mutual agreement and the final model form was decided upon. The next

task involved a review of LAP and its STMGEN subroutine to insure that

the model in its latest form was being implemented properly. After the

program was corrected for necessary changes, it was revised into a

simplified form with all extraneous statements in STMGEN that remained

from the GA version deleted. Finally, the input data for LAP as

supplied by GA was checked for accuracy. The entire process just

described that was done in preparation for the model investigation

was time-consuming but necessary. Although several mistakes were

located and corrected, no major' errors were discovered that completely

invalidated results obtained previously. The major benefit was an added

degree of confidence that could be placed in the steam generator model.

In the investigation of various model assumptions, the assumption

that the gas dynanics are negligible was questioned. The appropriate

62

modification to the model was made to include such effects but no

results have been obtained at this time.

As a first step in the investigation of the dependence on the

weight factors in Chang's model, some of the steam generator time

response results that he had obtained were verified. Although slightly

different due to the corrections made, the responses were still affected

greatly by the variation of W 3. The responses of main steam temperature

(TS4) and helium outlet node temperature (TG1B) to a 1°F helium inlet

step and a 1 Ibm/sec helium flow step are shown in Figures III.2-5. As

shown, the asymptotic values of some variations can more than double

when W, is increased from 0.45 vo 0.55. (Note: the GA value suggested

for use is 0.30. Some responses for this value are shown later.)

ORNL-OWG 76-7784

5.0 •

4.0 •

Method 1

Time (sec.)

Fig. III.2. Helium Inlet Temperature Perturbation (1°F).

!

r--u

J. 41;

63

OHNL OWTG 76-77«5

•-4 *0r the ^ i ' « » Outlet r̂ xJp 4v*r«9t ?e«Ger«tjr*, s**»tft

64

25

2 0 t

ORNL DWG "6 7:8?

Method 1

W, 045

15

l " t

4 1 69

h 0 9 0

r 0 5 t

40 80 ;20 160 Time Isec)

200 240 280 320

Fig. ill.5. Helium Flow Rate Step (1 lbm/sec).

This behavior led to an examination of the equations and the

program for an explanation of th is dependence. Included in the input

data, GA supplied steady-state values for the helium node boundary

temperature (TG.) but supplied no values for the helium node-averaged

temperatures (TG-). These were calculated in the subroutine by Chang

using whatever weight factors the user chose tc input. This meant that

steady state va'lues of TG.. were being varied as the weights were varied.

This change affected the heat transferred out of the primary side as well

as other t^rms where TG~. appeared in the equation coeff ic ients. So i t

seemed that the variat ion of TG". with weight was responsible for the

large dependence on weight factors.

65

To avoid th is problem, i t was necessary to calculate steady-state

values of TIT. that were not weight-dependent. This was accomplished by

calculating T5\ with the or ig inal 6A weight factors. In th is way,

the steady-state values of TG. and TG. were constant and the weights

could be varied independently. Results using this approach are shown

in Figures I I I . 6 - 9 . As shown, the time responses were no longer very

dependent on weighting factors. This method seemed adequate unt i l the

question was raised that var iat ion of weight factors whileTG- .. and TG. ^ 3 iss iss

were f ixed violated the relat ion i n Equation ( I I I . l ) . To avoid t h i s , i t

i t was decided to calculate TG~. with the or ig inal weights as before

but to recalculate TG. with each new set of weight factors input.

1

4.D

,3.0

= 2.0

1.0

Method 2

0 W3 = 0.43

x W3 = 3.65

40 JO

ORNL- DWG 76- 7788

= 1.62

- 1.55

12) 160 Tins (sec.

200 240 230 320

Fig. I I I . 6. Helium In le t Temperature Per turbat ion (1°F).

66

ORNL OWG 76 7789

• 2 .1

* £ _ l . i . ft 3 O

s * i -4

ttuod 2

O M

Fig. III.7. Helium Inlet Temperature Perturbation (1°F),

5.;.

«.C.

J.;.

.?.

67

ORNL-OWG 76 7791

'tethod 2

Tir» (sec!

F i g . I I I . 9 . Helium Flow Rate Step (1 l b m / s e c ) .

Results using this method are shown in Figures III.10-13. As shown, the

responses are more weight-dependent than those of the second method

(Figures III.6-9) but less weight-dependent than those of the first method

(Figures III.2-5). The third method is actually more closely related to

the first since weight changes lead to changes in various steady-state

parameters in both schemes. As a result, the flaws inherent in the

first method due to steady-state values being adjusted are present in

the third as well. However, from the results, the effect of changing T 6iss 1 S n o t a s 1 m P ° r t a n t a s changing TG.

The previous discussion of weighting schemes illustrates flaws in

all three methods. However, when dealing with a linear model, several

68

OHNt OttG 76 7 792

« ' t I ' * I =* A'

.55

Fig. III.10. Helium Inlet Temperature Perturbation (1°F)

letiiod i

•*, = 0.30

• *. ' 3.6i

ORNL OWG 76 7793

Fig. III.11. Helium Inlet Temperature Perturbation U°F),

1

69

ORNL DWG 76 7794

f.-.ti-Z i

5 , .

« . 4 , ..f.

r E

-V"

3 • - » —t— —•— —f " . J .

/

^ z2&z*=^ r 1 —I— - r : • ' = \ 3 =

4 : ;; \v. I6T ;M :•!"; ?ff! •J?i

Fig. III.12. Helium Flow Rate Step (1 Ibm/sec)

ORNL DWG 76-7795

2.b

w ' 1.30

121 K.T Tir» ( S I T . )

Fig. III.13. Helium Flow Rate Step (1 lbm/sec)

70

approximations and assumptions have been made already. The question

of importance is what additional approximations or assumptions are

implied when weighting factors or any other parameters are adjusted.

For example, the methods given above modify the model in ways

ranging from the amount of heat transferred from the primary side into

the tube metal (method 1) to the helium boundary outlet temperature

(method 3).

The next step in the investigation will be to classify the possible

weighting factor schemes along with the assumptions they imply and the

results they produce. In addition, methods need to be developed for

the variation of the weighting factors on the secondary side independently

from the primary. Examination of the relationship between primary and

secondary sides throi i the heat balance should produce these methods.

If such parameters as pressure and enthalpy on the secondary side could

possibly be weighted independently of each other, another weighting

factor effect could be observed. All of these results have led to a

broadening of the scope of the work in the project to a consideration of

the variation of other parameters besides the weight factors. A more

general sensitivity study will be performed where various parameters

that are fairly arbitrary (like weight factors) or not precisely known

(like heat transfer coefficients) can be examined and the accompanying

assumptions and consequences outlined.

Since two of the techniques discussed for variation of the primary

weight factors involved changing steady-state parameters, possibly

the most accurate way to determine the effects of such variations would

be to recalculate a completely new set of steady-state values for each

71

proposed variation. This method will b? investigated to see if it is

feasible to develop such a capability at UT.

In conclusion, results from the investigation of the effects of

variation of weighting factors on the steam generator response have

been obtained and have led to a broader sensitivity study of the

model. A more general classification of the assumptions made and the

results obtained when parameters are varied is desirable and will be

developed. Assumptions such as the neglect of gas dynamics have been

selected for further verification. These future studies should yield

valuable information about significant parameters and assumptions

used in this steam generator model.

72

CHAPTER IV NONLINEAR, NODAL SYSTEM MODEL

J. G. Thakkar

This section b r i e f l y describes work being done in formulating the

nonlinear system models, for the Fort St. Vrain nuclear steam supply

system. The helium circulator model has been completed and some

improvements were made in the reactor, steam generator and reheater

models.

IV.1 Steam Properties

ASME steam tables were added in the steam generator program using

available function generating capabi l i ty , in CSMP I I I , of the forms

y = f (x) and y = f ( x , z ) . This is a convenient way to calculate

thermodynamic properties of both saturated and superheater steam.

The results were found to be comparable with the previous model where

steam properties were calculated using polynomial f i t s . The comparison

of two cases for the change in main steam outlet temperature ( for 1°T

step change in Helium in le t temperature) is shown in Figure IV .1 .

IV.2 Reactor

The reactor model was reformulated using the prompt jump model '

i . e . , sett ing

dt i i

This gives an algebraic equation for the reactor power, which allows

us to take larger time steps while integrating the heat transfer

equations. For small react iv i ty perturbations (~5£) results from the

1

73

CBNL OWC 76 7796

— Reference Case (Polynomial f i t )

» ASHE Steai Tables

3.U

1.0

0.0

1 1

.. c :;

/

/

/ X * % (

/*

1 1 i 20.0 40.0 SO.O

Tine (sec-wls)

80.0 100.0

•ig. IV.1. 1°F Step Change in Helium Inlet Temperature.

prompt jump model give less than 2:= error compared with the reference

case. For larger reactivity perturbations (~40c) the error increases to

about 2(h. This is probably due tc the comparatively large neutron

life time ("1.85 x 10~ sec) in the HTGR. (The prompt jump model

gives accurate results for shorter neutron life times, i.e., in fast

reactors ' ) .

IV. 3 Reheater The reheater model was updated using new constants and steady

state values from the General Atomic memo (2) This is being tested

74

for perturbations in primary in le t temperature, primary f low, and

steam flow. The preliminary results are shown in Figures IV.2-3.

IV.4 Helium Circulator

Fort St. Vrain has four identical helium c i rcu la to rs , two in each

loop. Each circulator unit consists of an axial flow compressor, a

single stage steam-turbine main dr ive, and a single stage water-turbine

auxi l iary dr ive. The schematic of the system is shown in Figure IV.4.

The c i rculator steam turbine operates on cold reheat steam from

the exhaust of the high pressure turbine. The c i rcu lator turbines are

in series with the main turbine. Steam enters the blower turbine at

about 875 psia and 748°F and leaves i t at 685 psia and 611°F. Steam

from the exhaust of the c i rculator turoine flows to the reheater section

of the steam generator, About 5% of the steam flow bypasses the

c i rculator turbine, at f u l l load, to provide a control margin. The

bypass flow is adjusted to the desired value by maintaining a nearly

constant pressure rat io across the blower turbine.

Helium from the steam generator exi t enters the c i rculator at

about 638°F and 686 psia and leaves the circulator at 647°F and 700

psia and enters the reactor in le t plenum.

The mathematical model of the circulator is based on conservation (3) equations and performance maps for the turbine and the c i rcu la to r . '

The c i rculator dynamic characteristics are evaluated from the

conservation of momentum equation

r

75

•m + * • i i • i i "» i *> i i i * ' i

2HZ 1 1 1 1 1 i • • • I I I 1 | 1 1 I 1

: : i : : : i i i i t i • • t - » • * • • i i • | ( • • * • ! | i

1 • • • • i f I I

| I * » I 1 1 t 1 i * l 1 1 •

| a * 1 t 1 *

1 » i 1 1 1 1

I | 1 * 1 1

i * i i i i i * i i > i i

• • • i i •

r* ~t r i • i i i i tf v m 1 | » 1 1 I

i • i i t i • • • i | i • i i

« • * i i i t 1 - . - 1 i i i i i

i • i t i i

I f i i i 1 l • • i i

! : : : ! i ! •' 1 ; ! ! 1 I 1 • i 1

i i 1 : i i

: ; i ! i ; i i i > i i i i i i I 1

• • " 1 1 1 1 1 1 •» * - r • , i i | | * * * i i i i i i>* ** r* , | | | |

>

-a

O •T.

3) 31 C o a. en o

>

00

mi ^ I .

w 3 L* ^ a o ^ c o w w o o o o ^ c o * ; c o o > * « « * < ^ « « > ^ p I • . » » O

76

2 *

---• i i : : • * - * — i I 1 i i • 2 2 2 i i i t ; ;

i i i i i • i i i i i • • t i l l * i i i i i • I I I I I (

i : I : ; : i l l : ; : t i t • ( • i i i * < • » i i i , * . . * * i t i 1 • • * • • • i i i i i * * * • i i i _ i * • . . _ _ JL • i i i * * i i i i I i • i i

i : i . • : : : i • • • • i

i , 1 .-* i i i i i i i t i i i i • i i i ; * * i i i i i i • i i i i • i • • i i I * I i i • i i i i i i i * ( • • * : : . : : ; i i t * i ( i • i i i • i 1 * 1 i • i i i ( i i i i i i i ( i • » • i . i • i i i » i *

1 1 * t 1 1 1

* tr m i i l i 1 l I * I i i *

• • * i i i i i i - - - i i i , i i

: . * - * : * : : : : i i i i i t i I * I * i t

. t i • i i • i i i i I * I i i i

i S S i i 1 . 1 1 1

; ; : : : ( i ; i . t I 1 ' J ' ' . . . . ' i " ' " " — • - - - - - - • ^ - — - - - ( - . . - . . - . - Y — - - — - ~ i , - i » ! ' • . »

1 * 1 1 • : I • I f f I • • 1 1

r 1 1 f t I 1

» - — » * w * * w » — . • — - . . . , — _ - • . - - . . - - - _ .. — . • — _ . . . . . . , „ » . . . * - » . » - 4 — ~ v -1 I 1 • 1 1 1 1 1 1 t * f I 1

1 1 * • • • • 1 I 1 1 • • • • I 1 | i * i • * • i 1 i i

» • > i r i • • • i i i r i * i i i f i * i i i i i i i « i i

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i oo

i . o c o , ~ e» z" "z. e a „ © o :. o c T e c %i_ J C " " J t o i ' , O C T C O C I - or -~ a •» r " O c * 0 9 0 ~ ? o o u C O l s O w 'J O * - " - - - - - - - - - - - - - - - - - - - - - - - - - - - - -* t > C ^ - < - * * * 0 0

77

Reactor Core

Steam Generator

OHNL-WfG 76-7799

Steam In — * 748° F

875 PSIA

Blower Turbine

647°F 700 PSIA Helium Circulator

Helium In 638° F

686 PSIA

Steam Out 611 °F

685 PSIA

Fig. IV.4. Schematic of Blower Turbine Helium Circulator.

I = mass moment of inertia of the blower turbine rotor shaft w = angular velocity T = Torque exerted by steam T = Torque exerted by helium. T and T r are calculated from energy balances, as follows T t * Pt/w T c - Pc/w P t = power transmitted to the turbine P = power required by the compressor.

t steam hs T

7Q

wsteam = s t e a m f 1 o w r a t e t n r o u 9 n the turbine

"hs = l s e n t r ° P i c steam enthalpy drop through the turbine

iy = turbine ef f ic iency

P c = V * A h H e / r i c

w H = Helium flow rate through the compressor

Ah H = Helium enthalpy rise through the compressor

n = compressor efficiency.

n , i-j., A. , and AP„ are calculated from the compressor and the

turbine performance maps.

The computer program for the circulator model has been completed

and is being debugged and tested for the steam flow perturbation.

Future Work

1. Model the steam turbines

2. Review the steam generator model, to explain the differences with

the linear steam generator model currently used in LAP, particularly

for the primary inlet temperature perturbation.

3. Improve pressure drop and heat transfer correlations in the models.

4. Couple all subsystems of Fort St. Vrain to obtain an overall model

for the plant and test it for different perturbations and

sensitivity to changes in different system parameters.

5. Prepare for correlation of model predictions with trip test results.

1

79

References

1. SaDhier, D., L. W. Kirsch, etc. , "A Reactor Core Model for a Fast

Breeder Power Plant Simulation," 2nd Power Plant Dynamics,

Control and Testing Symposium, Knoxville (1975).

2. Tang, C. K., Fort St. Vrain Linear Analysis Program Documentation

with Data for 100% and 25% Loads, General Atomic Memo SAB:

OCT: 545: 75 (November 1974).

3. Versteegen, P. L., and D. A. Sargis, A Program to Evaluate the

Series Steam Turbine Helium Circulator Performance Under

Dynamic Conditions, Trpical Report SAI-75-562-LJ (April 1975).