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EIR-Bencht Nr 346
Eidg. Institut fur Reaktorforschung Wurenlingen
Schweiz
Rapid Depressurization of a Compressible Fluid: a comparison between usual 1 -D numerical analyses and a 2-D
experiment based on the shallow water analogy
M. Dang, J. F. Dupont, H. Weber
i? Wurenlingen, August 1978
EIR-Bericht Nr. 346
RAPID DEPRESSURIZATION OF A COMPRESSIBLE FLUID:
a comparison between usual 1-D numerical ana
lyses and a 2-D experiment based on the shallow
water analogy
M. Dang, J.F. Dupont, Dr. H. Weber
Wtirenlingen, August 1978
This report refers for work performed within the cooperation of
BBC Aktiengesellschaft Brown Boveri ft Cie, Baden
Brown Boveri ft Cie AG, Mannheim
Eidg. Institut für Reaktorforschung, Würenlingen
Gebrüder Sulzer AG, Winterthur
Hcchtemperatur-Reaktorbau GmbH, Köln
Kernforschungsanlage Jülich GmbH, Jülich
Nuclear-Chemie und Metallurgie GmbH, Wolfgang
Schweizerische Alurti'jiiiin AG, Zc.ich
on the development programme for nuclear power stations
with high temperature reactor and heliumturbine (HHT), which
is sponsored by the Federal Republic of Germany, the state
Nordrhein-Westfalen, and the Swiss Government.
Abstract
The rapid depressurization of a plenum is a situation fre
quently encountered in the dynamical analysis of nuclear gas
cycles of the HHT type.
Various methods of numerical analyses for a 1-dimensional flow
model are examined:
- finite difference method
- control volume method
- method of characteristics
Based on the shallow water analogy to compressible flow, the
numerical results are compared with those from a water table
set up to simulate a standard problem.
CONTENTS
Notation Page
Introduction 1
A. Basic Equations of conpressible fluid flow 6
A.l Conservation laws 6
A.2 Intrinsic constitutive equations 6
A. 3 Extrinsic constitutive equations 7
B. Numerical Methods of Analysis 8
B.l Finite Difference Method (PDM) 8
B.l.l local conservation equations 8
B.1.2 local one-dimensional equations 8
B.l.3 Explicit time derivative equations 9
for p, p and m
B.l,4 Space discretization 10
B.2 Control Volume Method (CVM) 12
B.2.1 Space subdivision into control volumes 12
B.2.2 Conservation equations for con- 13
trol volumes
B.2.3 Standard simplified form of the CVM 13
equations
B.2.4 Remarks 15
B.3 Method of Characteristics (CHM) 16
C. Experimental investigation based on the shallow 18
water analogy to compressible fluid flow
Page
D. Application to a Standard Problem 20
D.l The Standard Problem 20
D.2 Results 21
D.2.1 Shallow Water Analogy 21
D.2.2 Finite Difference Method 24
D.2.3 Control Volume Method 26
D.2.4 Method of Characteristics 26
D.3 Comparison 27
D.3.1 Comparison of the numerical methods 27
FDM, CVM, CHM
D.3.2 Comparison of the 1-D Compressible 27
Flow Model with the 2-D Shallow
Water Flow
D.3.3 Influence of the number of discre- 28
tization zones
E. Conclusions 29
Appendix 31
References 34
Notation
Symbol Description Units
c veloT*. vy of sound £ m/s ]
C- Characteristics
CHN Method of Characteristics
C specific heat at constant pressure [ J/kg/K 3
C specific heat at constant volume £ J/kg/K 2
CVH control Volume Method
D. hydraulic diameter of flow C m 1
FDM I finite difference method with dis
continuous treatment of the area
change
FDM II finite difference method with con-
tinous treatment of the area change
h depth of water [ m ]
HP high Pressure
i enthalpy per unit mass [_ J/kg ~]
v2 r i - i + -r "dynamic" enthalpy per unit mass L J/*9 J
Cw/n2 1
r N/m2 1
C N/m2 j
r w ;i L W/m3 j
j
LP
m
Nu
P V 2
P0=P+(K"1,2
Pr •
Q •
q
heat flux
low Pressure
mass flow
Nusselt number
pressure
Prandtl number
total heat current
heat current density
Symbol Description Units
R Gas constant
Re Reynold number
£ area of cross-section £ • 3
SHU Shallow water
t time L * l T absolute temperature Q if]
u internal energy per unit Mass £J/kg 3
v fluid velocity f a/s ]
Cr
heat transfer coefficient [ w/m /K ]
'P K = — adiabatic exponent
cv
A thermal conductivity £ W/M/K ]
p density [_ Xg/m J
5 friction coefficient
T friction tensor CN/,n J
- 1 -
Introduction
The analysis of rapid pressure transients within a gas tur
bine circuit has become a major safety concern for gas cooled
nuclear power plants using a gas turbine in a direct conver
sion cycle (HHT project in Europe ref /5/ and GT-HTGR project
in U.S. ref /6/ and /7/). The reason for this is that hypo
thetical accidents such as the total deblading of turboma-
chines Must be taken into account as design basis accidents
(Fig.l). Such accidents induce a very fast pressure equali
sation between the high (70 bar) and low (24 bar) part of
the gas turbine cycle (Fig.2).
Various computer codes have been developed which solve the
1-D flow dynamics equations with numerical schemes based
either on "finite difference" (ref /8/, /9/) or on "control
volume"(ref /10/, /ll/) techniques. The main problem is to
simulate the dynamical behaviour of the whole plant without
excessive computing time. The code PLAYGAS developed at EIR
was found to be a good compromise to meet the above contra
dictory requirements.
To assess the validity of computer models several available
dynamical measurements from fossile fired power plants with
closed air turbine cycles have been checked against numerical
results. The examined transients varied from "load following"
and "loss of load" operations (ref /ll/) up to an emergency
shut-down" with rapid air relief over exhaust valves (ref /10/).
For this last case mass flow inversions and depressurization
rates of up to 50 bar/s could be satisfactorily simulated. In
this way validity of the usual numerical models can be con
sidered as demonstrated for the above category of transients.
However, the accidents postulated for luclear gas turbine
cycles involve much higher depressurization rates in the high
pressure part of the circuit, about 200 bar/s in the large core
- 2 -
inlet or outlet plenum and even up to 2000 bar/s in some ducts
(Fig 2). The question now arises of the validity of the usual
numerical models for these extreme transients, i.e. the vali
dity of
• one-dimensional models
• "control volume" versus "finite difference"
methods
• large space discretization
• neglecting pressure discontinuities (shock
waves)
With this report it is intended to answer some of the fore
going questions by presenting:
• an overview of the usual 1-D numerical treat
ments of the gas flow dynamics
• a comparison between 1-D numerical computa
tional results and 2-D measurements obtained
with a shallow water analogy experiment for
the case of the rapid depressurization of a
large volume through a long pipe (standard
problem)
The "shock wave" has not been examined here since the stan
dard problem involves only depressuri*..tion transients and a
rarefaction wave can never lead to a shock wave. Consequent
ly some conclusions drawn here apply only to the high pres
sure part of the circuit. The main conclusions are the fol
lowing :
• no basic physical difference exists between
"control volume" and "finite difference"
methods. They even produce identical results
when the numerical schemes described here
are used and provided that the same space
- 3 -
zoning is applied to both Methods.
the number of discretisation zones can
be varied in a large range for the stan
dard problem without significantly affec
ting the results.
The 1-D numerical treatment is in a good
agreement with the 2-D experiment, par
ticularly for the pressure behaviour
within a large v*. li
Usual 1-0 flow dynamics computer models, such as PIAYGAS, can
be considered as a reliable tool for the analysis of very ra
pid transients within closed gas turbine cycles, at least for
the high pressure part of the circuit. The validity of these
models for the low pressure part of the circuit, when shock
waves are involved, should be further examined.
Zntnreoolar
Thexiwl Sh ie ld
E l«c . H»-Turbln« HP Oaqpcwaaor LF O a v m n r
FIG. l SCWMKTIC VIEW OF THE HELIUM FLOW PATH IN THE HOT OCNOVB PRESSURE VESSEL.
HYPOTHETICAL ACCIDENTS J TURBCMACHINERY LOSS OF BLADES OR MAfSXV. FAILURE
OF A PBCUPERATOR SUPPORT PLATE.
- 5 -
BO
70
60 -
50 -
40 -
30 -
20 0.2 0.4 0.6
Time (s) 0.8
2 -
± I 0.2
LP-Recuperator Turbine
Reactor Precooler
i.
HP-Recuperator LP-Compressor
— Intercooler HP-Compressor
J I 1 I I 0.6
Fig. 2
0.4 Time (s)
DEBLAD1WG OF THE HELIUM TURBINE: TIME
BEHAVIOUR OF SOME PRESSURES AND MASS
FLOW RATES.
0.8
- 6
A. Basic Equations of Compressible Fluid Flow
A.l. Conservation Laws:
The unsteady motion of a three dimensional compressible fluid
is governed by three conservation laws: mass - energy - and
momentum-Conservation. The integral form of the three con
servation equations applied to a fixed volume V with boundary
3V can be written as:
lp/7/pdV = - //pv»do A-l
V 3V
§£///(pu+p|-)dV = -//p(i+|-)v-do+Q A-2
V 3V
|t///pvdV = -//pv(v«da)-//pdo+//T«da
V 3V 3V 3V A-3
This system of equations is completed by the following two
types of supplementary equations:
A.2 Intrinsic Constitutive Equations:
The thermal and caloric equations of state for an ideal fluid
with constant specific heat are:
p - pRT A-4
u = CvT A-5
and the following thermodynamic relationships apply:
1 - u + £ - c_T A-6 P P
- 7 -
C = C + R A-7 P v
£ - « A-8 V
where n. is the gas constant of the fluid, C and C the speci
fic heats and * the adiabatic exponent.
A.3 Extrinsic Constitutive Equations;
For three-dimensional flow, the total heat current is given by
the well known Fourier's law:
Q - / / 8 V J • da = - // 3 VA |S dok k = 1, 2, 3
k
\ denoting the thermal conductivity of the fluid. This ex
pression is, however, nnsuitablr for one-dimensional treatments
and must be replaced by the empirical equation:
Q » // aAT | da | A-9
with a = 2H-2. , Nu = Nu (Re, Pr)
where a denotes the empirical heat transfer coefficient, and
D. the hydraulic diameter of flow.
On the other hand, the friction-tensor T for a Newtonian fluid
is given by:
dv. ?V. 3v, T i k - a < ^ + j ^ + b irt ' 6ik
- 8 -
for three-dimensional flow:
For one dimensional flows, friction is considered by the addition
of a pressure drop term of the form
«Pfr (x) = n *te& ax A-IO
to the pressure.
B. Numerical Methods of Analysis
B.l Finite Difference Method (FDM)
B.l.l Local Conservation Equations
A local form of the basic equations is obtained by applying the
theorem of Stokes to the surface integrals:
3p ,. -*• B-l ^ = - div pv
+2 *2 |^ (P (u + |-)) = - div (pv (i + |-)) - div 3 B-2
9 - (pv) - - div • (v p v) - grad p + div T B-3
B.l.2 Local one-('imensional equations
In a one dimensional coordinate system with voliw> element
dv = S (x) dx, the expressions for div and grad are:
div - > sTxT k (s(*> >
*rad ~> k.
- 9 -
Equations B-l to B-3 then become
H - - sW h «"s ««> ' "-*
It (»(» + 5-)) --ST50 fc «»vS(i4|-) ) + 6 B-5
I t « » » » - - s W S i l o v S - v ) - I f - f ^ ° i v i v - B-6
For the heat-flow term -divj, the 1-dimensional expression
A-9 in the local form with the thermal power per unit volume
u S(x) 3x
must be used. For the frictional forces the expression A-lO has
been inserted.
B.1.3 Explicit *--tne derivatives equations for time derivatives
p, po an m:
2 If the variables p, m = pvS(x) and p = p + (x-1) %r- are
O 2.
chosen as differential variables, the partial equations B-4 to
B-6 can be written as:
i£ - I 12! n 7 at " S 3x
^ - - ^ lx-'" < i +T" * «-»« B-8
If - - k «->-s If - 2Jri»i- "-'
- 10 -
The corresponding transformation of the equations, together
with the algebraic expressions for i, v and p are s.»own in
the Appendix.
B.1.4 Space Discretization
The finite difference method consists to replace the derivatives
in the local conservation equations by finite differences.
The space under consideration is first divided into discrete
zones and the physical variables are then determined at each
node. For one-dimensional flows in channels, axial zones of
length Ax are considered as shown below in fig. 3.
k-1 k+1 Zone k
Fig. 3 Space Discretization (FDM)
- 11 -
The equations B-7 and B-8 are now applied at the zone centres
to determine the dynamic pressures p . and densities p. by
substitution of the gradient terms on the right hand side of
the equations by finite differences of the corresponding values
at the zone boundaries:
pk I k k-1 _ * . _ i * *_± B-10 dt SR Ax
dP„ L. it.n ("v^ v_mt_i *„ i,_i > o,k _ _ (*-l) '"Vo,k "k-1 o,k-l' ,K ,. • ,,
~aT sk AX + ( 1 , qk B n
Similarly, the equation B.9 is applied at the zones boundaries
to determine the mass flows m. by substitution of the gradient
terms by the finite differences of the corresponding values at
the zone centers:
!mk (Pk+ i s k + i " pk Sk ) ( mk+ i v k + r m k V dt AX AX
B-12
A - Sk+l~Sk , C v - * i
+ Pk —Ax (2D^ 'mkfVk
It should be emphasized that the discretization scheme adopted
in deriving equations B-10, B-11 and B-12 was chosen in preference
to many other possibilities according to the following physical
requirements:
a pressure or momentum disturbance is propagated
either to the right or to the left of the flow direction,
a thermal disturbance is, however, only propagated
downstream.
- 12 -
A detailed derivation of the finite differences equations B-10,
B-ll and B-12 is given in the Appendix.
B.2 Control Volume Method (CVM)
B.2.1 Space Subdivision into Control Volumes
The flow channel is divided into 2 types of control volumes as
shown below in fig. 4.
Control volumes with
mass and energy con
servation
Control volumes with
momentum conservation
Fig. 4 Space Discretization (CVM)
- 13 -
B.2.2 Conservation Equations for Control Volumes
The integral equations of conservation A-l, A-2, A-3 applied to
the 2 types of control volumes considered in fig. 2 give:
ft "'k pdv " " (ik " "k-l>
. 2 dt ;//k P(»*T)«V = -(Vo,k-Viio,k-i) + ̂ k
^ ///pvdv = -(Pk+1sk+1-pksk) + pk (sk+1- sk)
I • I ~
- (mk+i vk+i • rak V - ^ f e r W ^ k n
where the indexed variables of the right hand sides are local
variables at the boundaries of the control volumes.
The following approximations are now introduced:
fffk pdV = Pk Sk Ax
v2 1 v2
/;/pk(u+^-)dv = ;— /;/kp dv + ///k P|- dv
, 2
" Z=I pk sk Ax + / ; / R P T dv
= r p . S, AX >c-l *o,k k
///k pvdV = mk Ax
- 14 -
Thus:
dp. , (m -m ) 31* " " ̂ A B"13
dt S. Ax
dfoik= . (^ ^ o ^ - ^ - ^ o , ^ + - B_14
dt S. Ax /4k
^ k = _ (Pk^l Sk+1 ' Pk Sk) . (mk+l Vk+1 " mkVk)
dt Ax Ax
A ~ (Sk-H ~ Sk} , s, , ,. ,
+ Pk A^ (25T> !mk'Vk k
where p. and (-r*-.) correspond to values averaged over the h k
control volumes k.
B.2.3 Standard Simplified Form of the CVM-Equations
For practical reasons which will be presented later, the following
simplifying hypotheses are usually made:
a in the control volumes k (boundaries included), the v2
kinetic energy per unit mass y- i s negligible compared
to the specific internal energy u = c T
2 2 Y_ << c T -=> ^— << c T ="> i . • i. * c T. 2 v 2 p o,k - k p k
2 v —— << c T =*•*=> P i * P, 2 v Ko,k - *K in the control volumes k, the cross section is constant
and the contribution of the mass flow gradient to the
- 15
momemtum balance (equation A-25) is negligible in
comparison to that of the pressure gradient
Sk+1 - Sk
i"2k+r"<U) K< (pk+i sk+rpk sk )
Pk+i sk+ i A x fix
(Pk-,i s k , i ~Pk V A ( n w v k + i - mk V
fiX AX
•2 (1 - L.) Sk (pktfPk' + \ pk+l pk
Ax S, Ax k
Thus we obtain the standard simplified form of the CVM
equations:
• 9
dp̂ ̂ Vvi ) dt vk
d t - = \ — + ( x - 1 , ( * k B " 1 7
,1 1_. ^ k ( P k + r P k ) S k m2 p k + 1
pk ,5 , •• ,
dF Tx sT ——* (2Dr> 'mk' \ B'18
k h k
B.2.4 Remarks
It should be noted that the CVM equations B-13, B-14, B-15
are identical to the Fl-'/ equations B-10, B-ll, B-12 i.e.
the control volume method and the finite difference method
result in the same final set of equations provided that
the discretization schemes are comparable. Thus, there are
basically no fundamental differences between the 2 numerical
methods.
- 16 -
b The simplifying hypothesis of constant cross sectional
axea was only applied to the control volumes
The area changes do not appear explicitly in the
equations B-16 and B-17 relative to the control
volumes . Hence the standard simplified form B-16,
B-17, B-18 of the CVM equations is very practical,
particularly for overall plant simulation, as it
constitutes a convenient way to deal with the diffi
cult handling of area changes: the flow channel is
discretisized in such a way that area changes only
take place in the control volumes
In order to distinguish between the 2 kinds of control
volumes, the flow channel is usually represented in
simulation scheme by a succession of plena (corre
sponding to the control volumes ) and pipes
(corresponding to the control volumes ) as shown in
fig. 5.
B.3 Method of Characteristics (CHM)
The Method of Characteristics uses the fact that the basic
partial differential equationsB-4 to B-6 are of hyperbolic cha
racter. This means that there exist lines in the t-x-plane,
the so called characteristics, where the dynamic variables of
a flow problem cannot be arbitrarily chosen, but are subject
to characteristic relations. The characteristic relations be
come surprisingly simple for an isentropic flow in a constant
cross-section. It is convenient to chose the velocity of sound
c as a thermodynamic variable, pressure and density are then
given by:
- 17 -
+ *k~l p k - l
-4
"Vl Pk
I
*k+l p k * l
+
*k+l
I
plenum Pipe
F i g . 5
- 18 -
2K
° co
2 r-1
° co
The differential equations of the characteristics C in the
t-x plane are
C : §r = v + c B-20 dt —
and the characteristic relations are:
i 2 * J - v ± —^r c * constant an C B-21
+ The constancy of J , the so called Riemann-Invariants, allows
an explicit construction of the flow-field in the t-x-plane,
starting from the initial and boundary-conditions. The purely
geometrical construction of the characteristics in the x-t and
v-c planes is discussed in many textbooks on compressible fluid-
dynamics, e.g. in (1). The same method can also be put into
numerical form and results in very simple and fast computer
programs; but this is not commonly done. A detailed description
of the corresponding numerical method can be found in (2).
C. Experimental Investigation by the Shallow Water Analogy (SHW)
The local equations of motion B-l and B-3 for an isentropic flow
of e compressible fluid in two dimensions can be written as:
- 19 -
|| = - div (pv) C-1
a — (pv) = - div • (v p v) - grad p C-2
On the other hand, the flow of an ideal incompressible fluid
under the influence of gravity on a horizontal plane in two
dimensions can be described by the same set of equations, when
the following conditions are satisfied:
a) The depth of the fluid h must be small compared to
the wave length of the phenomena studied.
b) The vertical velocity must be small compared with
its horizontal component.
c) The vertical acceleration must be small compared with
gravity g.
Let x denote the two-dimensional position and h (x, t) the
time-dependent local depth of the fluid. The equations of
motion for the fluid can now be written in the form:
|| = - div (hv) C-3
l ^ L = - div (vhv) - V (f) C-4
Equations C-1, C-2 and C-3, C-4 are the same when the following
substitution is made:
p • a - h C-5
- 20 -
p = 2* h2 C-6
a is an arbitrary constant. From C-5 and C-6, the following
equation of state can be obtained by elimination of h:
p = k ' p2 - p 0 * {7~)K' K=2 c_7
o
This is exactly the equation of state for an ideal gas with
* = 2, e.g. the shallow water analogy models the motion of an
ideal gas with * = 2. A more detailed description of the ana
logy can be found in (3).
D. Application to a Standard Problem
D.l The Standard Problem
A standard problem with a simple configuration (see below fig. 6)
is defined in order to allow a direct comparison between measure
ments from the shallow water te.ble (see section 2) and numerical
results obtained with the different methods already presented in
part A.
p = 20 m2
p = 60 bar
V • 200 m3
10 m
S - 1 m2
15 m
Fig. 6 Geometry of the Standard Problem
- 21 -
The geometry of the standard problem has been chosen according
to 2 requirements:
the ratio of the volumes of the reservoir to the exit
area is representative of that corresponding to the
HHT-reactor inlet plenum,
the configuration is simple enough such that the water
table could be constructed.
The type of transients studied is the rapid depressurization of
the system initially filled with a stagnant ideal gas at a
uniform pressure of 60 bar. The time dependent pressure at the
exit is an input function given by measurements from the water
table (see fig. 9, measurement position nr. 1) .
D.2 Results
D.2.1 Shallow Water Analogy (SHW)
The analogy between shallow water flows and compressible flows
provides a convenient way to study experimentally the defined
standard problem. A water table was constructed according to
the geometry of the standard problem scaled down to 1 : 10
(see fig. 7).
The depressurization takes place when the slider is suddenly
opened, t'ms connecting the high pressure section (initially
at 60 bar) to the low pressure section (initially at 30 b&r).
The pressures are measured at 6 positions as shown in fig. 7.
- 22 -
p = 60 bar
1 m
Pressure^ measurement
• * •
. jp = 30 bar'
I
^ Slider.
LJ I
1.5 m
Fig. 7 Geometry of the Water Table
- 23 -
The following fig. 8 shows the water table with its measure
ment equipment. A detailed description may be found in (4).
Fig. 8 The Water Table and the Measurement Equipment
The time dependent evolution of the pressures measured at the
6 positions is presented in fig. 9. The pressure at position 1
decreases from 60 bar to 35 bar in 4 ms. The expansion wave
runs along the pipe (position 2 and 3) toward the plenum and is
considerably damped out by the change in cross-sectional areas
between position 3 and 4. The pressure distribution in the
plenum remains astonishingly uniform* only slight differences
- 23 a -
Fig. 9 Experimental Results from the Shallow Water Analogy (SHW)
IK,
n,
ss.
s§.
45.
4f.
35.
m.
—
—
—
\
\ 1
1
POSITION M 1
1 1
\JT" Y i i
i
•—->
i
—
—
—
—
• ^ ^
•
55. _
S Sfl.
* m.
35. _
POSITION NR 2
T 1 1 r
J I I L
TINE IN MLLISECOND
•- 20. 4f. S». TIIC IN niLLISECOO
•. 2i. 4*. 60. THE IN niLLISECONO
I . 20. 4f.
TINE IN maiSECOND
0. 20. 4f. 60.
TIIC It. IflLUSECONO
65.
{ 55.
50. | _
45.
40.
•
—
—
1
1
POSITION NR 6
1 1
1 1
1
1
-
—
.
-2f. 0. 20. 40.
TINE IN MILLISECOND
- 24 -
between pressures at positions 4,5 and 6 are observed.
D.2.2 Finite Difference Method (FDM)
The finite difference equations B-10, B-11 and B-12 are derived
without any explicit hypothesis concerning the flow area. It is
however implicitly assumed in the 1-dimensional model that the
fractional rate of flow area change with respect to the flow 1 A ^
distance is small (- — << 1). This is not true for the sharp o AX
edged area change in the standard problem.
In order to resolve the difficulty, the discontinuous area change
can be treated as
the coupling of 2 pipes with different cross-sectional
areas,
or by approximating the flow area change by a continous
function S(x).
The standard system is divided into 2 distinct pipes with 2 2
cross sections S = 20 m and S = 1 m respectively and
the standard problem is solved by applying the equations
B-10, B-11 and B-12 to both. The following ideal con
ditions are used at the junction of the 2 pipes:
m.. = m_
2 2
(h + f - ) ^ (h + | - ) 2 D-l
'l = S2
- 24 a -
Fig . 10 Numerical Results from the F in i t e Difference Method I
1
,.
1.
1.
1
1
POSITION NR I
1 1 i
i
•
• i
• i
• i
w^
I s
POSITION NR 2
T 1 1 r
J I I L -20. 0. 20. 40. 00. 80. -20. 0. 20. 40.
TIIC IN mLLISECONO TIRE IN RILLISECOND
Q SHU O FDR I
POSITION NR 3 _ POSITION NR 4
«0 1
55.
50.
45.
40.
35.
90.
1
"
1
1
1
1
1
1
1
•
•
^
^
_-
^
•
-20. 0. 20. 40.
TIRE IN niLLISECOND
-20. 0. 20. 40. 00.
Tift IN niLLISECOND
• SHU O FOR I
m.
99»
50.
45.
40.
»
POSITION NR 5
1 1 1
1 1 1
1
1
•
—
-
' -20. f . 20, 40. 00.
TIRE IN mUISECOND
m.
m i
55«
50.
45.
40. •
1
. I
PD5ITII
1
•
3NNR6
1
1
1
] . .
•
a
—
—
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TIRE IN niLLISECOND
- 25 -
The results obtained with the FDMl method are shown in fig. 10.
b 92D£iDH2H§_**f:5_9!*?59S-IE9!L*Il.
The sharp edged area change of the standard system is
approximated by a continuous linear function S (x) as
shown in fig. 11.
21 22
0.2 m
Fig. 11
- 26 -
A very fire discretization (10 in our calculation) of the part
of the plenum where the flow area is continuously varying is
therefore required in order to satisfy the condition
— — << 1. No coupling conditions are necessary as the system is
considered as a whole to which the equations B-10, B-11 and B-12
are applicable.
The results of the FDNII method are presented in fig. 12.
D.2.3 Control Volume Method (CVM)
The standard simplified form B-16, B-17 and B-18 of the control
volume equations are used. As previously mentioned, no special
care has to be made for the treatment of the area change if
the system were discretized in such a way that it occurs in a
control volume where the equations B-16 and B-17 are valid.
The number of control volumes considered is equal to the number
of axial zones used in the FDMI calculation.
Fig. 13 shows the results in comparison with the measurements
from the water table.
D-2.4 The Method of Characteristics
The method of characteristics is applied to the two cylindrical
parts of the standard problem. These two parts have been coupled
by use of the junction conditions D-l. The details of the com
plete numerical algarithm have been described in (2). Fig. 14
shows the results of these calculations.
- 26 a -
Fig. 12 Numerical Results from the Finite Difference Method II
as.
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- 26 b -
Fig. 13 Numerical Results from the Control Volumes Method
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- 26 c -
Fig. 14 Numerical Results from the Method of Characteristics
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D. 3 Comparison
D.3.1 Comparison of the numerical methods FDH, CVK, CHM
The FDMI. CVM and CHM results are shown in fig. IS. The PDMII
results which correspond almost exactly to the IDMI results
are not presented.
At the positions 4, 5 and 6 in the plenum, only slight differen
ces are observed whereas at positions 2 and 3 in the pipe, where
the fluid velocity is much greater, the differences are quite
noticeable. These differences were, however to be expected since:
the CHM method of characteristics makes use of an
isentropic flow model,
the CVM control volumes method (which results in the
same final set of equations as the PDM method as previously
mentioned in part B, section 2.4) was used in our calcu
lations in its so called standard simplified form (see
part B, section 2.3) where the fluid kinetic energy and
the flow gradient were neglected in both the energy and
in the momentum balance respectively.
According to these additional simplifications introduced in the
CHM method and in the CVM method, the PDM should present the
best agreement with the measurements from the water table. This
is demonstrated by a comparison of the figures 10, 12, 13 "ind 14.
D.3.2 Comparison of the 1-D Compressible Plow Model with the
2-D Shallow Water Plow Model
The results obtained with the PDM method based on a 1-D compres
sible flow model have already been compared, in figures 10 and
12, with the measurements from the water table.
- 27 a -
Fig. 15 Comparsion of the results obtained from three different numerical methods with the same number of zones N = 25
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- 28 -
A very good agreement between the numerical results and measure
ments was observed in the plenum thereby, confirming the validity
of the one-dimensional flow model for studies of rapid depressuri
zations of a reservoir.
However, in the vicinity of the sharp eged area change, the
2-dimensional effects (particularly the flow area contraction)
were very acute and consequently resulted in appreciable differen
ces at positions 2, 3 and 4. Some of these 2-dimensional effects
(e.g. flow area contraction) could have been taken into account
in the 1-dimensional flow model using some refinements. Thus the
agreement between numerical results and measurements could be im
proved, as shown in fig. 16, by introducing an empirical flow
area contraction ratio.
D.3.3 Influence of the number of discretization zones
With the FDM and CVM methods, a reduction in the number of dis
cretization zones is desirable since it leads to a profitable
reduction in the computer running time. However, the influence
on the quality of the results has to be examined. For this pur
pose, 2 CVM calculations with 25 and 3 zones respectively as
well as 2 FDM calculations with 25 and 5 zones respectively have
been run. The comparison of these results is presented in
fig. 17.
It is evident that informal-ion is less detailed if the number
of discretization zones is reduced. In the 3-zone CVM run,
only pressures at position 2 in the pipe and position 5 at the
middle of the plenum are computed. Nevertheless the pressure com
puted at position 5 with fewer discretization zones (5 for FDM
and 3 for CVM) agrees favourably with that computed using 25
discretization zones for both FDM and CVM.
- 28 a -
Fig. 16 Numerical Results from the Finite Difference Method II with reduced cross-section of junction
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Fig. 17 Influence of the number of zones:
Fig. 17a: FDM I
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- 29 -
The depressurization wave is somewhat damped out at position nr. 2
if fewer discretization zones are used. This result is to be ex
pected according to the FDM equation B-17 and to the greater
volume of fluid involved in a discretization zone.
E. Conclusions
E.l The 1-dimensional flow model
For the types of transients studied (depressurization of a plenum)
the 1-dimensional flow model seems to be quite satisfactory as
far as the depressurization of the plenum is concerned.
In the pipe, where 2-dimensional effects lead to appreciable
differences between numerical results and measurements, the 1-
dimensional flow model could be improved by introducing empiri
cal requirements.
E.2 The different numerical methods
As previously emphasized in part B, section 2.4, no fundamental
differences occur between the FDM and CVM methods which would
result in the same final set of equations if the same discreti
zation schemes were chosen.
A very good agreement between results obtained with the FDM,
CVM and CHM methods concerning the pressures in the plenum was
observed. However, differences did occur in the pipe where the
fluid velocity was greater due to the additional simplifications
introduced in the CHM method as well as in the standard simpli
fied form of the CVM method.
Inspite of the isentropic flow hypothesis, the CHM method proved
to be very interesting due to its low computer costs.
- 30 -
The standard simplified form of the CVM method is particularly
convenient for an overall plant simulation code as it avoids
the difficulty of changes in flow area and is suitable for mo
dular programming techniques.
E.3 The number of discretization zones
A reduction in the number of discretization zones does not affect
sensitively the results concerning the depressurization in the
plenum. However, in the exit pipe, the amplitude of the pressure
oscillations is considerably damped out.
- 31 -
The t r ans fo rma t ion of e q u a t i o n s B-4 t o B-6 i n t o B-7 t o B-9 _ • p v
by introduction of t he v a r i a b l e s p , m = pvS(x) and p =p+(K-l) *-=—
i s as fo l lows :
From A-4, A-5, A-7 and A-8 u can be expressed as:
(1)
pu = —fy and therefore It *•* J.
•«*r> - A <p*«-" T > - £ r
B-8 is obtained from the multiplication of B-5 with *-l and the
substitution of the new variables p and m
mation of equations B-4 and B-6 is obvious
substitution of the new variables p and m = ovS. The transfor-
The finite difference equations B-10 to B-12 are obtained by
application of the discretization method illustrated in Fig. 1
to the partial differential equations B-7 to B-9. Since the dy
namical variables p, p and m are defined at different locations
in the discretization net, interpolation must be used in order
to calculate the other dependent variables: i . , p. (mv).
and p..
From A-4, A-5 and A-6
. ^ 1 2 K p 1 2 (2) 1 = 1 + T V = =- C + - V o 2 (c-1 p 2
is obtained. With help of (1) can be expressed in terms of p , p
and m :
2
1 - K=r-p0/P + (1"K) y b r 2 o)
- 32 -
p p and S are now interpolated at the junctions by
— Po,k+l * po tk - _ pk+l+pk ^- _ Sk+1 + Sk po,k = 2 ' pk " 2 ' 5k " 2
and i . is found by inserting these mean values into
— '2 K F£Ji + a-*) JS_ 2. (4)
o,k <-l 7^ 2 ( 5k Sk )
p can be expressed in p , P and m by solving (1)
2 j 2pS" P " Pn " <*-D 7^2 (5)
p is a zone-variable, p and p are already defined in the zone-
centers, m at the zone-centers must be interpolated by
(m) k = K
2 (6)
By introducing (6) into (5), p is found:
•2
Pk = Po,k " «'»' I^sf (7)
(mv). is defined as the value of mv at the zone centers. From k »2
mv = —- , (mv). can be found as: pS K
(mv). = — ~ - (8) k pk Sk
Two further junction variables are needed in B-12: p and (mv). .
- 33 -
These are directly formed as ari thine trie mem of the adjacent
zone variables p. and (m v).:
Pk + pk 1 — <mv\, + tmv)
pk= j - ' (mv)k — - Inv),. = S__̂ ^-± (9)
Finally (r̂ r-) is the local value of the expression *r- at the 2Dh k 2Dh
junction k and q. is the mean of the heat-input per unit volume
and time at the zone k.
- 34 -
References
(1) G. Rudinger,
Wave Diagrams for nonsteady Flow in Ducts,
D. Van Nostrand Company, New York, 1955
(2) H. Weber,
Anwendung der Charakteristiken-Hethode fur die Kreislauf-
dynamik,
Swiss Federal Institute for Reactor Research, Wiirenlingen,
Switzerland,
Report TM-ST-392, 1976
(3) P.A. Thompson,
Compressible-Fluid Dynamics,
McGraw-Hill, New York, 1972
Chapter 11
(4) H. Weber,
StrSmungsversuche mit dem Wassermodell:
Beschreibung der Versuchseinrichtung,
Swiss Federal Institute for Reactor Research,
WUrenlingen, Switzerland
Report TM-ST-421, 1976
(5) HHT-Project
Referenzanlage, Phase IB
Report HHT-27, HRB-BA1671
October 1977
(6) C.F. McDonald,
Large Closed-Cycle Gas Turbine Plant
GA-A 14311
General Atomic company,
May 1977
- 35 -
(7) Gas turbine HTGR
A technology assessment
Report NUS-3041, NUS Corporation Maryland
October 1977
(8) A. Tiberini
Das dynamische Verhalten eines gasgekühlten
schnellen Brutreaktors mit direktem Gasturbinen
kreislauf, dargestellt am Beispiel des Rohrbruch-
Unfalles in einer 1000 MWe-Anlage mit Rekuperator.
EIR-Bericht Nr. 263 und Dissertation ETH-Zürich Nr. 5329
Juni 1974
(9) M. Waloch
Anlagenstörfalldynamik von Hochtemperaturreaktoren
mit direktem Gasturbinenkreislauf
Institut für Dampf- u. Gasturbinen
RWTH - Aachen, 1977
(10) J.F. Dupont, R. Jeanmonod, H.U. Frutschi
TUGSIM-10, A Computer Code for Transient Analysis
of Closed Gas Turbine Cycles and Specific Applications
Nuclear Engineering and Design 40 (1977) p.421-430.
(11) G. Krey
Das dynamische Verhalten von einwelligen geschlossenen
Dampfturbinen,
Institut für Strömungsmaschinen, Technische Universität
Hannover,
Bericht 288/74, 1974.
- 36 -
(12) P. Quell,
Die Bedeutung von Bruchstorfallen bei Kernkraftwerken mit
Hochtemperaturreaktor und Heliumturbinenkreislauf in
nicht-integrierter Bauweise
KFA-Jiilich, Institut fur Reaktorentwicklung
Bericht KFA Nr. 1244 und Dissertation RWTH-Aachen
October 1975
(13) J. Dean, G. Preinreich, G. Soubelet,
Control and Safety Studies for an HTR Gas Turbine System.
International Conference on Nuclear Gas Turbines
BNES, London (1970), Paper No. 10
(14) J.F. Dupont, G. Cina, M. Dang,
PLAYGAS, a computer Code for the Transient Analysis
of Nuclear Gas Turbine Power Plants
First European Nuclear Conference, April 1975, Paris
and EIR-Bericht Nr. 284.
