On the hydrodynamic behaviour of parallel boiling water
channels
Citation for published version (APA):
Vonderen, van, A. C. M. (1971). On the hydrodynamic behaviour of parallel boiling water channels. Technische
Hogeschool Eindhoven. https://doi.org/10.6100/IR70309
DOI:
10.6100/IR70309
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Published: 01/01/1971
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ON THE HYDRODYNAMIC
BEHAVIOUR OF PARALLEL
BOILING WA
TER CHANNELS
A.C.
M. VAN VONDEREN
Laboratory for Heat Transfer and
Reactor
Engineering
Eindhoven University of Technology
- The
Netherlands
ON THE HYDRODYNAMIC BEHAVIOUR OF PARALLEL BOILING WATER CHANNELS
ON THE HYDRODYNAMIC
BEHAVIOUR OF PARALLEL
BOILING WATER CHANNELS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. IR. A.A.TH.M. VAN TRIER, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP DINSDAG 23 MAART 1971 DES NAMIDDAGS TE 4 UUR.
DOOR
ANTONIUS CHRISTIANUS MARIA
VAN VONDEREN
GEBOREN TE BEUGEN c.a.
Dit proefschrift is goedgekeurd door de promotor PROF. DR. M. BOGAARDT
Catah Wo:rds
instabili ties parallel channels non-linear model
Contents
Nomenclature Abstract 1. Introduction
1. 1. Previous work
1.1.1. Single channel models 1.1.2. Parallel channel models 1.2. Experimental work
1.3. Aim and scope of this study
9 13 15
2. Description of the apparatus 24
2.1. Experimental set-up 2. 1 . 1 . The loop
2.1.2. The test section 2.1.3. The power supply
2.2. Measuring- and recording equipment 2.2.1. Temperature
2.2.2. Mass flow rate 2.2.3. Pressure 2.2.4. Void fraction 2.3. Analysing equipment
3. Experimental results - Steady-state data 3. 1. 1. Introduction
3. 1.2. The inlet velocity 3.1.3. The void fraction 3.2. The slip ratio
3.2. 1. Slip correlations 3.2.2. Present slip data
4.
Experimental results - Dynamic data4.1. Threshold power, frequency and mode of oscillation 4.2. Further analysis and discussion
5, Theoretical study 5.1. The correlations
5.1.1. Correlation for two-phase friction 5.1.2. The slip ratio
5.1.3. Subcooled boiling and heat distribution parameter
39
5.2. The mathematics
5.2.1. The conservation equations 5.2.2. Boundary conditions 5.2.3. Solution technique
5.3. Model's results and comparison with experiment 5.3.1. Steady-state inlet velocity and void fraction 5.3.2. Dynamic calculations
5.3.3. Prediction with a single channel model
5.4.
Selected items5.4.
1. Terms of momentum equation during instability 5.4.2. Residence time of a water particle5,4.3. Void-flow feedback mechanism
6.
Conclusions Acknowledgements List of References List of Figures List of Tables 119 122 123 129 131 Appendix A Derivation of the two-phase conservation equations 132 Appendix B Experimental data for steady-state condition 138Nomenclature a Atot A . rJ Ao'"."" A12 b0 , b 1, b2 c
c
c~c
2 d, D DF e E. de) f f1, f2 F g G G Gs hw h hs hsat .w 1 j js jw k , kc k. k~n in ktot K kB ki, F j k a constant in eq. 3.35. amplitude of V. . complex notatihg J for defined by eq. 4.3. amplitude ~f Vtot cross section, areachannel j appendix A 5. 13. cross section of terms defined in constants in eq. specific heat constant in eq. 3.32.
constants in eq. 5.9. and 5.10. diameter
constant in eq. 4.9. arbitrary variable
error, defined by eq. 5.37. arbitrary function of e friction factor
distribution functions of a and j, eq. 3.31.
friction term in momentum equation acceleration due to gravity total mass flow rate
=
G +G mass flow rate of steam s wphase mass flow rate of water phase heat transfer coefficient specific enthalpy of steamspecific enthalpy of water at Tsat specific enthalEy of water
complex unit; i
=
-1total volumetrie flux density
=
j + jv5lume~ric flux density of steam
phase
volumetrie flux density of water phase
constants in eq. 2.6. exit loss factor
loss factor in eq. 4.17. inlet loss factor of channel j eq. 2. 5.
total inlet loss factor, defined by eq. 3. 1.
parameter in slip equation, eq.3.18. parameter in slip equation, eq.3. 11.
m/s J/kg.°C m 2 2 kg/m s N/m2 2 2 kg/m s m/s2 2 kg/m2 s kg/m2 s kg/~ s W/m .°C J/kg J/kg J/kg m3/m2 .s m3/m2 .s m3/m2.s
óp ópa ópf ópg q
~
J Qtot Qthres r re R Re RF s Tscb Tsub uv.
J. vin yM s v SW vtot v w x parameter in parameter in parameter in parameter in length slip slip slip slip equation, equation, equation, equation, eq.3.28. eq.3.l5. eq.3.25. eq.3.37. distance between channel inlet and water level in steamdrum heated lengthexponent, eq. 3.7. pressure
pressure, indicated in figs 2.2. and 2.3.
pressure difference
pressure difference due to acce-leration
pressure difference due to pressure difference due to tational forces
heat flux
heat input per unit volume heating power for channel j total heating power
=
LQ. threshold power J exponent, eq. 3.15.friction
gravi-real part of complex number two-phase friction multiplier Reynolds m.unber
acceleration force of steam, eq. 4.9.
distance from wall/radius, in circular tube
slip ratio time
time derivatives, defined in appendix A
temperature
temperature at which steam bubbles detach from wall
temperature at which void fraction increases rapidly
temperature at which subcooled boiling begins
subcooling
=
(Tsat)con- Tin specific internal energywater inlet velocity in channel j velocity of mixture, fig. 3.9. cross-sectional averaged velocity of steam phase
vapeur drift velocity, eq.3.30. total inlet velocity
=
LV.
./3
cr6~sJsectional averaged velocity of water phase
steam quality by weight
m m m N/m2 2 N/m2 N/m 2 N/m2 N/m 2 N/m2 W/m3 W/m
w
w
w
oc
oc
oc
oc
oc
J/kg m/s m/s m/s m/s m/s m/sxth z z 0 Greek symboZs et B K p <j>. nJ Ul a À T ' t \JdL Subscripts c con d ex in m r s sat w Superscripts + MisceUaneous < > SPF TPF
thermodynamic steam quality, eq. 5.21.
axial distance m
unheated length at channel inlet m
void fraction
parameter in eq. 5.14. heat distribution parameter specific density
phase angle of channel j dynamic viscosity angular frequency surface tension loss factor, eq.
4.9.
shear stress oC(W/m2)-0.25 kg/m 3 0 Ns/m2 rad/s N/m 2 kg/m .s N/m2 residence time of fluid particle s equivalent length of downcomer mcenter of channel condens er down corner exit of channel inlet of channel mixture ris er steam saturation water complex number local value mean value vector
averaged over cross section single-phase friction two-phase friction
Abst:ract
In this study the results of an experimental and theoretical in-vestigation into the hydrodynamic behaviour of parallel boiling channels, are reported. The experimental rig could be operated in natural or in forced circulation, employing water as a test fluid. The test section comprised three, uniforrn.ly heated, inside-cooled, boiling channels, which were connected to a common inlet and
out-let plenum. Experiments have been performed at 15.5 and 30 bar.
In the experimental part of the study, most attention has been paid to the following subjects:
a. the influence of pressure, inlet subcooling and different
com-binations of channel orifices, on the steady-state and dynamic hydraulic characteristics,
b. the influence of a pump on the stability of parallel channels.
The inlet subcooling has been varied from 1 to 35°c; the values of
the inlet loss factors were 1.4 and 8.0 velocity heads.
In the theoretica! part, it has been tried to predict the experi-mental results with a non-linear computer program that describes the hydrodynamics of three parallel boiling channels.
Good agreement between experimental results and predictions is
re-ported. Besides this, it has been examined if, and how far,
paral-lel channel stability characteristics can be predicted with a
com-puter model for a single channel.
Chapter 2 describes the experimental boiling water loop. The
measur-ing, recording and analysing equipment is described. Special
atten-tion is paid to the measurement of the void fraction inside a
heat-ed channel,without interruption of the heatflux. Calibration of the
void gauge in a separate calibration loop under full scale
condi-tions, is discussed in detail.
Chapter 3 covers the steady-state experiments. Inlet velocity, and void fraction at two positions along the channel axis are presented
as a functjon of pressure, inlet subcooling, power and channel
in-let loss factor. The experimental data are analysed with regard to
the void fraction. Comparison of the present data with existing
slip correlations is made. Good agreement is reported with Jones'
slip equation, both for 15 and 30 bar.
In Chapter 4 the dynamic experiments are reported. A criterion for
determining the onset of instabilities is proposed. Instability
threshold power, oscillation frequency, and the mode of oscillation
are presented for the same experimental conditions as reported in Chapter 3. Some of the experimental results are compared with pre-dictions from engineering rules of thumb. The validity of p redic-tions with a computer model for a single channel, is discussed. Burn-out measurements have not been performed. When the amplitudes of the inlet velocities approached the mean value of the inlet ve-locity, then the experirnents were stopped.
In Chapter
5
a non-linear computer model is presented, which des-cribes the hydrodynamic behaviour of the three parallel channels. Correlations for slip ratio, two-phase friction and subcooled boil-ing are discussed extensively. A small power disturbance is imposed to assess stability. Predicted threshold power, oscillation fre-quency, and mode of oscillation are compared with experimental re-sults for both natural and forced circulation. Goed agreement be-tween experiment and model is reported. Predictions from a single channel model are compared and discussed; it is indicated when a single channel model may be expected to predict correctly, and when not. Some selected results from the model are compared with experimental werk from ethers. The parallel channel model, as well as the single channel model, have been written in the CSMP lan-guage. An IBM-360-75 computer has been used for the computations. In Chapter 6, the main conclusions that can be drawn from the present study, are summarised.In Appendix A the conservation equations, as applied in the com-puter model, have been derived.
In Appendix B the experimental steady-state data have been listed. In these data corrections have been made already for the exterior heat losses of the channels, and also for the influence of inlet subcooling on the void meter reading.
Chapter 1. Introduction
In conventional steamboilers and water-cooled nuclear reactors, with forced- or natural circulation, two-phase flow is of'ten pre-sent in the heated channels. In modern steamboilers, and to a les-ser extent in nuclear reactors, many boiling channels are connected to the top- and bottom plenums, thus fonning a set of parallel chan-nels. In order to obtain amore economical design, it is f'requent-ly attempted to increase the heatflux. The increase in heatflux, however, mey have among ethers, the following two undesirable con-sequences:
(1) a sudden local break-down in heat transfer, called the boiling crisis or the departure from nucleate boiling (DNB}. This DNB results in excess temperatures of the heated surface and pos-sible danger of melting of the heated wall. The physical back-ground of the DNB phenomenon is not yet understood. In general local temporary vapour blanketing is thought to occur; Kirby
et al. (1), on the contrary, observed that during DNB a liquid
layer is still present on the heated wall. A large number of experiments (2), (3), have been performed in recent years in order to determine the DNB heatflux for typical geometries and physical conditions.
Several investigators
(4), (5)
have derived DNB correlationsfrom their experiments, but the agreement between the different
correlations is still poer
(6).
(2) The occurrence of spontaneous, sustained oscillations in mass
flow, void fraction, etc. As early as 1953 experiments
perform-ed by Ruddick (7) indicated that the DNB heatflux could be
re-duced by 40% when the flow was oscillating. Lowdermilk's in-vestigations (8) in 1958 confirmed Ruddick's conclusions. So there is a real chance that oscillations induce a premature
DNB. Moreover, the oscillations mey introduce mechanical
vi-bration of components, or they can cause problems of system control.
In a natural-circulation loop the flow of coolant is generated by the difference in mass density in riser and downcomer. Oscillations will occur when the heatflux reaches a critica! level. This flow instability is of a combined thermoclynemic and hydroclynemic nature. In a single channel forced-circulation loop, osçitlations in mass flow can be suppressed by a sufficiently stiff \+ pump/valve com-bination. For typical operating conditions, Dijkman (10) measured in a natural-circulation loop a stability threshold power of about
200 kW, whereas, with forced-circulation at same conditions, even
at 700 kW of heating power no oscillations occurred. The maximum power was now limited by DNB.
In parallel channel systems with forced or natural circulation things can be completely different. If the main circulation pump is sufficiently strong, the total mass flow will be constant. The parallel channels, however, can interact through the top and bot-tom plenums. The resu.lt is, that even with forced circulation, parallel channel oscillations can occur at relatively low .power levels ( 18).
In boiling single channel natural-circulation systems as well as in two-phase parallel channel systems oscillations can occur at heatfluxes lower than the DNB heatflux, so the oscillations can be the limiting factor for the maximum power level. In connection with this, Wallis and Heasley (11) stated:
"What is now needed is a logical theoretical basis for the prediction of system performance in advance, and for systematic optimization of the various para-meters. The problem is not a simple one. Even steady-state theoretica! predictions of two-phase flows are still inadequate for many design purposes."
Commonly flow instabilities are subdivided into aperiodic and pe-riodic instabilities.
Aperiodic instahiZity
As early as 1938 Ledinegg (12) derived an analytical expression for the steady-state pressure drop 6p across a boiling channel as func-tion of the mass flow G. An excursive flow instability can occur when the
vs mass flow curve has a part with a negative slope, sa
<
o.
In 1952 Ledinegg's theory was affirmed by experimental results of Weiss (13). The aperiodic instability, however, seldom appears in practice and can be avoided by suitable operating conditions (30).
(+) With stiff is meant that the pressure drop across the mass flow regulating valve minus the pumphead is very sensitive to small va-riations in mass flow. This implies mostly that the pumphead is large compared to the pressure drop across the heated channel.
Periodic instability
Periodic instability or sustained oscillations can occur when the Ledinegg criterion is not fulfilled. The oscillations can take pla-ce in the entire loop or in one or more of a number of parallel channels. The periodic oscillations can be subdivided into:
density waves pressure waves
thermal waves
flow transition waves
the period is in the order of a fluid particle residence time
the period is much langer, the oscilla-tions are accompanied by violent pres-sure pulses
the period is also much langer than for density waves, the waves occur at high heatfluxes and are accompanied by large wall temperature fluctuations
these waves can occur close to the point
of transition between bubbly flow and annular flow
The pressure waves and thermal waves have been observed by Stenning
et al (26), using a Freon loop. These types of oscillations will not normally take place in a water loop as DNB will occur.
Wallis (25) has demonstrated experimentally that the pressure drop
can decrease when the flow regime changes from bubbly to annular in horizontal flow. This pressure drop characteristic can be considered to be the cause of the flow regime transition waves.
The oscillations, this study is mainly concerned with, are of the density wave type. Simplified the density wave phenomenon can be
described as fellows:
a decrease in mass flow raises the void fraction, this increases
the buoyancy driving force and with some time delay, the mass flow.
The higher mass flow in its turn decreases the void fraction and
thus the driving force and, - again with some time delay - the mass flow. Now the cycle is completed. Whether the oscillations will be sustained or not depends on the relative magnitude of all parameters that are involved. Neal (9) calls the cycle the flow-void and void-flow feedback.
The steady-state and dynamic behaviour of the coolant mass flow is
governed and described by conservation laws. These time and space
dependent mass, momentum and energy conservation equations are known
rather well and are generally accepted. Mostly the equations are simplified to the one-dimensional space. Together with correlations for slip, friction multiplier, subcooled boiling and heat transfer, the resulting set of equations is non-linear.
Up to now one did not yet succeed in finding an analytical solution.
Two approaches are possible:
i. linearise the equations and calculate a small perturbation
response. Two techniques are mostly used:
~ Laplace transfonn, :2. Lagrangian notation.
ii. direct approximated solution of the non-linear equati.ons with the help of computers.
It will be tried to give a brief review of some of the models that
have been developed during the last decade.
1 • 1 • l'l'evious Wol'k
1. 1. 1. Single ChanneZ ModeZs
NatuPal-CircuZation Loops
In 1960 Wallis and Heasly (11) made a theoretical study of instabi-lities in a natural-circulation loop with a single boiling channel. They considered the loop as a dynamic system of non-linear time de-lays, storage elements and resistances. With many simplifications,
the partial differential equations were integrated; the boundary
conditions were then included in the equations. Small perturbation
technique was applied to linearise the equations, the Lagrangian notation was used. The dynamic stability of the system was tested with the Nyquist locus concept. The results can be used for
quali-tative pTedictions only, due to the large number of simplifications
and assumptions that have been made.
In 1965 Boure (21) stated that the causes of instabilities can be
studied with a very simplified model only. He investigated the
me-chanism of density waves with such a model; the equations were writ-ten in terms of time delays.
It was concluded that the oscillations are sustained owing to the
interaction between a disturbance in mass flow and its influence on
the density. Later on in this study this item will be discussed
further.
Spigt (14) has developed a model with non-linear steady-state
equa-tions, the dynamic equations were linearised using Lagrangian nota-tion. The steady-state program was solved with a finite difference technique. The initial conditions for the dynamic part were provid-ed by the steady-state solution.
Reisch and Vayssier (15) were amongst the first who used IBM's CSMP
language to simulate an analogous integration on a digital computer.
They solved their non-linear dynamic equations with finite
diffe-rences in space, and applied a CSMP integration procedure for a
"continuous" integration in time. The system's response to an
im-posed power disturbance was calculated to determine stability.
Several other models have been developed, the differential equations
mostly do not differ very much, the models commonly differ in inte-gration- and solution technique, and in the correlations for slip and two-phase friction.
When a channel region with thermal non-equilibrium is taken into account, two correlations must be added, viz. a correlation for pre-diction of the point where subcooled boiling starts, and a
correla-tion that describes the energy distribucorrela-tion between the two phases. Two comparative studies of different models should be mentioned, the first by Neal and Zivi (16) in 1965, the second one by Bj~rlo
et al (17) in 1967. Both studies agree in their conclusion that the predictability of loop physics by all models is still rather poor.
1.1.2. Parallel ChanneZ Models
In 1959 Quandt (18) made an analysis of parallel channel transient response and flow oscillations. He assumed that the Ap across each channel is constant, because (quoted):
"Reactor channels are so constructed that a disturbance of the mass flow in one chan.nel can be absorbed almost equally by several of the neighbouring channels. Hence, a flow disturbance in one channel results in a lower order change in the total pressure drop across the com-bination."
As a matter of fact, Quandt's model acts with only one single chan-nel with a constant Ap. The response to a small transient in power is calculated. The linearised equations are obtained from a pertur-bation technique, the Laplace transformation Îs applied.
The result is a transfer function of the form:
AV AQ
as 2 + bs + c 2
s + ds + w2
Constants are determined by the steady-state conditions.
When d=O the response is considered to be oscillatory. Predictions of qualitative influence of system pressure and friction factor agree with experiments.
In 1966 Davies and Potter (20) applied Quandt's constant Ap concept to their model. With the assumption that the water and steam phase have equal velocities, and with the neglect of subcooled boiling, they can integrate their linearised equations spatially. Laplace transformation is applied with regard to time, control theory (Nyquist locus) is used to assess stability.
Though some basic assumptions, necessary for an easy mathematical treatment, permit only qualitative predictions, the model can be very useful for gaining insight in the physical processes governing the dynamic behaviour of a boiler tube.
In 1967 Carver (22) developed the POISE computer code for parallel channels. Again Quandt's constant Ap assumption is applied, so the POISE code actually is a single channel model. The three conserva-tion equaconserva-tions are written in finite-difference form, the equaconserva-tions of mass and energy are integrated simultaneously along the channel,
the simplified momenttun equation provides the link to the next time step.
Correlations for slip and two-phase friction are used; however, subcooled boiling is not taken into account. The system's response to an imposed power disturbance is calculated, a damped response
means a stable operating condition. Poise results - so far as
fre-quency and threshold power are concerned - have been compared to
d'Arcy's experiments
(28)
with three identical parallel channels.The agreement is goed at a 70 bar pressure, for a 35 bar pressure deviations are considerable.
At Rescona Ltd. a computer program for a large number of parallel
channels is under development
(23).
A number of different, heatedor unheated channels, are connected to.the top and bottom plenums,
the top plenum of a number of channels can act as the bottom ple-num for another set of channels. The channel equations are based on ( 14).
An error is defined for the steady-state solution, which is zero when the boundary conditions for each channel are fulfilled. The Canyon process (24) is used to minimise this error.
For the dynamic behaviour the equations are linearised, the
steady-state condition is used as initia! condition for this dynamic part.
All the "parallel" channel models mentioned before - with
excep-tion of the Rescona model
(23) -
act with a constant óp across thechannel and are essentially single channel models. In a single
channel model the downcomer is taken into account, so the óp across
the channel can vary, in a "parallel" channel model this óp is ta-ken constant.
The magnitude of this óp however, is a function of geometry, pump characteristic and of the mass flow rate of every channel.
Interference effects for non-equal channels as observed by Mathisen
(29) cannot be predicted by these models.
H. Christensen in his chairman's review on a session on parallel
channel stability of the Eindhoven Symposium in September 1967,
stated:
"A very much more difficult problem, which really involves the consideration of several parallel channels, developes if óp is not constant in time. To treat this problem the power distributions and orifices for each channel (or
suitable groupings of channels) must be specified. The
downcomer circuit is then added on as an extra "upcomer"
with negative flow velocity and a negative pressure drop
representing the pump. I know of no computer program
free-ly available, which can treat this problem."
1.2. E:rperimentaZ Work
Quandt was one of the first who observed parallel channel
instabi-li ty in the
a(öpchannel - öppump)
Although his experimental results are not available to the present author, the analytica! work (18) that followed Quandt's observa-tions has been mentioned before.
Crowley et al (27) investigated the hydrodynamic behaviour of three parallel channels with refrigerant 113 as coolant. The natural- and forced-circulation closed loop comprised three inside cooled paral-lel verti.cal channels. A constant pressure drop was maintained a-cross the heated channels by a parallel by-pass, which took about 85% of the coolant mass flow. Care was taken that the loop was ope-rated in the
region.
The test section I.D. was 10 mm, the by-pass diameter 75 mm. The inlet subcooling was varied from O up to 45°c, the maximum liquid velocity at the test section inlet was about 1.5 m/s. The pressure was slightly above atmospheric pressure.
Some of Crowley's most important results are:
Increased subcooling generally increases stabili ty, but can be destabilising ove~ a certain range of Tsub·
Increased subcool1ng at constant power
level and mass flow increases the oscillation period. Increased mass flow increases stability.
Increased power decreases the period.
Crowley et al have extended and adapt2d the model of Davies and Potter (20) to their own apparatus, but significant results are not yet available.
As part of a preparative study of a boiling light water reactor of the CANDU type, d'Arcy (28) investigated the phenomenon of parallel channel instability in a boiling water test rig up to 70 bar. The experimental apparatus contained three equal parallel annular flow channels, internally heated, and was operated in forced-circulation wi th a mass flow rate up to 1 360 kg .m-2s-1. The test section heated length was about 3 m, the inner shroud diameter 20.8 mm, the outer heated rod diameter 15.2 mm. Subcooling was varied from 6 to 8ooc. Though it is impossible to compare the absolute values of d'Arcy's results to those of Crowley, they qualitatively agree. The desta-bilising effect of subcooling over a limited range, however, has not been reported. This might be a consequence of the relatively high operation pressure.
Three different modes of oscillation were observed, e.g. the three channels oscillating with equal amplitude, but 120 degree phase shift
two channels oscillating with equal amplitude and 18o0c
phase shift, the third channel being stable
two channels oscillating in phase with equal amplitude, the third channel 180° phase shi~ed with twice the amplitude.
It has not been indicated which mode of oscillation occurred as function of operating - or loop - conditions. The oscillations ob-served were of the density wave type, a goed correlation was found between the measured periods and the computed particle transit times.
Mathisen (29) performed some experiments on the hydrodynamic
sta-bility of two parallel channels. His test section consisted of two
inside cooled channels with 15 to 35 mm I.D., the heated length
was 4.3 m. The experiments have been performed under natural-circu-lation, the pressure mostly was 50 bar.
Some of the experimental results are:
a general qualitative agreement with Crowley's experiments (27) for equal channel geometry and power level both channels had equal amplitudes, 180° phase shifted.
The main liquid flow in the downcomer was not affected by the oscillations
an increase of the downcomer restriction increased the threshold
power for a single channel, but decreased this power level for
two parallel channels
an equal increase of both inlet restrictions increases the
thres-hold power for two parallel channels as well as for a single
channel
if, for equal channel powers, the ratio of inlet restrictions
k,Jk2 is varied from 1 to 0 wi th constant k 1 , the threshold
po-wer lias a maximum for 0.3< k 1/k2 < 0.5.
if, for k 1
=
k2 , the channel power ratio Q1/Q2 is varied from 0to 1, the quantity (Q1 + Q2 )has a maximum for
o.8
<Q/Q2 < 1.Mathisen's experiments support Christensen's comments on parallel
channel models. A "parallel" channel model with the constant tip
concept cannot predict the interference effects mentioned under
Mathisen's experiments.
1.3. Aim and Scope of this Study
During the course of this study experiments have been performed in
order to obtain a better understanding of the behaviour of boiling
channels and of the interaction between parallel channels. The test
facility comprised three uniformly heated parallel boiling channels,
the maximum operation pressure was 40 bar.
The influence of system pressure, subcooling and different inlet
restrictions on parallel channel <lynamics has been investigated.
The loop has been operated in natural- and forced circulation.
In order to provide the póssibility of prediction and extrapolation
of the experimental results, a mathematical model which describes
the hydrodynamic behaviour of three parallel channels has been
de-veloped.
It isstressed that the applied correlations for slip, friction multiplier, subcooled boiling etc. have been taken from the
gene-~al available literature, they have not been reduced from own ex-periments and then used to recalculate the experimental results.
The simultaneous one-dimensional partial differential equations
have been solved in space with a fourth order Runge Kutta proce-dure with variable step length, and in time with a finite
diffe-rence technique. An iteration technique - based on the Taylor Series
for more independent ve.ribales - has necessarily been used in order to satisfy the dynamic boundary conditions for each channel. The code has been written in the CSMP language (Continuous System Modeling Program). The model's predictions have been compe.red to
the experimental results in a quantitative sense.
Chapter 2. Desal'iption of the Apparatus
2. 1 • Expel'imentaZ Set-Up
This section consists of a brief .description of the experimental equipment. The loop is constructed of stainless steel throughout,
with a maximum operation pressure of 40 bar. Power for heating is
supplied by a 1 MW electrical rectifier. The thermal energy is
transported by a steam-water mixture to a condenser and a cooler in which it is again removed.
2 . 1. 1. The Loop
On fig. 2.1. a flowsheet of the loop is shown. The loop consists of
four main components, viz. the pressure vessel with a steamdrum, the condenser, the subcooler, the preheater and the pump. The pres-sure vessel contains the test section, that means the combination of the three channels. When the pump circulates the coolant, the water in the test section flows upward to the steamdrum, then down-wards through the annulus between pressure vessel and the tube that surrounds the test section (the downcomer) and via the cooler to the pump. The water level in the steamdrum is about 14 cm above the test section exit. A waterdrum with a relatively large cross sec-tion is connected to the steamdrum in order to reduce changes in the water level due to thermal expansion and void generation. The water level can be viewed by means of a level gauge at the control panel, and another level gauge near the steamdrum.
When heat is supplied to the test section, the flowing coolant eva-porates partly. The water and steam are separated in the steamdrum and the steam continues to the condenser. After condensation has taken place the condensate returns to the downcomer.
In the natural-circulation loop (pump is by-passed) the driving
force is gravity, and the flow is generated by the difference in
mean density in the test section (riser) and the downcomer.
The subcooler consists of a stainless steel vessel with four coiled tubes. Secondary cooling water is pumped through these tubes. The cooling capacity depends on the water velocity in the tubes and on the number of tubes that are cooled by the flowing water.
The preheater is mounted in series with the subcooler. A 9 kW
elec-trical heating element is inserted in the pump delivery line. A
variable transductor unit supplies the heating power, which can be read off from a Voltmeter and an Ammeter.
1' • • • • • • • • + t
e:~'.:lroèc
pUIT.p l':'!u.itiC'lsno:nelr.r
2. 1. Flow sheet of the experimental loop
The condenser consists of 16 parallel tubes, the steam flowing in-side the tubes, the outin-side of tube being cooled by water from two reservoirs with a perforated wall. The wetted length of the tubes and thus the condenser cooling capacity depends on the water level in the reservoirs. In turn this water level depends on the cooling water mass flow.
Pressure control is achieved by automatic adjustment of the conden-ser cooling capacity. The steam temperature in the steamdrum is compared to a preset reference temperature by means of proportional, integrating and differentiating control. The condenser cooling wa-ter mass flow is regulated according to the temperature deviation.
2.1.2. The Test Section
The test section, composed of commercially available stainless steel tubes, is schematically shown in fig. 2.2. Three inside cool-ed tubes are silver soldercool-ed into flanges near the top- and bottom sides. Copper electrodes are connected to these flanges in order to supply the electrical heating power. The three inside cooled tubes constitute the parallel channels, all three of them having a uni-form heatflux.
Around the three heated channels there is a thermal and electrical insulation. Each channel can have its own orifice just at the chan-nel inlet. The three chanchan-nel inlets are connected to a common inlet chamber. Three holes in the chamber wall act as a joint inlet ori-fice for the three channels. In each channel the inlet mass flow is measured as well as the void fraction at two different positions along the channel. The total amount of coolant mass flow is rneasur-ed separately. In order to get an impression of the channel exterior heat losses, the temperature drop across the wall of the tube that surrounds the thermal insulation of the test section, is measured. Corrections for the exterior heat losses (maximum 5 kW) have been made.
Same basic dimensions of the loop and the test section are collect-ed in table 2.1.
2.1.3.
The Power
SuppZyThe DC heating power is supplied by a transformer/rectifier unit.
A~er the rectification a small 300 Hz component still is left.
The transformer is connected to the 10 kV mains, the output vol-tage can be controlled continuously up to 50 Volt. The maximum cur-rent is 14.400 Amps. By connecting a second stepwise controlled transformer/rectifier unit in series, the maximum voltage can be increased up to 70 Volt.
The current is measured with the help of a 3000:1 current trans-former. The accuracy of the transmission ratio proved to be
with-in .:!: .1%. For steady state operation the power is measured with a
light spot Wattmeter (class .2) and verified by a Voltmeter and an Ammeter. The light spot Wattmeter reading, and the power calculated
thenn.1!1.l insull!l.lion detail A 0 12 •22 •32
."
0 21 0312.2. Schematic of the test section
channcl wa.l l
thcnn&.l insulntion tube
a = void 111.eler
J!I • pressure lap
T • lempcrature
from Voltmeter and A.mmeter readings agreed within + .5%.
Correc-tions for the effect of the 300 Hz component on th; power reading have therefore not been made.
As the voltage drop is measured between the top- and bottom
elec-trode, instead of over the test section, a small error in the
po-wer reading is introduced. Calcu.lated on the basis of the
electri-cal resistances of the test section and the electrodes, this error
is about .4%. Correction for this error has not been made.
A current input in a transductor system at the primary side of one
of the transformers provides a continuous dynamic control of the
heating power. During dynamic power supply, the power can be
mea-sured with a Hall generator, a device that produces an output
vol-tage proportional to the product o.f voltage and current at the
en-trance. The Hall generator is calibrated against the Wattmeter in
Table 2. 1. Basic Loop Data and Dimensions max pressure
pressure vessel I.D. max I.D. steamdrum max electrical power max electrical current heated channel I.D./O.D. heated surface (one channel) max preheater power
k. velocity heads (eq.
2.5.)
k~:t velocity heads (eq. 3.1.) a.xial positioninlet heated channel inlet pressure tapping reference void gauge begin heated length, z=z0 void gauge 2
DNB detection wire void gauge 3 end heated length channel exit
water level in steamdrum, z=L
40 bar 150 mm 350 mm 1000 kW 14.400 Amp 20/23 mm O. 1601 m2
9
kW 1 .4 and 82.5
z m 0 0.03 0.075 0.090.73
1. 502.64
2.64
2.72 2.882.2.
Measuring and Reaording EquipmentThe locations where the physical quantities are measured are indi-cated in fig. 2.2. The following quantities will be reviewed:
temperature mass flow pressure void fraction
2.2.1. TemperatUPe
Temperatures of water and steam as well as temperature differences are rneasured with Chromel-Alumel type thermocouples with a 1 mm Inconel sheath around the wires. The sheath is insulated frorn the wires by Al2
o,.
The thermocouples pass through a pressure wall by rneans of ConaJt sealings. Thermocouple sheath and Cona.x sealings are electrically insulated frorn each other. The thermocouples are calibrated within an accuracy of +.25°c.
For ternperature diffe-rences two thermocouples with the-best identical calibration cur-ves are selected. The cold junction is placed in a Dewar flask fil-led with a water-ice mixture. The thermocouples respond as a first order system with a time constant of ,35 s. Conventional millivolt recorders are applied for printing the absolute temperature, tem-perature differences are written continuously on a one pen recor-der. Both types of recorders have a rather slow response.Excessive temperature rise of the heated channels is prevented by means of a burn-out detector (39). This safety device is used to monitor the ratio of the electrical resistance of the upper- and
lower sections of the heated elements. When DNB is occurring, this
ratio will change rapidly, due to the local temperature rise. The
heating power is then switched off by the burn-out detector within
about 70 ms. This shut off time is short enough to prevent damage
to the heated tubes.
2.2.2. Mass Flow
For one-phase flow the pressure drop is a function of the rnass
ve-loci ty. As pressure differences could be rneasured easier than
wa-ter velocities, the rnass flow measurements have been replaced by
pressure difference measurernents. In fig. 2.3. the way in which the
pressure differences are measured is illustrated. The pressure drop
across the orifice of the common inlet charnber is represented by
p - p4, and is therefore a measure for the total amount of coolant
m~ss
flow. The statie pressures p1 throughp~
are measured 3 cm from the channel inlet. In general the equation for the channel inlet pressure drop reads:k. in w in p
v.
2+~
dt Pw ( 1v )
in ' eq. 2.1.with 1
=
distance between channel inlet and point where pchannelis measured.
In this equation the friction is assumed to be incorporated in k. ;
k. depends on the Reynolds Nurnber. When the mass flow oscillate§~
V~n may be approximated by in
v.
in V. + a sin w t.in eq. 2.2.
~and V. are constants. If ais assumed to be small compared to
V in then in' and <!> arctan 1 V. k. in in sin(wt+cj>) eq. 2.3. eq. 2.4.
29
channel
coD'IDlon i nlet. chamber (boltOlll plenum)
cross sect.ion A-A
channel pressure inlct. cha:nber prcssure Ps
2.3. Detail of bottom part of the test section
For w
=
6
rad/s; 1 0 ,05 m; kin 1. 5 and Vin 1 m/s; the term1.02 and ~ 11 degrees.
î
Usually the influence of the acceleration term is neglected. This seems acceptable as far as the amplitude is concerned. When, hov-ever, inlet velocity and Ap. are confounded with respect to the phase shifts of the void frägtion (40}, one should be on guard for possible deviations.
For the present experiment, 1
=
.03 m, so ~ vill be smaller. Be-sides this the inlet velocities of the three channels are mostly compared with each other, and then the effect of the acceleration term diminishe~ even more.kin and ktot have been obtained from steady state calibrations,
so k.
in eq. 2.5.
The water is circulated with a pump, the pressure differences are read off from a multimanometer, and the mass flows of the indivi-dual channels are determined with the help of a balance and stop-watch. In fig. 2.4. kin for a channel is plotted vs the Reynolds
Number. 4
kin is Reynolds Number dependent; for Reynolds > 5.10, however this dependency is very weak.
During experiments at high temperatures the Reynolds Number is
al-ways in the order of 105, sok. in can be considered to be constant.
-
--
-1
-.
1 1. 7 " " 1 " 0 ~ ~ 1.6.
1.
0 ~.
f - - - - -- - - -M f - - - --~
~
.
f'..._.
" 1. 5 ~--
-
. . .
·
.
.
.
.
.
'".
.
1. 3 10-4 Reynolds nanber2.4. Calibration curve of inlet loss factor
2.2.3. Pressure
The absolute pressure is measured by two Bourdon type pressure gauges. They are connected to the steamdrum. The pressure gauges have been calibrated with an accuracy of about !. .1%.
Pressure differences in steady state condition are measured with a multimanometer. The manometer liquid used has a density of 1750 kg/m3. The meter can be read off with an accuracy of !. 1 mm,
that means an absolute óp accuracy of+
7.5
N/m2 . For an inlet ve-locity of 1 m/s, the óp measured across the inlet is equivalent with about 80 mm manometer liquid, so the relative error is about + 1,25%. Parallel to the multimanometer, dynamic pressure diffe-rence transmitters are mounted (type SEL). These transmitters have been calibrated against the multimanometer in steady state condi-tion. The linearity is within 1%. Up to two Hz the amplitude and phase need no corrections as pointed out in(36).
The transmitter output first passes through a low pass filter (12,5 Hz), and then is recorded on an 8 pens Sanborn recorder and on an Ampex magnetic tape recorder.
Signals have been stored on this 12 channel Ampex recorder (FM type) in order to perform the dynamic analysis a~erwards.
2.2.4. Void Fraction
Because measuring the void fraction is always a difficult matter, special attention will be paid to this subject. Several different techniques have been applied up to now. The gamma-ray technique is considered to be one of the most reliable methods. The void frac-tion can be calculated ·from the attenuation of a gamma-ray that crosses the water-steam mixture (31). The application of this me-thod, however, is strongly restricted by the safety measures that must be taken. Besides this, the method requires rather long count-ing times, soit is suited for steady state measurements only; for this reason the gamma-ray technique cannot be applied to the pre-sent experiments.
Many other techniques exist (32), but they all have their specific disadvantages, o~en the response is too slow, or the sensor can-not be mounted conveniently. It was decided to apply the impedance technique for two reasons:
the response is very fast;
a very simple sensor could be designed, and it can be mounted in the heated channel without interruption of the heatflux. In fig. 2.6. a drawing of the sensor is given. The gauge consists of three radial silver plates, the angle between two subsequent
plates being 120°. The three plates are interconnected in the
cen-ter of the channel and electrically insulated from the channel wall by means of small ceramic bars. A wire, insulated by teflon tubing, is connected to the sensor and passes the channel wall by means of a minisize Conax sealing.
The measuring principle is based upon the dependence of impedance on the void fraction. In fig. 2.5. the impedance measuring system is drawn. An oscillator provides a 3000 cps voltage to the two electrodes, viz the channel wall and the star-shaped gauge. The voltage drop over a resistor R depends on the conductivity of
the steam-water mixture and is a s measure for the void fraction.
After amplification and rectification the signal can be read off from a meter or be sent to a recorder. An AC voltage is applied for two reasons:
void gauge r---, 1 1 1
f\i!
L _____ .J trans.tonner 1:1 ,__...,.~~~~~~~~•
,
meter/
2.5. Schematic of the impedance void measuring system
it prevents polarisation of the water;
AC can be separated from DC, which is supplied to the heated channel wall, even if a 300 Hz component is superimposed. In 1881 Maxwell (33) made an analysis of the dielectric constant of a suspension of small spheres in a carrier fluid. The system was considered to be located in a homogeneous magnetic field. Max-well expressed the dielectric constant of the suspension as a
function of the relative content of spheres, and the dielectric constants of the carrier fluid and of the spheres.
Analogous to Maxwell's theories a fonnula for the conductivity of the medium can be derived. This formula reads:
k - k a c k + 2k a c
=
a k. - k 1 c k. + 2k 1 c eq. 2.6.~hei~o~~~c~~~~~;v~fv~!~:c~~~
!
~~tt~
~
:
~
~=~=:~ki'
kc and ka arecarrier fluid and mixture.
~rbeck (34) was the first who translated this formula toa
water-steam mixture, a being the steam volume fraction. He assumed k. to be very small compared to kc.
Eq.
2.6. reduces to the fonn 1k a k c - a + a/2 eq. 2. 7.
k and k are to be measured.
a c
In two-phase steam-water flow Maxwell's assumptions,however, are not always satisfied. Steam bubbles are not necessarily small com-pared to the distance between the electrodes of the sensor; the electrical field is not homogeneous; and last but not least, the steam bubbles are not homogeneously distributed. This distribution depends mainly on the flow regime. Lacey (37) investigated the different flow regimes and ascertained a streng dependency of type of flow regime on steam quality. In the heated channels the steam quality changes duri~g experiments, so the flow regime can change too, and the void fraction distribution probably is not constant. For all these reasons equation
2.7.
is not correct under the prevailing conditions and a calibration of the impedance gauge is always required.After 0rbeck the impedance method has been applied by many ethers (35), (36), they all calibrate the gauges. 0rbeck calibrated with perspex spheres in water, Spigt (36) used an atmospheric air-water system in which the air content was calculated from pressure rnea-surements and the water level.
The conductivity of the steam-phase (see eq. 2.6. ), however, is not equal to that of air, and besides this, it depends on the sys-tem pressure. Moreover, the flow regimes in air-water and steam-water experirnents rnay be different, for equal "steam-quali ties". So it is at least questionable if air-water calibrations can be used for steam-water experiments.
For all these reasons it was decided to calibrate the void gauge
in steam-water flow. System pressure.and geometry had to be the
same as during the experiments. In order to perform this calibra-tion, a separate loop has been built. For a schematic drawing of this loop, see fig. 2.6. It is a small natural-circulation loop, in principle identical to the boiling loop of fig. 2. 1. The test section of this calibration loop is a full scale model of one chan-nel of the parallel chanchan-nel test section. Downstream of the heated tube two quick closing valves are mounted, at a distance of about
1 m from each other. These valves can be closed simultaneously within .1 sec. In open position the valves do not disturb the ri-ser cross section. The tube between the valves is unheated, so the void fraction is approximately constant in that area. Three void gauges are mounted as drawn in fig. 2.6. The reference gauge at the heated channel inlet provides a reference signal representing the impedance of
100%
of water (k frorn eq.2.7,).
The calibration procedure is as fellows: cThe loop is operated at a certain power level and pressure. When steady state conditions exist, the void gauges are read off, then the valves are closed quickly and simulatenously. The water content of the tube between the valves, and thus the void fraction,
can now be read off from the gauge glass drawn in fig. 2.6.
Cor-rections are made for temperature changes between valve closing moment and read off point of time, and for the water content of the gauge glass.
condens er va.lve T voic! gauge W'&ter gau8e valve void gauee
rfJf'
.L
1J
î
!
1 void gauge cera.mic insu.lation he&ted channel2.6. Diagram of void gauge and calibration loop
During the calibration experiments gauge 1 proved,after a small
correction for the pressure difference, to give the same signal as gauge 2, so an influence of the electrical current through the test section tube on the void gauge signal could not be detected. Calibration tests have been performed at 10, 20 and 30 bar with steam-water flow, and at atmospheric pressure with air-water. In
the latter case air is injected near the orifice. In fig. 2.7. the
calibration data for 10 bar steam-water flow have been plotted. A curve through these calibration data has been calculated with the least square fitting method. In fig. 2.8, curves for calibrations at 10, 20 and 30 bar have been drawn.
It has already been mentioned before that for these calibrations the geometry and pressure are equal to experimental conditions. The inlet velocity, however, may sometimes differ about 10% from the experimental value. In order to investigate the effect of a different inlet velocity a number of calibration runs have been performed with a 20% lower inlet velocity. The lower mass flow was achieved by mounting a downcomer restriction. Any influence of the changed inlet velocity on the void gauge calibration curve could, however, not be detected.
1.0 u " ~ ~tl .9 t' > ~ -à
3
~~
.6 • 7 .6 . 5 .4 • 3 .2 1 0~
.1~
P = 10 bar T • 179°C s&l.
~
.
I~
:·.
~
,\
~
:
'\
~
·~
~
·~
~
~
~
.2 . 3 .4 . 5 .6 • 7 .8 .9 1.0 void fract.ion o2.7. Calibration curve of void gauge at 10 bar
In fig.
2
.
9.
the atmospheric air-water calibration data have been plotted together with the theoretical MaJ{\{ell curve from eq. 2.7. and an air-water calibration curve of Akesson(35).
Akesson applied the impedance method with a very similar gauge; the only differen-ce is that his gauge has four blades with a90°
angle in between them.The maximum attainable void fraction during the present air-water calibration experiments was
62
%
.
For higher void fractions, slug flow occurred, and as a consequence the void meter could no more be read off accurately. It is supposed that the same happened in Akes-son 's experiments, and that therefore his calibration curve deviat-es strongly from the straight line for n >60
%
.
It will be evident from a comparison of fig.
2.8.
with fig.2
.
9.
that atmospheric air-water calibrations cannot be used for steam-water experiments at higher pressure. A more extensive description of the experimental set-up and the calibration procedure is given in (38).During steady state conditions the void signal is read off from a meter. For dynarnic experiments the void signal is recorded on an
8
pen Sanborn recorder. The latter is suited for dynamic signals up to 150 Hz.2.8.
2
.
9
.
.
>~
1.0 .6 .L . 3 .2 .1~
~
'
I
~
~' - - - 10 bar T t = 179°c 20 bo.r Tsa • 211°c~
sat ~---
-
---
30 bar T sat. • 233°c'
~
\
0_
~
,
~
,
.
~
''~
~
'~ ' ... ""':::"~
.1 .2 . ~ .L .6 .1 .8 .9 1.0 void fro.ction aCalibration curves of void eauge at 10, 20 and 30 bar
1.0
,,
\~'
Akesson' s curve ..!.-·-.-.!.--Eind.hoven curve - ·'\
~~
- --- - M8.X"-'ell curve eq . 2.1.'\'
~'
•'""''\
'
...
~',,"""~
,• ' ' ' .9 .8 . 1 .6 • 5'
'
'\"
•
\
'"'
~'~
""'
f - - .-·-',
""
..._
"
-~
.L . 3 .2 .1 .1 .2 . 3 .L . 5 .6 . 7 .9 1.0 void tl-act.ion aCalibration curve of void gauge with atmospheric air-water mixture
2.3. AnaZysing Equipment
Transfer Function Analyser (T.F.A.).
The T.F.A. is a special purpose analogue computer. It calculates simultaneous the power spectra of two different signals as well as the cross spectrum. The power spectrum is a measure for the energy content of the signal at a certain frequency w within a bandwidth + ~w. The cross spectrum provides insight to which extent the two signals are correlated. From the auto-spectra and the real and ima-ginary components of the cross spectrum are further calculated the phase angle between the two signals, and the amplitude ratio ex-pressed in decibel.
The T.F.A. has been used to analyse the signals that have been stored on the Ampex magnetic tape recorder, however, only in those cases where direct analysis of the signals from the Sanborn record-ings was considered to be too inaccurate.
Amore extensive description of the apparatus and the way the sig-nals must be handled is given by Wamsteker (41).
Computers
For the purpose of steady state data reduction several small com-puter programs have been used. The programs were written in Algol 60 for the EL-X8 computer at the Technological University. For the theoretical study a computer code was written in the CSMP language. Computations with this code were performed with an IBM 360-65 from the Technological University of Del~ and later on with the IBM 360-75 computer, owned by the Philips Company in Eindhoven.
Chapter 3. Expel'imental Results Steady-State Data
3.1.1. Introduction
Before starting a series of experiments, the loop has to be
pre-0 pared carefully. An extensive description of the start-up
proce-dure has been given in (36), and will therefore not be repeated here in detail. Very briefly, the routine is as fellows:
the loop is filled with demineralised water; the power is switched on;
the loop is deaerated at a pressure of about 2 bar; all meters and instruments are checked.
Now the experiments can start. The experimental sequence is outlined with the help of a flow diagram in fig. 3.1.
A review of the experimental conditions is given in table 3.1. kin is defined by equation 2.5.; ktot by the equation
with
The measured quantities (see figs 2.1. and 2.2.) are: the the the the the the
velocities vin 1' vin 2' vin 3' vtot
temperatures Tin' Tsat' Tsub' T7-T6, T4
total channel power Qt t
void fractions a 12 , a
2
~, a 32 , a 13 , a23 , a33 condenser pressure pwater level in the con steamdrum
eq. 3.1.
In discussing the results a distinction has been made between steady state and dynamic experiments. The steady-state results will be pre-sented in this chapter, the dynamic results are reported in chapter
4.
During natural-circulation experiments the inlet velocity is physi-cally determined by a pressure balance along the closed circuit steamdrum-downcomer-heated channel-steamdrum:
Table 3. 1. Experimental eonditions
peon T sat T sub k. in k. in 2 kin 3
v
tot groupbar oc oc m/s 15,55 200 1;10;20;35 1.4 1. 4 1 .. 4 n. c. a 30 234 1;10;20;35 1. 4 1. 4 1. 4 n. c. b 200 1;10;20;35 8.o 8.0 8.0 n. c. e 200 1; 10;20;35 1. 4 1. 4 1. 4 f. c. + d 200 1; 10;20;35 1. 4 8.o 8.o n.c. e 200 10;20;35 1. 4 8.0 8.0 f. e. + + f 200 10 1. 4 8.0 8.o f. e. 1. 3 g 0 eq. 3.2.
ópdown
=
peon - Ps is only slightly dependent of V. , because the' downcomer cross section is muchin lareer than the heated channel cross section. The pressure drop across the
heat-ed channel consists of three main components, respectively due to
gravity, friction and axial acceleration:
eq. 3. 3.
When heat is supplied to the channel, the water in the channel will begin to boil. This results in a decrease of the mean density in the channel, and thus in a decrease of óp . The reduction of óp introduces a driving force that makes theg water circulate; g ópf and óp a (being functions of V. in ) restore the balance in eq. 3. 3.
+ resp ++ Vtot is adjus~ed so that.the Vtot vs Qtot curve of expe-riment a, respectively experiment e is
followed.
n.c. natural-circulation f.c. forced-circulation
For convenience during reading a copy of the above shown table is attached to the last page of the report.
