Further Design Features of the Nuclear Reactor at North Carolina State College
FURTHER DESIGN FEATURES
OF THE
NUCLEAR REACTOR
AT
NORTH CAROLINA STATE COLLEGE
PHYSICS DEPARTMENT
of the
The material presented in this report has
resulted from the combined efforts of various
members of the Physics Department. Almost
all members of the department, in one way or
another, have made noteworthy contributions
to the reactor project. Members of the other
departments in the School of Engineering,
have also made significant contributions. The
assistance of these men and the value of their
contributions are gratefully acknowledged.
The Authors
INTRODUCTION
A proposal to construct and operate a small nuclear reactor on the campus
of the
a reactor which would be suitable for this purpose have been presented in previous
reports.a1a2a3 The
projecta4 and of the general design of the proposed reactor. It was recognized at
the time of approval, of course, that a great deal of work yet remained to be done
on the details of the design of the reactor before construction could begin.
Additional general plans of the reactor facility and further details of the
actor
ing
ideas of the
systems have been borrowed from the successful
from the reports on reactor studies at
particular conditions and needs of the Raleigh Research Reactor. It has not been
the intention of the
make original contributions to reactor technology. Rather, it has been the purpose
to design and construct as simply and quickly as possible, a safe, flexible nuclear
reactor with maximum adaptability to instructional and research purposes. The
staff of
technical personnel of
ideas which they have contributed to the Raleigh Research Reactor project with
unfailing generosity and cooperation.
The present report presents a general description of the Reactor Building,
plans of the Reactor components, further discussion of potential hazards which
may be involved, and the anticipated start-up procedures and operating policies.
The Raleigh Research Reactor is to be located in the southern half of the
Court of Ceres, an open quadrangle near the center of the
College
ing
actor
tion
highest locations on the college campus and in the entire city of
buildings housing Physics, Chemistry, and Engineering Research, whose
ties
cent
several reasons the site chosen seems to be an advantageous one.
The plan of the Reactor Building is shown in Figures 1 and 2. A sketch of
the external appearance is shown in Figure 3. The octagonally shaped reactor
assembly is below ground level in the center of the 57 foot diameter Reactor
Room. This room, in turn, is in the center of the building. On three sides of
the room at ground level, are fourteen laboratory rooms to be used for
tion
Reactor Room, are five additional utility or laboratory rooms.
On the fourth side of the Reactor Room are the Control Room and the
servation
and water-filled glass windows, 8" thick. In front of the Observation and
trol
The photographic dark room, the counting laboratory and the Control
Room are air conditioned. The entrance lobby and the offices are provided
only with ordinary window ventilation. A ceiling exhaust fan furnishes
tion
be generally accessible to visitors and observers.
It is not anticipated that radioactive contamination of these areas of the
building will occur at any time. Plumbing facilities here are connected directly
to the city system in the usual manner. The remaining areas of the building,
i.e., the laboratories on upper and lower levels and the Reactor Room, may
become contaminated with radioactive materials and therefore special
tion
Separate systems are provided for supplying filtered air to and removing air
from the laboratories. The intake air system is made up of two parts: one for
the west side and one for the east side of the building. In each of the two parts,
air is drawn through a bank of disposable paper filters located under the
ing
above the ceiling of the locker rooms and from these the air is distributed by ducts
to each laboratory. The blowers in the supply system can be run either at full or
at half capacity. In normal operation, each blower will operate at half capacity,
which is 6,275 cfm. At full capacity, each system can supply 12,500 cfm, or a
total of 25,000 cfm.
The air is delivered to each laboratory through a control damper mounted
on the inside wall of the room. In normal operation, the air is changed once every
10 minutes.
Providing comfortable temperature conditions in cold weather, with air
ing
ing
exchangers in the main air supply duets insure a minimum air temperature of 65oF.
In addition, coils of hotwater pipes mounted in the ceiling or walls of each room
provide radiant heat to the extent needed for comfort.
The Exhaust Ventilation System withdraws the air from each laboratory
through the hood in that laboratory and through the filter in the exhaust line under
each hood, then through a system of ducts underneath the floor, to the filter room
at the rear of the building. Air is supplied to and removed from the Reactor Room
through louvered ventilators mounted in the wall of the room. Ducts from the
haust
System. Two 12,500 cfm centrifugal blowers, arranged in parallel, draw the air
through the bank of filters and discharge it through a 48 inch diameter, 110 foot
stack located at the rear of the building. In normal operation, only one of the two
blowers is used. If desired, however, both blowers can be used simultaneously,
in which case, 25,000 cfm are withdrawn through the exhaust system. At times
it may be necessary to maintain the flow of air up the stack, when ventilation of
the building is not desired. To permit this, a "by-pass" from the outside wall at
the rear of the building to the blower intake is provided. If this by-pass is
pletely
air, or only part of the total, coming from the building itself. Operation of the
by-pass can be either automatic or manual.
The hoods in each room are of a special, downdraft design, having removable
filters located in the exhaust duct under each hood.
It is not intended that any quantity of radioactivity above maximum
tration
building. Short-lived radioactive materials above this concentration will be stored
until sufficiently decayed, and long-lived materials above this concentration will be
stored for disposal in other ways. Due to emergencies, accidental spillage or
misoperation, however, above tolerance concentrations of activity may occasionally
escape into the drain lines. For this reason, therefore, no drain lines from the
work areas of the building connect to the city sewer system, except through a
tion
All drain lines from the Reactor Room and the work rooms on the lower level
discharge into a sump near the air filter room. A pump lifts the material from the
sump to the drainage system of the laboratories above. The liquid at the sump is
continuously monitored, and if excessive amounts of activity are found, the material
is not pumped to the drainage system above. It should be necessary to use the drains
on the lower level only seldomly and hence the sump pump will need to be used only at
infrequent intervals.
The drainage system from the ground level laboratories passes through
system of monitors, holding tanks, and valves designed to prevent discharge of
radioactive materials above concentrations permitted by
city sewer system. (Fig. 4)
In normal operation, laboratory drainage passes a radiation monitor and a
cut-off valve into one of two 550 gallon tanks, arranged in parallel. The effluent
line of this tank is halfway from top to bottom, so that the tank is always half full.
As liquid leaves the tank it passes a second radiation monitor, located in the exit
line, and then flows through an automatic, pneumatically operated valve to the city
sewer system. When desired, the exit line from the midpoint of either tank can be
closed and the tank becomes a holding tank with 275 additional gallons of capacity.
In normal operation, the activity of materials discharged down the drains
will be positively controlled within permissible limits before release. The
tion
amounts of radioactivity reaching the city sewer system in case activities of above
tolerance levels are inadvertently released into the drains.
The monitored holding tank system described above is intended to operate
as nearly as possible as follows: If an amount of activity above the maximum
permissible level goes down the drain, it passes the first radiation monitor, which
sends a signal to the Control Room and also sends a signal to the pneumatically
operated valve in the exit line of the hold-tank. The valve closes and the
active
second tank and this one is used in the drainage system while the decision is made
as to the method of handling the material caught in the first tank. The radiation
monitor in the exit line of the flow tank serves as a double check on the first
tor
level reaches that point.
In actual practice, it may prove quite difficult to obtain radiation monitors
capable of performing with reliability as described above, because the levels of
activity intended to be discharged are extremely low. (Actually, the exact values
of the maximum permissible concentrations have not yet been determined.
ther
in addition to the plan outlined above, in which the best monitoring procedures
available will be used, spot checking of liquid samples withdrawn from various
points in the system and other means as necessary will be instituted to insure
that activities above safe levels are not discharged. If all possible methods of
continuous operation of the system proves unsatisfactory, the two tanks can be
used entirely as hold-tanks. That is, material can be discharged into one tank
until it is full, and then the flow be diverted to the other. Meanwhile, sampling,
monitoring and analysis of the material in the first tank should be made to
termine
active
Each tank is provided with a gas vent line connected to the exhaust system
of the building, a valved overflow line connecting to the city sewer system, a
valved drain-line (normally closed) connecting to the city sewer system for
pletely
monitoring of the tank contents can be achieved.
Since 1944, the
nuclear reactor of the "water-boiler" type.b4 At first the reactor was operated
at a level below 1 kilowatt, then in step-wise changes in the design over a period
of 6 years, at 5, 10, and now 25 kilowatts. Operation at each level has yielded
knowledge and experience on the stability, behavior and use of a reactor of this
type. With added experience, increased confidence and satisfaction with the basic
principles on which the unit operates and its inherent safety, adaptability and
utility have been gained.a5
The reactor being built by
water-boiler type, and its fundamental principles of operation are identical with
those of the
corporated
satisfactorily to its proposed location on the college campus, and its proposed
usage in research and instructional programs.
The
U235, of 90% isotopic enrichment. The uranium will be in the form of uranyl
sulfate in water solution, and will be contained in a stainless steel cylinder
10-3/4 inches (i. d.) in diameter and 11 inches high (inside dimensions). The
cylinder will be enclosed in 20 inches of purified graphite reflector and this, in
turn, will be surrounded by six feet of a special, high density concrete. (Fig. 5)
The graphite reflector is extended out in one horizontal direction to form a
thermal column five feet square and five feet long. Four internal cooling coils
in the fuel cylinder provide heat removing capacity to enable the unit to operate
at a power level of 10 kilowatts. Besides the cooling system, instrumentation
for operation and control and a "gas disposal system" for handling the gaseous
by-products from the reactor, are the chief auxiliaries.
The concrete shielding performs two essential functions: Absorption of
radiation from the reactor and protection of the fissionable fuel from danger of
theft or sabotage. The shielding must be arranged so that samples may be placed
inside for irradiation, or radiation beams may be allowed to emerge for external
use. Also, the shielding must permit convenient access to the internal components
for repair and maintenance.
For absorption of radiation, the concrete is made 6 feet thick, and is composed
of special ingredients.a6 To increase the density above that of ordinary concrete and
hence enhance its absorption of gamma rays, Barium Sulfate (Barytes Ore) is used
as the coarse aggregate. A boron containing ingredient (finely ground colemanite
ore) is used as the fine aggregate. The boron increases neutron absorption in the
concrete and hence neutron activation of the shield is reduced. The boron content
of the finished concrete is 1% by weight.
Final composition of the concrete, per cubic yard, is:
| 4200 lbs. Barytes - | 95% between 1/16" and 3/4" size 5% smaller than 1/16" |
|
| 423 lbs. Colemanite ore - | 100% larger than 100 mesh sieve size and smaller than 20 mesh sieve size. |
|
| 882 lbs. Portlant cement - | Type 3. | |
| 51.7 gallons water. |
The colemanite is soluble in water, hence considerable difficulty is encountered
in its use. There appears to be a competition for the water between the dissolving
action of the colemanite and the normal process of cement setting; the setting of the
cement is delayed enormously. The barytes appears also to be sufficiently soluble
to have some effect on the setting process. Two proceduresa7 were found to be
ful
second mixing process after some of the primary setting processes had commenced
(40 to 45 minutes later). The concrete made in this way was found to have almost
double the strength of normal concrete. An overall density of 3.4 g/cc was achieved
in contrast to 2.4 g/cc for that of ordinary concrete.
When assembled, the concrete shield is octagonally shaped in horizontal
section
the reactor, the reflector and the auxiliary components. A 36" concrete slab covers
the top of the assembly. (Figs. 6, 7) The shield can be partially disassembled by
removal of various interlocking concrete blocks making up the assembly. The
crete
"cap-stones" on top of the assembly must be removed first, and then the
locking
When all the portable concrete blocks are removed, there remain the two
massive sides of the octagonal shield, separated by a five-foot gap. Thus,
ibility
subsequent maintenance and repair. (Fig. 9)
There are 11 exposure ports extending through the shielding and reflector
to the surface of the reactor or across the thermal column. All external
ings
to the interior of the shield is prohibited. Each port (Fig. 10) is closed externally
by a combination-lock safe door, the frame of which is imbedded in the shielding
concrete. The outer 18" of the steel lined port is 7 inches in diameter. From
this point to the inner end of the port the diameter is 6-1/2 inches. This one
quarter inch annular offset in the port prevents escape of radiation through the
crevices surrounding the concrete plugs which fill the port when it is not in use.
One special port consists of a 1 inch vertical tube extending from the top
center of the concrete shield, downward to the reactor and re-entrant into the
center of the reactor. Small samples may thus be placed at the center of the
actor
Attenuation of radiation by the shielding, and the levels of radiation
pected
in Table 5, in Section III, below.
The reactor at
solutions of uranyl sulfate and of uranyl nitrate as the nuclear fuel. From the
standpoint of neutron absorption, the sulfate has some advantage. From the
standpoint of solubility and corrosion rate, the nitrate appears to have a slight
advantage, particularly at high temperatures and pressures. It appears,
ever
reactor, and there is at present no strong reason for choosing one instead of the
other.
Uncertainty in the exact chemical composition and geometry of the fuel
tainer
and re-entrant tubes, and other factors of uncertainty make it impossible to
culate
known critical mass of the
ence
cooling coils and control rods, and estimating the effect of all other different
features, it appears that 715 grams of U235 in uranium of 90% isotopic purity,
as uranyl sulfate in water solution, are required to produce criticality at room
temperature. To this must be added the amount required to overcome the negative
temperature effect at operating temperature and the "working excess" needed in
experimental procedures.
A tabulation of various calculated quantities and characteristics relating to
the nuclear fuel is presented in Table 1.
TABLE 1 - CHARACTERISTICS OF THE REACTOR FUEL
| Estimated critical mass at room temperature | 715 gm U235 |
| Added U235 for temperature coefficient | 55 gm |
| "Experimental Excess" | 20 gm |
| Total U235 content | 790 gm |
| Solution density (9% UO2SO4 by weight) | 1.08 gm/cm3 |
| Hydrogen to U235 atom ratio | 450 |
Type 347 stainless steel is used throughout the reactor system wherever
tact
in preference to other means of connection.
A volume of 14 liters is provided in the reactor cylinder for the fuel solution.
In addition to the fuel volume, the reactor cylinder (Fig. 11) contains numerous
connecting and re-entrant tubes and an empty space at the top to allow for frothing
and expansion of the fuel solution.
Data relating to the fuel cylinder are contained in Table 2 below:
TABLE 2 - DATA RELATING TO FUEL CYLINDER
| (347 stainless steel) | |
| Diameter o.d. | 10-7/8 inches |
| i. d. | 10-3/4 " |
| Height, outside | 11-1/8 " |
| inside | 11 " |
| Wall thickness | 1/16 " |
| Weight of steel in walls | 2.3 kg. |
| Total inside volume | 15.6 liters |
| Inside volume occupied by re-entrant tubes | 0.71 liters |
| Weight of steel in inside re-entrant tubes | 1.9 Kg. |
| Liquid depth | 9.92 inches |
| Liquid volume | 14.0 liters |
| Depth of space above liquid | 1.08 inches |
| Volume of space above liquid | 0.90 liter |
In case a leak should occur in the fuel cylinder, radioactive vapor or liquid,
or perhaps both, depending on the location of the leak, would escape into the
rounding
the reactor is enclosed in an envelope of 1/16" aluminum. (Fig. 12)
All tubes, control rod sheaths, and thermocouple leads connect to the reactor
on its top surface, and project vertically upward inside the reactor envelope. The
motor drives and electromagnetic couplings of the control rods are located inside
the top portion of the reactor envelope. Tubes and electrical wires leave the
velope
The reactor envelope which is snugly fitted to the reactor is, in turn,
closed
Calculations show that a 20 inch layer of graphite around the reactor core is 90% as
effective in reducing critical mass as an infinitely thick layer. Two to four inches
of lead are placed around the graphite reflector, inside the concrete shielding, as
a primary gamma ray shield. (See Section III for calculations of attenuation.)
On one side of the reactor, outside the lead shielding, an additional 5 foot
cube of graphite forms the thermal column. (Calculated values of thermal
tron
and across the thermal column to provide means of using the neutrons for
perimental
A layer of lead four inches thick is placed across the end of the graphite
column. This is followed by a layer of boron to absorb the neutrons reaching
that point. The boron, in the form of finely. ground boron carbide, is impregnated
into a layer of paraffin 3/4" thick, in sufficient concentration to form a boron
layer of 3 gms/cm2 across the end of the column. A layer of concrete 12 inches
thick is placed outside the boron layer.
Specially prepared graphite, obtained from the
in the reflector and thermal column. Data relative to the graphite are presented
in Table 3, below.
TABLE 3 - DATA ON GRAPHITE USED IN REFLECTOR AND THERMAL
COLUMN
The graphite portions of the assembly are built up of successive layers of
graphite bars 4 inches square in cross-section and in lengths up to 48". All joints
are fitted to ± 0.002".
Due to the negative temperature coefficient of nuclear reactivity of the
actor
limited by the rate of heat removal from the solution. If less heat is being
moved
mal
is reduced so that a heat balance is established.
Dissipation of heat from the reactor through the external walls of the vessel
will hardly exceed half a kilowatt. For operation at higher levels, therefore,
auxiliary cooling of some sort must be provided. The system devised consists of
four symmetrically arranged coils of 1/4" (i. d.) stainless steel tubing immersed
in the reactor fuel. One gallon per minute of refrigerated water flows through
each coil. A 7 foot length of each coil is immersed in the solution. The wall
thickness is 1/32 inch.
Water from the city mains flows through refrigerated coils in the air
ditioned
manifold inside the reactor assembly and thence to the four reactor cooling coils
(Fig. 13). The coils are adjusted for equal flow before installation, but after
stallation
Temperatures are measured and recorded at the inlet manifold and at the exit of
each coil.
In its passage through the reactor, the cooling water becomes somewhat
active
amount will not be large and most of this will be short-lived. (Table 4) The
normal time required for the transit of an average water molecule through the
actor
calculation of the data presented in Table 4, a transit time of 2 seconds and a flux
of 1012 thermal neutrons were conservatively assumed. Average analyses of
water in the
ment
TABLE 4 - ACTIVITY INDUCED IN THE REACTOR COOLING WATER
| Target Isotope | Form of the purity | Avg. Conc. ppm | Active Isotope | Half Life | [lambda] sec-1 | [sigma]a barns | Abund. of Target Isotope (%) | N cm-3 | Result. Activity d/sec/cm3 |
|||||||||
| Si30 | SiO2 | 9.8 | S31 | 2.7h | 7.1 x 10-5 | 0.12 | 3.1 | 1.0 x 1016 | 0.017 | |||||||||
| Mg26 | Mg | 1.4 | Mg27 | 9.58m | 1.2 x 10-3 | 0.05 | 11.3 | 5.3 x 1015 | 0.64 | |||||||||
| Na23 | Na+K | 4.2 | Na24 | 14.9h | 1.3 x 10-5 | 0.6 | 100 | 1.4 x 1017 | 2.2 | |||||||||
| S36 | SO4 | 11.0 | S37 | 5.0m | 2.3 x 10-3 | 0.14 | 0.0136 | 5.0 x 1013 | 0.032 | |||||||||
| Cl35 | Cl | 4.9 | S35 | 87.d | 9.2 x 10-8 | 0.34 | 75.4 | 1.2 x 1017 | 10-2 | |||||||||
| Cl37 | Cl | 4.9 | Cl38 | 38.5m | 3.0 x 10-4 | 0.6 | 24.6 | 4.0 x 1016 | 14. | |||||||||
| F19 | F | 0.1 | F20 | 12s | 5.8 x 10-2 | 0.009 | 100 | 3.35 x 1015 | 3.5 | |||||||||
| O18 | H2O | 106 | O19 | 29.4s | 2.4 x 10-2 | 0.0002 | 0.204 | 6.8 x 1019 | 653 |
Negligible activities were found for a number of additional elements likely to be
present. These are:
Fe54, Fe58, Ca44, C13, S34, Cl35, N15
From this table, it is seen that the O19 activity (0.018 micro curies per ml),
initially, is dominant. Since the half life is only 29.4 seconds, however, this
activity will be reduced below 1 disintegration per ml/sec in 5 minutes, and thus
no hazard would be anticipated.
The next most significant activity is that of Cl38 (3.8 x 10-4 micro curies/ml)
which is of the order of
hazard would be expected to result from discharge of this material into the sewer.
Inasmuch as the power at which the reactor operates is intimately associated
with the rate of cooling, an interlock with the control rod system is provided
which insures that the cooling water is flowing before the reactor can be operated.
The exit line carrying the cooling water from the reactor passes through an
underground, concrete shielded trench under the building to an underground tank
of 250 gallon capacity located outside the building, and from there to the city sewer
system. Radiation detectors continuously monitor the activity of the water as it
enters and leaves the tank. The same plan of operation will be followed for this
system as that described in Section I, C, for the laboratory drainage system.
As in the drainage system, an attempt will be made to use a continuous flow method
of discharging the reactor cooling water, through a monitored tank, to the city sewer
system. But if the level of radiation proves too high, or if reliable monitors cannot
be found for the low level, of radiation involved, then another tank will be installed
and the two will be used alternately as hold tanks The cooling water will then be
discharged batchwise to the city system after sampling and analyses of each batch.
When the reactor is operating at 10 Kw, from 1500 to 2000 cc of gas per
minute are evolved. More than 99% of this gas volume will consist of gases
ing
gas volume will be fission product gases which, though negligible in volume,
greatly complicate the disposal problem, because of their high radioactivity.
Various possible methods of disposing of the gases were briefly outlined in
the previous report, "Program Administration and Installation Design of the Nuclear
Reactor Project at
this problem.
The scheme developed by Los Alamos and used thus far with complete
faction
for the
components has been incorporated for adaptation to the new reactor but the
mental
is based on a very ingenious combination of ideas and operating principles which
achieves freedom from explosion hazard, simplicity of apparatus and mechanical
components, low maintenance requirements, and satisfactory handling of
active
The gas disposal apparatus as adapted for the
consists of a closed circulating gas system, with a small exhaust gas bleed to the
stack and a small inlet make-up flow. The flow path (Fig. 14) includes, in
quence
with refrigerated water, a steel-wool-filled filter, a circulating blower, a
platinized-alumina catalyst bed, a water cooled steam condenser, and a pipe
nection
is filled with air (94%), and gases from the reactor (6%). Fifty to one hundred ml
per minute of mixed gases are bled from the circulating system through a series
of hold-up tanks to the stack.
A small volume of non-condensable gases from the reactor (SO, SO2...)
plus a small inflow of make-up air to the system is sufficient to maintain constant
circulating inventory. The inbleed of make-up air occurs to some extent around
the pump shaft, and the additional amount required is admitted through a small
adjustable leak located on the inlet side of the pump.
The essential features of this system are:
The details of construction and arrangement of the chief components in the
gas disposal system and other details are shown in Figures 15, 16. and 17.
About 50 to 100 ml per minute of gas are bled from the gas circulating-
recombining system during periods when the reactor is in operation. The point
of withdrawal (Fig 14) is located immediately downstream from the exit end of the
steam condenser where the H2-02 content is lowest.
The gas bled from the system cannot be discharged directly into the
phere
discharge into the atmosphere sufficiently to permit radioactive decay to a safe
level. Calculations showing amounts of gaseous fission products expected, their
decay characteristics, and the adequacy of the proposed hold-up system are
sented
The plan of the gas withdrawal system is shown in Fig. 18. A 1/4"
less
within the reactor shielding, then horizontally under the floor of the reactor room
and out underneath the building to a series of gas-holding tanks immersed in an
underground water tank (for shielding) outside the building The bleed gas is
drawn through the holding tanks by a small pump, and is discharged from the
pump into the 12,500 cfm ventilation exhaust stream from the Reactor Building.
A constant flow of gas is maintained through the system by a suitably sized
critical flow orifice located between the holding tanks and the exhaust pump. A
water trap, through which the gas must bubble, is located between each holding
tank. Eight-hold-up tanks in series, each having 100 gallon capacity, are,
vided
of eight such tanks is on the order of 700 for Xenon133, which is the most
troublesome gaseous fission product. By-pass valves are arranged so that any
tank can be removed from service without disrupting operation of the others.
The instrumentation system of the reactor facility has four functions:
(1) to indicate and record the level of radiation flux and the rate of increase or
decrease of the neutron flux in the reactor; (2) to provide data for and means of
safe manual and automatic control of the reactor; (3) to provide safety
isms
area and facilities monitoring to safeguard personnel from radioactivity
hazards and to prevent inadvertent release of radioactive materials.
The first three of these functions are performed by the instrumentation
system provided for the reactor itself. In general plan, this reactor
tation
located in the graphite reflector of the reactor assembly, which are connected
by coaxial cables and power lines to their respective power supplies, amplifiers,
indicating meters, recorders, and control devices located in the Control Room.
Operation and control of the reactor are handled entirely from the Control Room.
Twenty feet of space and a wall of 12 inches of masonry or 8 inches of water
tween
operator from stray radiation. Viewing advantage is furnished the operator, and
some added protection from stray radiation, by the floor of the Control Room
ing
Within the Control Room, there are two primary assemblies having to do
with operation and control of the reactor: The Rack of Data Recorders and the
Control Console (Fig. 19). In normal operating position, the operator sits at the
central panel of the Control Console, with the water-window to the Reactor Room
to his right. Directly in front of the operator, above and about 5 feet beyond the
central panel of the Control Console, the 12 data recorders are in full view. The
six central recorders, which handle data of primary importance, have illuminated
scales. Along the wall to the left of the operator are instruments and recorders
which indicate and collect data of secondary importance to the operation of the
reactor.
The fission chamber sensing elements of the instrumentation system are
similar in design to chambers successfully used with the homogeneous reactor at
0.002" aluminum foils which, in turn, are coated with a layer of U235 (0.35 Mg/cm2)
of above 90% isotopic purity. The rate of fissioning is proportional to the neutron
flux from the reactor, and hence the current from the fission chamber is directly
related to the power level of reactor operation.
It is anticipated, on the basis of
bers
a neutron flux of 1010 neutrons/cm2/sec.
In the reactor instrumentation system, there are three independent
channels having responses linearly related to the neutron flux. (Fig. 20).
(a) In the first linear channel, the fission chamber is connected directly,
without amplification, to a galvanometer located on the central operating panel of
the Control Console. The galvanometer sensitivity is matched to the fission
ber
range, and operation at 1 watt level produces full scale deflection on the most
sensitive range. There are 8 intermediate ranges.
Four meters, in series with the galvanometer and with each other, are
cated
reactor operation is visible to persons at any location in the Reactor Room.
These four meters have only one sensitivity and indicate 10 Kw at full scale.
(b) In the second channel with linear response, the fission chamber current
is fed into a preamplifier and then into a Brown Recorder. The level of radiation
flux is continually recorded. The sensitivity range of the recorder at any time is
identical to that being displayed simultaneously on the galvanometer of channel (a).
(c) The third channel with linear response is used in automatic control of
the Reactor. The output of the fission chamber is amplified and balanced against
the reference voltage from a Rubicon potentiometer. The latter voltage can be
controlled at will by the operator. The difference between these two signals, if
any, is amplified and applied to the motor which controls the motion of the reactor
control rod. The motion of the motor is always in the direction which would
crease
difference in signals is also displayed on a small cathode ray tube mounted on
the Control Console.
There are certain safety features common to each of the above systems. In
each system, a safety interlock is provided such that, in case the neutron flux
exceeds a pre-set level, the current to the electromagnets supporting the control
and safety rods is turned off and the rods are dropped. In addition, between
channels (b) and (c) there is a signal comparator which also causes the rods to
drop in case the initially matched signals differ at any subsequent time by a
set
not fail without knowledge of the operator.
It is necessary to provide for the operator a set of instruments in which
the flux over a very wide range of values can be shown. This is accomplished in
duplicate channels by feeding the output current of ionization chambers into
fiers
The logarithmic characteristic of a diode is used to obtain this logarithmic
ponse
(a) In the first channel, the ionization chamber is logarithmically amplified
and is then recorded on a Brown Recorder Safety features are described below.
(b) The second channel is an exact duplicate of the first, except the current
is not recorded.
In either of these channels, if the neutron level exceeds a pre-set value,
the current to the control and safety rods is interrupted and the rods are dropped.
In addition, between these two channels there is a signal comparator which also
causes the rods to drop in case the initially matched signals differ at any
quent
The currents from these two channels provide the input signals to the two
respective channels described below.
The "period" of a nuclear reaction is the time required for the power
level to increase or decrease by a factor of e. It is very important that the
period.
The logarithmically amplified currents from the two channels described
above are fed into parallel electronic circuits respectively, where these
mic
period of the reactor.
(a) The period as measured by the first of these channels is indicated and
recorded on a Brown Recorder. Safety features are described below.
(b) The second channel is identical to the first, except the output is not
recorded.
In either of these two period-measuring circuits, if the period becomes
shorter than a pre-set value, a thyratron interlock causes the safety and control
rods to drop. In addition, between these two period measuring circuits, there is
a signal comparator which also causes the rods to drop in case the initially
matched signals differ at any subsequent time by a pre-set value.
A Master Range Changing Switch is provided for the logarithmic and period
channels which permit simultaneous change from one range to another on these two
systems.
In the U235 fission and boron ionization chambers used in the channels
described above an indeterminate proportion of the current output is caused by
ionization of the gas in the chamber by gamma rays from the reactor. The neutron
fluxes indicated are therefore too high. To provide information on the magnitude of
this effect, a gamma-compensated channel is included in the reactor instrument
system. This channel consists of two identical "fission" chambers, except one
chamber contains no U235. The difference in the signals from these two chambers,
which is displayed on the Control Console, is a measure of the neutron flux
tially
information only, and is not connected to automatic safety mechanisms of the reactor.
While a nuclear chain reaction is in progress in the reactor, the neutron
detectors described above furnish adequate information on the level of radiation
ing
fission and the neutron detectors indicate cessation of activity. The gamma
tion
continues at a relatively high level. To follow the level of this activity after
down
mentation
connected to automatic safety mechanisms of the reactor.
This channel consists of an ionization chamber feeding through an amplifier
to a recorder.
The Control Console consists of a 3-section desk, each section being 24"
wide and set at an angle of 135o to its adjacent section (Fig. 19). The top of each
section slopes toward the operator position at an angle of 30o to the horizontal.
The operator is thus able to view the apparatus on each panel with maximum con-
venience and minimum parallax.
The central panel contains apparatus of primary importance in the operation
of the reactor. Included on this panel are:
The right hand panel contains the adjustment dials and galvanometer of the
Rubicon potentiometer used in automatic operation, the shim rod positioning
switches, shim rod extreme position indicating lights, and various meters
cating
The left hand panel contains pairs of red and green lights (red indicating
non-operating and green indicating normal operating condition) for numerous
auxiliary instrumentation systems; e.g.:
| Stack radiation monitor | Gas recombiner flow meter | |
| Campus radiation monitors | Recombiner coolant water flow meter | |
| Coolant radiation monitor | Reactor coolant flow meter | |
| Stack flow meter |
Also on the left hand panel are located the switches for trip-testing the
automatic safety mechanisms in the neutron measuring channels, the meters
indicating the recombiner gas circulation rate and the recombiner coolant
perature
The Recorder Rack contains 12 Brown Recorders. The data recorded on
each are as follows: (Single channel, curve-drawing instruments used unless
otherwise indicated.)
Two vertical boron rods moving inside of sheath tubes which are re-entrant
into the Reactor through the top surface, and two vertical cadmium strips located
on the external periphery of the reactor cylinder, provide the means of controlling
and adjusting the level of reactor operation (Fig. 22).
The rods are of thin-walled stainless steel tubing filled with boron powder.
The two rods and their remotely controlled actuating accessories are completely
independent, but are identical in construction and in relative reactor location, so
that the rods may be used interchangeably to perform respective functions as
Safety and Control Rods. When fully lowered, the rods extend (inside of their
re-entrant tubes) 9-1/2 inches into the reactor solution, to within 1/4 inch of the
bottom of the reactor. When fully raised, the lower ends of the rods are 1-1/2
inches above the top surface of the reactor. The length of travel is 12 inches.
With the motor at maximum speed, the rate of rod movement is 5 inches/minute.
This may be controlled at any lower rate desired.
The boron rods are connected to their respective motor-driven elevating
screws by a direct current electromagnetic coupling. If any one of the
ous
suspended rods are released and they fall by their own weight into the reactor.
A shock-absorber slows the rate of fall over the last 1 inch of travel. The
electronic safety circuits pre-set to release the rods and the electric circuit to
the magnet are so adjusted that the rods are completely released by the magnet
in 0.02 seconds after the initial signal appears, if the reactor is operating above
the 100 watt level. At lower levels, the time of release is somewhat longer.
The two boron rods are so connected by a micro-switch interlock that one
rod must be completely poised at its upper limit of travel before the other rod
can be raised. The excess reactivity in the reactor is so adjusted that either of
these two rods alone, completely inserted, will stop the chain reaction. Thus
one rod is always "cocked" in Safety Rod position before reactor operation can
be initiated by
The boron rods are located on a diameter of the reactor cylinder on alternate
sides of the center, each 2-11/16 inches (on center) from the center, with their
axes parallel to that of the reactor cylinder. The rods are thus at position of
proximately
of these rods, as compared to similar rods similarly placed in the spherical
Alamos
excess uranium, above that required to produce criticality, will always be less
than this amount.
The Shim "rods", two in number, and likewise independent in operation
and identical in construction, consist of 4" wide strips of 1/32" Cadmium, 10"
long, are located on the periphery of the reactor cylinder in a vertical position
(Fig. 22). The shim rods are positioned by a variable speed, motor driven
mechanism exactly similar to that of the control rods, except no electromagnet
coupling is provided between the motor driven screw and the shim.
In lowered position, the lower ends of the cadmium shim "rods" are even
with the bottom of the reactor cylinder. In fully raised position, the lower ends
of the shims are 1 inch above the top of the reactor. The length of travel is
12 inches; the maximum rate of travel is 5 inches per minute.
The shims are positioned with their centers 5-3/8 inches apart, on one
side of the diameter on which the control and safety rods are located, and
symmetrically spaced with respect to the Control-Safety Rod diameter. The
shims thus are each 64o from the Control and Safety Rods, and 52o from each
other. The Control-Safety Rod diameter is perpendicular to the direction of the
thermal column from the reactor, and the shims are on the opposite side of the
Control-Safety diameter from the thermal column. The shims are thus in
tion
flux entering the thermal column.
It is calculated, again, by comparison with the somewhat similar shims of
the
is "worth" 25 grams of uranium.
The shim rods are intended essentially to "normalize" the excess reactivity
of the reactor so that the control rod operates at its position of maximum
tivity
trol
or both shim rods (in succession) are withdrawn until a sustained reaction begins.
The reaction is thus achieved with the control rod at the desired position.
The shim rods are necessary because the excess reactivity of the reactor may
change from day to day as exposure samples are inserted or withdrawn, as
tor
frequent additions or removals of nuclear fuel might be necessary to keep the excess
reactivity adjusted to safe and workable limits.
It is not anticipated that enough radioactivity will be released from the
reactor facility to cause significant increase in the normal level of background
activity. To insure positively that this is the case, a series of instruments designed
to measure continuously the level of radioactivity at various locations on the campus
are provided. Part of this system was placed into operation in September, 1951, in
order that a continuous record of normal background radiation might be obtained
over a period of several months before the reactor is placed in operation.
Beta and gamma monitoring stations are located at 5 positions on the campus.
A mobile monitoring station will be used in making periodic check of the area. The
stationary positions are in a deliberately chosen pattern with respect to the reactor
location: Four positions are each about 550 feet from the reactor, in northeast,
northwest, southwest, and southeast directions, respectively. The fifth station is
1200 feet northwest of the reactor. The prevailing wind in this area is toward the
northwest; therefore there are two monitoring stations in the direction of the
vailing
pectively
Two types of monitoring instruments have been chosen: A G-M tube
meter
electrometer system. Two monitoring stations are equipped with one type and two
with the other. The fifth station, namely; the near station in the direction of the
vailing
Each electrometer is provided with an automatic zero drift
automatic calibrating device which operate at regular intervals. The latter consists
of a clock mechanism arranged to withdraw a known radioactive source from a
shielded position and place it in a pre-determined position near the detector. A
similar calibrating mechanism is provided for the G-M units. Thus, on the recorded
chart of the background radioactivity measured by each monitor, there also appears
an hourly check of the sensitivity of the instrument and a zero recalibration if any
drift has occurred. The recording apparatus for all monitoring stations is located
in the Control Room of the Reactor Building. Signals are sent from each outlying
station to central recorders so that the operator can be informed at all times of any
change in the level of radioactivity at any of the monitoring stations.
A number of radiation detecting and measuring instruments, in addition to
those described above, will be provided in the reactor facility. The following are
included;
When the reactor is fully assembled, at room temperature, with graphite
reflector in position, only removal of a shimor control rod) addition of nuclear
fuel, or substitution of Be or heavy water for some of the graphite reflector could
cause an increase in the reactivity. All other changes which could occur would
reduce the potential reactivity of the reactor. An increase in temperature,
sertion
mit
are all factors which would reduce the potential reactivity of the reactor.
The decrease in reactivity as the temperature increases is one of the most
important features. There are four factors which contribute to this overall effect:
(1) As the temperature increases, the fuel solution expands, the density decreases,
and the reactivity is reduced. (2) As the temperature increases, the average
energy of the neutrons, most of which are in thermal equilibrium with the atoms
of the solution, also increases. Increasing the average neutron velocity causes a
larger percentage to be captured (without resultant fission) by the U238 atoms
present, because of the large resonant absorption cross-section in U238 at 7 ev.
This tends to reduce the reactivity, though for small temperature increases and
for high U235 enrichment this effect is small. (3) If the temperature increases
due to an increased rate of energy release by fission, the total radioactivity in the
fuel solution increases and, a proportionally larger amount of decomposition
gases--mostly H2 and 02--are released throughout the volume of the solution.
The bubbles of these gases rise rapidly to the top, but their presence in the
tion
activity
solution, vapor bubbles of the solution are formed. These bubbles also lower the
solution density and decrease the reactivity. A steady state temperature in excess
of the boiling point of the solution cannot be achieved, since the pressure is
tained
observed for the Los Alamos Homogeneous Reactorf1 varies from 0.7 gm U235/oC
at low temperature and low power to about 1.25 gm U235/oC at higher powers and at
temperatures near the, boiling point. The temperature coefficient for the Raleigh
Research Reactor is expected to have similar values.
As water is removed from the reactor, by evaporation, entrainment, or
composition
of hydrogen atoms to uranium atoms, initially adjusted to optimum, deviates from
optimum, decreasing the reactivity. In fact, whether water is added or removed,
the reactivity tends to decrease. The optimum H/U235 ratio is not a sharply
fined
from H/U235 = 300 to H/U235 = 500, without great effect on reactivity. Hence,
rather large amounts of water must be added or removed before this factor
creases
uranyl sulfate tends to precipitate out of solution. This changes the homogeneous
distribution of U235 in the solution, and any such change causes a decrease in
activity
A change in the acidity of the fuel solution resulting from radiation
position
precipitation than is the removal of a small amount of water. Hence, the acidity
of the solution is checked frequently and adjusted as necessary.
It is estimated that 715 grams of U235 at 90% isotopic purity are required
for criticality at 20oC. About 770 grams are required at 80oC, the normal
erating
moved
grams in excess of the critical amount at room temperature and 20 grams excess
at operating temperature. The control, and safety rods are each "worth" 80
grams of U235, and the shims 25 grams each: a total of 210 grams. Thus either
control rod alone can control the total excess reactivity with the reactor at room
temperature, and any shim or control rod alone can absorb the excess with the
reactor at operating temperature.
It is of interest to explore the behavior of the reactor if all interlocks and
instrumental safety devices should fail as the control rod is being removed.
Consider first the case in which the reactor is at room temperature and the rod
is removed slowly, so that nuclear and thermal equilibrium are maintained.
Case 1. Slow removal of control rod.
As the rod is removed, the nuclear reaction begins, heat is released, and
the solution temperature increases. The negative temperature coefficient tends
to reduce reactivity, but as the rod continues to be removed, this effect is
ridden
will be removed by the cooling system of the reactor, but if sufficient excess
uranium is present, the amount of heat released as the rod removal continues
will eventually be more than the cooling system can remove, and the
ture
If enough excess uranium is present in the reactor to permit the reaction
to continue, despite the negative temperature coefficient, at the boiling point of
the solution, then, as the rod is removed, the temperature will increase until the
liquid boils. If the rod is removed further, more vigorous boiling will
Eventually, due to "evaporation" of water or decomposition of acid, the reaction
would cease. Vigorous boiling would probably result in large fluctuations in
activity
If the excess uranium present in the reactor is not sufficient to over-ride
the negative temperature coefficient up to the boiling point, as will be the usual
situation, then, as removal of the rod causes release of more heat than the
ing
the k of the reactor drops below 1. As the reactor cools slightly, the reaction
recommences, the temperature increases and k again drops below 1. Thus,
the temperature oscillates around the critical value with reactor operating at
such level that the rate of heat released is just equal to that removed.
It is thus clear that the steady state amount of power released by the
actor
that about 0.5 Kw of heat can be dissipated from the
conductivity and radiation into the reflector, with no auxiliary cooling. The
ing
when the reactor is at 80oC.
Case 2. Fast Removal of the Rods.
If the rods and shims are removed instantaneously from the reactor, while
the temperature is at 20oC; an excess reactivity of 0.016 results. With this
cess
e-folding time of about 0.02 seconds (and a total elapsed time of about 3-1/2
seconds) before the negative temperature coefficient can begin to control the
power downward. Another 1-1/2 seconds are required to reduce the excess
activity
In making these estimates, a constant negative temperature coefficient of
2.0 x 10-4/oC is assumed, and no consideration is given to the effect of bubble
formation in the solution. It is known that bubbles will be formed, and that more
favorable temperature coefficients will exist over a portion of the event. Both
of these factors would tend to slow the rate of rise, lower the value of peak
power achieved, and hasten the return of the power level to nominal values.
The net results of an event of this kind would be (1) the temperature of the
solution would be increased, probably to boiling, and (2) the levels at the outer
surface of the reactor shield would flash to a few thousand times continuous
posure
tainly
If all the rods and shims are removed from the cold reactor (20oC) at a
uniform rate in a total of 5 second, (the normal time required is 2 minutes) a
flash up to a power level of 14 megawatts results. (Fig. 23). As above, a
conservative value of the temperature coefficient and no effects of bubbling are
assumed. The flash up and return to normal levels again occurs within three or
four seconds. As shown earlier, with slow rate of rod removal (i.e., in times
longer than 15 or 20 seconds) the power rises smoothly to equilibrium value with
no flash up.
It is anticipated that the normal maximum power level of operation of the
reactor will be about 10 Kw. Calculations have been made of the radiation fluxes
to be anticipated at various points in and around the core, when the reactor is
erating
TABLE 5 - CALCULATED RADIATION FLUXES AT 10 Kw. OPERATION
| Fast n | Slow n | Gamma | ||||
| (n/cm2/sec) | (n/cm2/sec) | ([gamma]/cm2/sec) | ||||
| Inside Core | 2 x 1011 | 5 x 1011 | 6.7 x 1011 | |||
| Surface of Reactor Envelope | 7 x 1010 | 3 x 1011 | 4.5 x 1011 | |||
| Surface of Reflector (20" from reactor) | 8 x 109 | 8 x 1010 | 1.8 x 1010 | |||
| Outside 4" lead shielding | " | " | 1.1 x 108 | |||
| Four feet from center of reactor on axis of thermal column | 2.5 x 107 | 1.4 x 1010 | 6.0 x 107 | |||
| One foot from end of thermal column (on axis) | 2.0 x 105 | 1.5 x 109 | 1.2 x 106 | |||
| Outer surface of concrete at end of column | 70 | |||||
| Average radiation over outer surface of concrete shield | 0.8 | negligible | 0.15 |
After the reactor has been in operation for some time, the solution will
possess a considerable amount of radioactivity, due to the build up of fission
ducts
how much radioactive gas is evolved.
If the reactor were operated for a known time at a known power level,
culation
For intermittent operation at random levels and for various lengths of times,
calculation of the composite accumulation of non-volatile and gaseous products
becomes impossible. To determine the upper limit of activity to be expected,
a schedule of operation was postulated which results in a build up of activity not
expected to be exceeded by any actual schedule of reactor operation. The
tulated
(1) The reactor is operated at 10 Kw for the first 6 hours of every
cessive
period. (At times, the reactor may be operated continuously for periods
siderably
by correspondingly longer "down" periods, however, so that the average
sult
produced by the schedule described.)
(2) Assume that the gas disposal system operates as described in Section
II G for the 6 hours that the reactor is in operation, plus 1 additional hour after
shut down, and then does not operate over the remaining 17 hours of each 24
hour period. The' gases which accumulate in the reactor during the 17 hour
"down" period, however, are withdrawn during the subsequent operating period
along with those produced during the period.
(3) The refrigerated condenser of the gas disposal system lowers the gas
temperature to 13oC, and only those fission products remaining volatile at this
and lower temperatures escape from the reactor.
Two questions are asked: After the reactor has operated for many days
on this schedule, so that equilibrium has been reached, (a.) what is the level of
the radioactivity in the reactor solution, and (b) what is the quantity of
active
period?
(a) Radioactivity of the Fuel Solution.
According to information presented in an article by K. Way,a8 the amount
of radioactive energy released by the fission products in a reactor immediately
after shutdown, after the reactor had been operating for an extended period at a
10 Kw power level would be 2.6 x 1015 mev/sec. It is difficult to interpret
this in terms of curies. A rough approximation may be made by assuming that
one beta and one gamma, each of 1 mev energy are given off in each "
tion
It may be shown by rather simple calculations that the fission product activity
of shutdown in a reactor operating on a regular schedule of 6 hours per day at a
10 Kw level is about 75% of the activity at shutdown in a reactor operated
uously
Reactor is about 2. x 1015 mev/sec., or about 26,000 curies. The decay in
activity after shutdown is rapid. After 18 hours, the normal down time in the
postulated daily schedule, the activity would be only about 2.6 curies.
It is of interest to note the thermal effects of the fission product activity
after the reactor is shut down. If the reactor cooling water should cease to
circulate when the reactor should be closed down, low much increase in
ature
For data presented in the above mentioned article,a8 it may be shown that
the thermal energy resulting from the beta and gamma activity of fission
ducts
where t is the time in seconds after shutdown, and To is the time
in seconds the reactor had operated prior to shutdown.
By application of Way's relation for activities at various times after
down
reactor, the rate at which thermal energy is imparted to the fuel is
If the reactor should be at 80oC, 500 watts for about 15 minutes (
ing
estimated, on the other hand, that about 500 watts of power will be lost from the
reactor by condensation conduction when the temperature is at 80oC. Therefore, heat
from the fission products would hardly cause any increase in temperature, and
certainly would not cause the solution to boil.
(b) Daily Gaseous Radioactivity Produced.
Most of the primary and the daughter elements produced in the fission
process are neither gases nor vapors at the temperatures prevailing in the reactor.
A few, however, are sufficiently volatile to escape from the 80oC fuel solution in
the reactor. A careful examination of the, list of product elements leads to the
clusion
many times more activity than all others. Without the use of the refrigerated
denser
a considerable portion of the iodine might find its way out of the reactor solution,
through the disposal system, and out into the atmosphere. This would probably
necessitate a revision of the disposal system, for the tolerance concentration of
radioactive iodine in the atmosphere is very low. (10-7 micro curies/cc)
Available evidence indicates, however, that the iodine will be returned to the
actor
system.
This leaves the daughter product, Xe133, to be handled in the gas disposal
system, for all xenon produced would be expected to escape from the reactor. The
calculation of the amount of Xe133 produced by the intermittent reactor operation
schedule described above is performed by essentially standard methods. The
growth-decay equation
is solved for the number of atoms of the principal primary source (I133, having a
22 hr. half-life) resulting from a single 6 hour operating period. Here g, the rate
of generation by fission, is proportional to the power level. The contribution of
all previous days is added by a series method, with the effect of decay accounted
for, to yield the equilibrium level of iodine. The gaseous Xe133 (5.3 day half-life)
that is produced by the iodine over a 24 hour period may then be determined. No
account is taken of the distribution of fission products by atomic number, and the
predecessors of I are taken to have effectively zero half-life. An outline of the
calculation is given below. For a given operating period, say from time zero to
time [tau], the solution of the above equation is
For the down-time between [tau] and 1 day, the equation reduces to
Since the contribution to the number of previous days' operation is given by the latter
expression with time t - [tau] + 1, t - [tau] + 2, etc. inserted, a geometric series
maybe written. This is summed by the relation l + x + x2 + ---- = 1/1-x.
During the period of reactor operation, an additional growth term must be added
to this solution.
The equilibrium iodine level is different from that for continuous operation
N1 = g/[lambda]1 by a factor
tions
derived from the solution of the equation
where the last term is the fluctuating supply from iodine decay.
For the chosen pattern of reactor operation, i.e., 6 hours on and 18 hours
off in each 24, the equilibrium level of I133 is computed to be around 365 curies,
and, hence, the daily Xe production is 11 curies. To reduce the 11 curies of
Xenon activity to a value below 1 curie, a hold up of about 4 half-lives, or 21 days,
is necessary. The system designed to accomplish the desired hold-up of the Xenon
actually achieves reduction of the activity to a value many times lower than this.
In this system, the gas from the recombination unit is bubbled through a water
trap to the first of several holding tanks arranged in series. A water trap is
cated
these traps. The delay of the gas in reaching the stack after traversing this system
of hold-up tanks is calculated according to the analysis below.
Three types of holding systems are considered: (1) a single long continuous
tube, (2) a single large tank, (3) a sequence of small tanks.
Since the effectiveness of a given holding volume can be shown to be dependent