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fA_«EB RESEARCH ANIIIFIIIE!EI.IIPMENT REPORT ORNL-1508
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DESIGN CALCULATIONS FOR A MINIATURE HIGH-
TEMPERATURE IN-PILE CIRCULATING FUEL LOOP
“ea
M. T. Robinson and D, F. Weekes \ A1
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ORNL-1808
This document consists of 31 pages.
Copy’ 36 of 209 copies. Series A.
Contract No, W-7405-eng-26
SOLID STATE DIVISION
DESIGN CALCULATIONS FOR A MINIATURE HIGH-
TEMPERATURE IN-PILE CIRCULATING FUEL LOOP
M. T. Robinson and D, F. Weekes
With An Appendix on Analog Simulation by
E. R. Mann, F. P. Green, and R, S, Stone
Reactor Controls Department
DATE ISSUED
SEP 19 1955
OAK RIDGE NATIONAL LABORATORY
Operated by
UNION CARBIDE NUCLEAR COMPANY
A Division of Unioh Carbide and Carbon Corpeoration
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CONTENTS
Development of Loop Model ... 1
Basis of Heat Transfer Calculations .......ccooiiiiiiiiicieini i 3
Derivation of Heat Transfer EQUations ..o 4
Solution of Heat Transfer Equations ..o, 7
NUMEEICA] DB . oottt et s et e ea e sat s sae s smae b e b e b e e e e e s s b rna e pat e 8
Selection of Best Loop Model .........c.ooviiiiriniciii 8
Results of Design Calculations for the Mark VI Loop ... 10
Miscellaneous Calculations for the Mark Y Loop ... 11
Appendix A. Importance of Reynolds Number in Mass Transfer ..., 16
Appendix B. Analog Simulation in an In-Pile Loop—Liquid-Salt Fuel Study .......ccccoonviviminninn, 16
Appendix C, Time-Constant Modification through Positive Feedback ..o 25
DESIGN CALCULATIONS FOR A MINIATURE HIGH-TEMPERATURE
IN-PILE CIRCULATING FUEL LOOP
M. T. Robinson and D. F. Weekes
DEVELOPMENT OF LOOP MODEL
The calculations presented in this report have
been carried out as part of the development of a
small circulating fuel loop designed for a study
of the effects of reactor radiation on the inter-
action of fused fluoride fuels with container
materials, and particularly with Inconel.! One of
the principal purposes in making a small loop is
to simplify construction and operation of the
equipment to the extent that many loops can be
run under a variety of conditions. Another con-
sideration is that a small loop can be examined
more easily after irradiation. Finally, a suf-
ficiently small system can be inserted into any
one of several vertical irradiation facilities in
either the LITR or the MTR. This will allow high
thermal- and fast-neutron fluxes to be used and
will simplify experimental work by eliminating the
need for elaborate beam-hole plugs.
Two important considerations governed the
choice of the basic design. Safety of both the
reactor and the personnel dictated that no portion
of the loop be left uncooled. This is especially
important in the region of high thermal-neutron
flux, since, in case of fuel stagnation, the temper-
ature of the fuel will rise at a very great rate
(650°C per second for fuel composition 30 in the
MTR is a typical figure). Since the time neces-
sary for taking appropriate corrective action is
appreciable, a considerable amount of heat should
be extracted from the high-flux regions of the
loop to avoid melting of the container and release
of fuel and fission products into the cooling air
or into the reactor itself. It is impractical to use
two cooling circuits so that there will be one for
emergency use only, since such an emergency
circuit cannot be operated sufficiently rapidly.
The second consideration was simplicity, both of
construction and of design calculation.
1Eor details of the mechanical design, development
tests, etc., see reports by J. G. Morgan and W. R. Willis,
Solid State Semiann. Prog. Rep. Feb. 28, 1954, ORNL-
1677, p 30; W. R. Willis et al., Solid State Semiann,
Prog. Rep. Aug. 30, 1954, ORNL-1762, p 44 ff.
As a result of these considerations, the basic
design chosen was a coaxial heat exchanger, bent
into a close-limbed U-shape. Fuel is pumped
through the inner member in a loop closed by the
pump, and cooling air is forced through the sur-
rounding annulus. Studies have been made of
several different ways of arranging the cooling-air
circuit relative to the fuel circuit, as illustrated
schematically in Fig, 1. Detailed calculations
to be presented later show the marked superiority
of the type 3 arrangement, when viewed as to the
amount and pressure of air required to maintain
the desired operating conditions.
Studies have been made also of the effects that
the diameter of the fuel tube and the length of
the loop will have on the thermal behavior of the
UNCLASSIFIED
SSD-A-1052
ORNL-LR-DWG-3772
Uy
Ty
INNER LOOP SHOWS DIRECTION OF FUEL FLOW, OUTER LOOP
SHOWS DIRECTION OF AIR FLOW
Fig. 1.
Models,
Air Flow Patterns in Miniature Loop
TABLE 1, SUMMARY OF LOOP MODELS
Inside Diameter of Outside Diameter of Inside Diameter of a ) .
Mark No. Fuel Tube Fuel Tube Air Tube Length Cooling-Ar
(in.) (inJ) (in.) (cm) Type .
IA 0.100 0.200 0.500 100 1
1B 0.150 0.250 0.500 100 1
NA 0.100 0.200 0.500 120 1
1B 0.200 0.300 0.600 120 1
ne 0.100 0.200 0.500 200 1
Iv 0.200 0.300 0.600 200 1
v 0.200 0.300 0.600 d 1
7 0.200 0.300 0.600 200 4
Vil 0.200 0.300 0.600 400 1
Vil 0.200 0.300 0.600 200 3
IX 0.200 0.300 0.600 200 2
%The length given is the value of the quantity S, defined in section on *'Solution of Heat Transfer Equations.’” The
length of the U is about one-half this value.
bSee Fig. 1.
“Mark 11l was inserted to the bottom of the active lattice.
d
Mark V was unsymmetrical; entering the reactor, the length was 60 cm; leaving, it was 140 cm.
system. The air-annulus spacing was not usually
SSD-B-1064
ORNL-LR-DWG-3784 .
I — — T varied but was kept as small as seemed .
- MARK Y1 \ ] .. .
T — ] consistent with ready fabrication. Only a little
N —— MARK ¥ attention has been paid to positioning of the loop
€ S ———————— , ¥, ¥, AND : :
5 ——————— ARk TA ANbTg O L AND X relative to the thermal-neutron flux in a reactor,
¢ C—————— MARK 14 AND 1B since one such study showed that little experi-
3 MARK I .
L o™ mental advantage accrued from a change in po-
x .. . - .
3 3 1 — sition. Table 1 summarizes the dimensions and
u /] AR—VERTFCAL MIDPLANE . . .
z /Y o TR other pertinent data on the 11 configurations
2 - - - -
5 / | 0P OF AGTIVE examined in this report. The locations of the
z / LATTICE . .
2 1 L \ various models with respect to the thermal-
z e o
z [ neutron flux distribution in position C-48 of the
£ o Ll LITR h in Fi
O 2¢ 40 60 80 100 120 140 160 180 200 220 =240 are shown In Flg. 2,
DISTANCE f{cm)
Fig. 2. Location of Miniature Loop Models in
Position C-48 of LITR Compared with Thermal-
Neutron Flux Distribution.
BASIS OF HEAT TRANSFER CALCULATIONS
The calculation scheme detailed in the following
section is based primarily on the principle of the
conservation of energy in a system which is in
a steady (time-independent) thermal state., How-
ever, a number of simplifying assumptions are
required, both to allow derivation of appropriate
differential equations and to permit their useful
solution.
In the derivation of the equations, two important
assumptions are made. First, all heat is assumed
to be removed into the cooling air. Transfer of
heat by radiation from the outer wall of the fuel
tube and by conduction to external parts of the
system is specifically neglected. A correction
could be made for radiative heat transfer, but this
does not seem to be worthwhile in view of the
approximate nature of the entire calculation. The
second important assumption is that all heat flows
radially out of the fuel tube; this assumption is
at least reasonable, since the thermal resistance
radially through the fuel-tube wall is certainly
small compared with the axial thermal resistance.
Furthermore, the radial temperature gradient is
expected to be far greater than the axial one.
These two assumptions result in a somewhat
exaggerated calculation of the fuel temperature
profile and an overestimation of the amount of
cooling air required.
In the course of solving the heat transfer
equations, it is necessary to calculate the heat
transfer coefficients which govern the flow of heat
between fluid and container wall. For this calcu-
lation the customary empirical correlations de-
veloped by engineers for heat exchanger design?—4
have been employed. These relations apply only
to cases where so-called established flow con-
ditions prevail in the fluid. Away from the
entrance to the cooling annulus, these conditions
prevail in the cooling air, but the velocity distri-
bution is changed because of the large rate of heat
transfer.® This will result in somewhat greater
turbulence in the air stream and, perhaps, in larger
2See, for example, W. H. McAdams, Heat Trans-
mission, 2d ed., McGraw-Hill, New York, 1942.
3M. Jakob, Heat Transfer, Wiley, New York, 1949.
4). G. Knudsen ond D. L. Katz, **Fluid Dynamics and
Heat Transfer,’’ Engr. Res, Inst. Bull. 37, Univ. of
Mich,, Ann Arbor, 1954,
Slbid., p 45 ff.
The situation in the
fuel is more complex, due to the presence of the
large volume heat source. It seems likely that
this will cause a substantial increase in turbu-
lence in the fuel and will probably increase the
heat transfer coefficients between the fuel and
the tube wall.
It has also been assumed that the physical
properties of air and of fuel could be regarded as
being independent of temperature. A detailed
justification of this assumption is presented in
connection with the solution of the equations.
In any case, it is felt that the importance of this
assumption is minor, especially in view of the
large uncertainty (+25%) in the fission power
generated in the fuel,
heat transfer coefficients.
To aid in interpreting the results of the calcu-
lations which were made at ORNL, certain design
criteria were adopted. Conditions which matched
as closely as possible the behavior of an actual
reactor would have been attractive; however, we
thought it more important to design an experiment
sufficiently flexible to allow a real analysis of
the effects of several variables on the interaction
of the fuel and the container. The important state
variables are believed to be the flow velocity of
the fuel, the intensity of the fission heat source,
and the temperature range through which the fuel
moves. These variables are not all independent.
in particular, for a given fuel flow rate, the
fission heat source largely determines the temper-
ature range through which the fuel moves. The
flow rate of the fuel is specified in terms of its
Reynolds number, Ref (see Appendix A), the
intensity of the fission heat source in terms of
P,, the fission power generated per unit volume
of fuel at the maximum thermal-neutron flux (in
terms of the notation in the section entitled *‘Deri-
vation of Heat Transfer Equations’’ P, = BP P max)s
and the temperature range experienced by the fuel
in terms of AT, the difference between the
maximum and minimum values of the fuel temper-
ature., We have attempted to design a loop which
would have the values
Ref Z 3000 ,
Py 2 1000 w/cc ,
AT, 2 100°C ,
and which meets the requirement that all fission specified conditions can indeed be met in a
heat be removed in the loop proper so that no variety of ways, provided an ample supply of
additional heating or cooling of the fuel would be cooling air is available.
necessary in the pump. It was found that the
DERIVATION OF HEAT TRANSFER EQUATIONS
An element of a concentric tube heat exchanger external sink. The purpose is to find the steady-
is shown in Fig. 3. Fission heat is generated in state temperature distribution in the system.
the liquid fuel flowing in the central tube, is trans-
ferred through the wall to the air flowing in the The symbols employed in the calculations are
annular space r, < r < 74, and is removed to an defined below.
Nomenclature
Coordinates and Dimensions
s = loop axial coordinate (the loop is defined by 0 § s S $)
r = loop radial coordinate
7y = inside radius of inner (fuel) tube
Ty = outside radius of inner (fuel) tube
Ty = inside radius of outer (air) tube
Physical Properties =
Cpf = specific heat of fuel at constant pressure
Cha = specific heat of air at constant pressure -
kf = thermal conductivity of fuel
k, = thermal conductivity of air
kl = thermal conductivity of Inconel, averaged over T, 272 T,
By = viscosity of fuel
K, = viscosity of air
Py = density of fuel
p, = density of air
Other Variables
v, = linear velocity of fuel, averaged over g 2
. . . < <
v, = linear velocity of air, averaged over T, =1 214
- 2
W, = fuel current, a7y
. 2 _ 2 .
W, = air current, 7r(r3 1)V, P,
b] = heat transfer coefficient at r = "
b, = heat transfer coefficient at r = ro -
Tf = fuel temperature, averaged over r ,S_ T
Other Variables {continued)
T? fuel temperature at s = 0
1A
T = air temperature, averaged over ) § r
a 3
Tg = air temperature at s = (
T, = temperature of fuel tube at r = 7,
T, = temperature of fue! tube at r = )
y = heat removed radially from fuel per unit time and unit of s
¢ = thermal-neutron flux
3 = fission power generated per unit thermal flux and per unit fuel density
PO = fission power generated per unit volume of fuel at the maximum thermal-neutron flux
Abbreviations
=Yy T Y2t Y3
G, = 1/We, = 2/mr k;(Prf) (Ref)
Ay = V/Wye,, = 2/m(ry + 79) k4 (Pra) (Rea)
By = mrpB/W,c, = 20,Bp;/k; (Prf)(Re)
yy = V27r by = Vmk(Nuf)
Yo = (nry/r)/ 2k,
Ya = V2mrphy = (r3 ~ r9)/mrok , (Nua)
a; = (A = a,)/a,
a3 = B/
a, = (GL3 + az)/a]
Dimensionless Heat Transfer Numbers
(Rea) = 2{(r, - 72)PaV o Pa Reynolds number of air
(Ref) = 2r, pfvf/p.f Reynolds number of fuel
(Pra) = poe,q/k, Prandtl number of air
(Prf) = 'ufcpf/kf Prandtl number of fuel
(Nua) = 2(ry — 1,) by/k, Nusselt number of air
(Nuf) = 2r\h,y/k; Nusselt number of fuel
(S¢f) = b1/pfcpf”f Stanton number of fuel
In the steady state, in unit time, a mass of fuel taking with it a quantity of heat
W, enters the fuel tube of the element at temper- W/Cpf[Tf + (dT /ds) ds] .
ature ij bringing with it a quantity of heat
W/c T relative to T, = 0. The same mass of The fission heat generated in the element is
fuelp leaves at a temperature T, + (dT /ds) ds, ,qupfm-% ds. A quantity of heat y ds flows radially
UNCLASSIFIED
$50-A-1053
ORNL-LR-DWG-3773
Fig. 3. Element of a Cylindrical Heat Exchanger.
out of the fuel tube into the annulus. Conservation
of energy then requires
de
(]) flfffigfipf - waP/_ -y =’0 .
A similar argument can be made for the cooling
air in the annular element, which receives a
quantity of heat y ds in the steady state in unit
time, If the air flows in the same direction as the
fuel, then
aT,
(2q) Y = Wiep, =0 .
If the direction of air flow is opposite to that of
the fuel, then
dT
a
(26)
y+Wac =0,
ra 4
For the transfer of heat from the fuel to the tube
wall at 7y and from the tube wall to air at Tos
Newton’s law of cooling may be written in the form
(3) y = err]b](Tf—Tl) = 2ar,b,(T, ~T,) .
The usual relation governing radial conduction of
heat in a cylinder may be written for the temper-
ature drop through the wall of the fuel tube as
2mk (T ~ T,)
4 -
) Y In (r2/r|)
If the abbreviations given above are introduced
into Egs. 1 through 4, the following set of
equations results:
dT/
(5) o B¢ — oy
aT,
(6a) —— = a,y (parallel flow of fuel and air) ,
ds
dT,
(66) —— = ~a,y (counter flow of fuel and air) ,
ds
(7) Tf - T, =y,
(8) T'| - T, = YaY
(9) T, - T, = y3y -
The set of Eqs. 5 through 9, under appropriate
boundary conditions, describes the dependence of
the various temperatures on the physical properties
of the fuel and of air, on the several state
variables, and on the geometry of the system.
The solution of these equations is difficult
because of the dependence of the various coef-
ficients on temperature. On the other hand, the.
specific heats of ionic liquids (fused salts) are
generally independent of temperature. Therefore
it can be assumed that a, is constant, since
conservation of mass requires that W, be constant,
regardless of changes of temperature or of
dimensions of the system. The quantity @, varies
only as does the specific heat of air. This
variation, about 6% from room temperature to
800°C, may safely be neglected because of other
approximations made in the calculations, par-
ticularly if Cpa 1S evaluated at the mean air
temperature,
The lack of sufficient data demands that k/ be
taken as constant. Since the variation of p, with
temperature is small (about 3% per 100°C), it, too,
may be taken as constant. If, in addition, changes
of the dimensions of the system with temperature
are neglected — an assumption justified by the
very small coefficient of expansion of Inconel — the
quantities y, and y, may also be assumed to be
constant. (An average value for the thermal
conductivity of Inconel has already been used;
see Nomenclature above,) Also, Y; may be con-
sidered to be constant, if b, is evaluated at the
mean air temperature. The variation of b, with
temperature at constant mass flow is not large,
NUMERICAL DATA
Heat is transferred by forced convection in a
cylindrical exchanger in accordance with empirical
relations of the type
(Nu) = [[(Pr), (Re)]
obtained by dimensional analysis of experimental
data, The systems with which we are concerned
here are characterized by large ratios of length to
diameter, and, as was stated earlier, the physical
properties of the fluids are evaluated at suitable
mean temperatures and then treated as constants,
For the case of laminar flow, use has been made
of the formula derived by Seider and Tate:?
2,1 1/3
(15) (Nu) = 1.86 [(Pr) (Re) T} (Re <2100)
For fully turbulent flow, the expression proposed
by Dittus and Boelter® has been used:
(16) (Nu) = 0.023 (Pr)2/5 (Re)4/5 (Re >10,000) .
In the intermediate region of incompletely de-
veloped turbulence (2100 < Re < 10,000), the
suggestions of Sieder and Tate’ have again been
followed. The relation they recommend is presented
in a graphical form which has been reproduced in
various places.? Under the special assumption
used here as to invariability of physical properties
with temperature, this plot permits determination
of (St} (Pr)2/3 for any (Re) and any length-to-
diameter ratio. From this, (Nz) has been calcu-
lated through the relation
(17) (Nu) = (St) (Pr) (Re) .
The physical-property data used in the analog
calculations were obtained from various sources.
The data for the physical properties of air are
those derived from McAdams.'® The physical
properties of fluoride fuel 44 were obtained from
Poppendiek!! of the Laboratory. The thermal
conductivity of Inconel is from the data of
Haythorne.'2 The data are summarized below.
The thermal-neutron flux used is that measured
in position C-48 of the LITR.'3 (In earlier calcu-
lations, values slightly different from those
shown below were used for ¢/, oy and p;. The
difference is only significant m the case of/ Cpf J)
Air (200°C)
B, = 0.00025 g/cm-sec
Cpa = 1.05 joules/g-°C
k, = 0.0004 w/em:°C
Fuel 44 (815°C)
He = 0.072 g/cmssec
1.00 ioule/g-OC
k; = 0.0225 w/em-°C
p; = 3.28 g/cm’
B = 54.0 x 10713 joule-cm?/g
Incenel
k, = 0,250 w/cm:°C
7E. N. Sieder ond G. E. Tote, Ind, Eng, Chem. 28,
1429 (1936).
8F. W. Dittus and L. M. K. Boelter, Univ. Calif.
(Berkeley) Publs, Publs. Eng. 2, p 443 (1930).
See, for example, ref. 3, Fig. 26-1, p 549.
IOSee, for example, W. H. McAdams, Heat Trans-
mission, 2d ed., McGraw-Hill, New York, 1942.
My, F, Poppendiek, private communication te D, F,.
Weekes.
12 A. Haythorne, Iron Age 162, 89 (1948).
M. T, Robinson, Solid State Semiann, Prog. Rep.
Feb. 28, 1954, ORNL-1677, p 27. See also Fig. 2 of
this report.
SELECTION OF
Some typical results obtained for the Mark IA
configuration are shown in Figs. 4 and 5. While
none of the curves shown meet the desired
boundary condition (Eq. 13), several conclusions
can be drawn. If (Ref) is sufficiently high for
some flow turbulence to be assured, AT/ will be
very small, say 10°C or less. In order to get
BEST LOOP MODEL
valves of AT, near 100°C, it is necessary to use
very small flow velocities, well within the laminar
region. It is clear that insufficient fission heat is
available, If the fuel tube is increased in size, a
given value of (Ref) will be attained at lower
linear velocity, allowing fuel to spend a greater
time in the high-neutron-flux region of the loop.
This change was made in the Mark 1B and Mark IIB
models. In the latter case, the problem now was
removal of the fission heat. Apparently, too
little heat exchange capacity was available, This
was remedied by increasing the length of the loop,
forming the Mark IV configuration,
A study was next made of the effects of the
cooling-air pattern. Four loop models were used,
one for each of the patterns shown in Fig., 1.
The total quantity of air was held constant, giving
lower values of (Rea) for the two cases with a
divided cooling annulus. Other things being equal,
the divided annulus would be preferred, since
the air pressure drop for a given total air flow is
much less than for the single-annulus models,
Some results of the calculations are shown in
Fig. 6. Due to an error in computing f3,, the
curves do not correspond to irradiation in position
C-48 of the LITR but to a flux of identical shape
and doubled magnitude. The Mark VIII model
appeared to be most suitable, since, in addition to
the lower air pressure drop, it had the highest
value of ATf. The final air temperature would
A
$SD-A-10T
CRNL- LR-DWG-3794
860
840
820
ref 2
800
Rea = 10,000
FUEL COMPOSITION No.48 IN C-48 OF THE LITR
T;, MIXED-MEAN TEMPERATURE OF FUEL (°Ch
o 20 40 80 80 100
5, DISTANCE FROM INLET (ecm)
Fig. 4. Effect of Fuel Flow Rate on Fuel
Temperature Pattern in Mark 1A Miniature Loop.
be lower for this model, also. On this basis, the
Mark VIl configuration was adopted for the de-
tailed study discussed in the next section,
SSD-A-1069
RNL-LR-DWG-37
840 o DWG-3789
¢80
@
n
O
. 10,000
@
o
O
= 20,000
Reg
s 30 000
Pey .
40. Oop
-~
@
Q
Ref = 3000
T¢w MIXED-MEAN TEMPERATURE (°C)
~
o
o
FUEL COMPOSITION No.44 IN C-48 OF THE LITR
4
7 o0 20 40 60 80 100
5, DISTANCE FROM INLET {cm)
Fig. 5. Effect of Air Flow Rate on Fuel Temper-
ature Pattern in Mark 1A Miniature Loop.
S5 D«A-
ORNL-LR-0OWG-3793
500 T T T
Rea =10,000 FOR MARK ¥I AND YII
880 F Reo = 20,000 FOR MARKIV AND IX .
Ref » 3000 /;
/|
860 o~
7/
o N/
) 5
s/
7;, MIXED-MEAN TEMPERATURE OF FUEL {°C)
&
760
0 25 50 75 100 125 150 175 200
s, DISTANCE FROM INLET (em)
L\
Fig. 6. Effect of Cooling-Air Pattern on
Miniature Loop.
RESULTS OF DESIGN CALCULATIONS FOR THE MARK VIil LOOP
The Mark VIII loop model is shown in Fig. 7.
The v #Mulations are discussed in detail in
Appendix B, The results are summarized by the
plots of 9/ and of y vs s given in Figs. 20 to 27
(see Appendix B). The quantity 9/ may be defined
as T, = T,9, The numerical data are given above
in "“Number Data’’ and a plot of ¢ vs s is shown
in Fig. 17 (see Appendix B).
Several qualitative remarks can first be made.
The positions of maximum and minimum fuel
temperature are quite insensitive to the velocities
of the two fluids. This allows placement of control
thermocouples to be made with some precision.
The value of AT/ decreases with increasing (Ref),
but not indefinitely. The principal resistance to
flow of heat is at the interface between air and
fuel tube. At sufficiently high fuel velocities,
further increases have only a small effect on the
quantity of heat transferred from the fuel.
From the results in Appendix B, those pairs of
fluid Reynolds numbers must be selected which
correspond to ‘‘realistic’ conditions, namely,
1. initial air temperature, Tg, near room temper-
ature,
2. maximum fuel temperature near §15°C.
From the data of Figs. 24 to 27, the values of
y(O)oremelected, from which 79 — Tg is calculated
by Eq. 10. The values of T - Tg are plotted vs
(Rea) for several values of (Ref) in Fig. 8. The
conditions above allow calculations of the
“realistic’’ values of this quantity from
(18) (19 - 19
The values of (T/ - To)m" were obtained from
the curves of Figs. 20to 23. The resulting set of
“realistic’’ conditions is indicated by the dashed
line in Fig. 8.
It is a simple matter to extend these calcuy-
lations to a fuel wattage other than the one used
here, as long as the shape of the thermal-neutron
flux is not changed. In fact, if the fuel wattage,
P o is replaced by a new value, nP,, the curves
real — 785 - (T/ - T?)max .
10
of Appendix B still apply, but with the y and 0/
scales simply multiplied by a factor n. Extension
to a five-times-greater fission wattage is made in
Fig. 8 (this corresponds roughly to the use of
fuel composition 44 in position A-38 of the MTR).
The results of calculations for three different
wattages are shown in Table 2.
UNCLASSIFIED
SSD-B-988
ORNL-LR-DWG-2252
FUEL OUTLET FUEL INLET
§=0
AIR INLET = ~ = AIR INLET
— 5:=5 —_ )
1 1
4
it 1 -, 4
] ] 43N A
7 ¢ T | 7
/ 12 1
1 / 4 [y ¢
FUEL LOOP —4 1 4
[ r 1 / /
’ [
/ i f
AIR COOLING [/ ] g
ANNULUS
¢
i
75 )
\Ks,
AIR QUTLET
\
Fig. 7. Mark VIl Model of Miniature Loop.
SECRET
SSD-B-1067
ORNL-LR-DWG-3787
10,000
8000
6000
5000
4000
3000
2000 THESE GURVES FCR
2700 w/ce
1500
LOGI OF REALISTIC Zor,
o
o 800
™
600
500
400
300
200
150 540 w/cc
100
10 152 3 45 7 10° 152 3 45 7 10°
Rea
Fig. 8. Estimation of Cooling-Air Requirements
for Mark VIl Configuration of Miniature In-Pile
Loop.
TABLE 2, CALCULATED THERMAL BEHAYIOR
OF THE MARK Yill MINIATURE LOOP
Effect of Yolume Heat Source
P e w/cc 540 1,540 2,700
(Ref) 3,000 3,000 3,000
(Rea) 16,500 70,000 167,000
Atr,, °c 49 134 245
Effect of Fuel Flow Rate
(Py = 540 w/ce)
(Ref) 1,500 3,000 6,000 10,000
(Rea) 19,000 16,500 15,500 15,500
AT,, °c 102 49 25 25
(Po = 2700 w/cc)
(Ref) 1,500 3,000 6,000 10,000
(Rea) 368,000 167,000 118,000 118,000
AT, °C 510 245 125 125
MISCELLANEOUS CALCULATIONS FOR THE MARK VIil LOOP
A decision to select one or another set of
operating conditions for the miniature {oop depends
in part on a number of things other than the heat
transfer calculations discussed in this report.
The necessary additional calculations are summa-
rized in this section.
Air Pressure Drop, Velocity, Volume, and
Temperoture Rise, — The pressure drop in the
air-cooling annulus was calculated by the con-
ventional expression
2 2
, 2RT ,u-(Rea) P /s
(9 p2-pd =t ity L
! 2 2 p 4(r, - r.)
(ry —1,) 2 3772
where
p, = pressure at air inlet,
p, = pressure at air outlet,
R = gas constant,
S = length of loop,
f = friction factor,
and the other symbols are defined in ‘‘Derivation
of Heat Transfer Equations.’”’ The friction factor
was calculated from the relation of Koo:14:13
0.125
(20) f = 000140 + ——— .
(Re)0.32
The outlet pressure, p 5 was assumed to be 1 atm.
The resuits of the calculation for the Mark VIII
loop are shown in Fig. 9.
The volume of air required to attain various
Reynolds numbers in the Mark Viil loop is shown
in Fig. 10. The linear velocity of the air was
calculated, assuming the pressyre to be the
average of the inlet and outlet values, The
results are shown in Fig. 11. The scale of Mach
H4E . C. Koo, Thesis, MIT, 1932 (see ref. 2, p 119).
lsThe relations proposed by Davis and others specifi-
cally for annuli differ somewhat from the one used here.
No appreciable error in the calculated Fressure drop
results from this source (see ref. 4, p 134 tf).
1
numbers was calculated® by using the velocity
of sound in air at 1 atm pressure at 200°C,
1500 ft/sec. it will be noted that the velocity
of air is always subsonic in the Mark VIl loop.
The rise in temperature of the cooling air may be
calculated from Eq. 10. The maximum air tempera-
ture is given by
21 T > >
(21) zlmax) =Tf—2——a]y-§—.
There are two values of this quantity, since the
two branches of the cooling-air circuit experience
different conditions. The calculated results for
Y4 andbook of Chemistry and Physics, Chemical
Rubber Publishing Co., Cleveland, Ohio, p 2257,
27th ed.
UNCLASSIFIED
SSD-A-1058
-LR- 3778
PRESSURE DRCP (psi)
6 8 10° 2 4
Rea
10 2 4
Fig. 9. Cooling-Air Pressure Drop for Mark VIl
Configuration of Miniature In-Pile Loop.
12
UNCLASSIFIED
SSD-A-¢t059
779
VOLUME (std. ¢fm)
o
6 8 10° 2 4 6 810°
Reag
4
10 2 4
Fig. 10. Air Volume Required for Mark VIl
Miniature Loop.
UNCLASSIFIED
550-A-1057
ORNL-LR-DWG-3777
2 T T 1 1 T 1
g
2408
* s
- o
E 6 z
X
g 2
-
o 4 =
>
e
<
P Lol L1
6§ 810 2 4 & 810
Rea
Fig. 11. Air Velocity for Mark VIl Configuration
of Miniature Loop.
the highggienir temperature for each of the cases
of Table 2 are shown in Table 3.
Fuel Pressure Drop, Velocity, and Volume, — The
pressure drop in the fuel tube was calculated from
the expression
#fz(Ref)z L.s
Ap = ——oeo——
@)
41""1'l P/
where L is the effective length of the loop; the
other symbols have been defined before. The
effective length was calculated by adding to the
actual length (205 e¢m) an amount, 1507, to account
for the bend at the tip of the loop. The friction
factor was taken from McAdams.!? The results
are shown in Fig. 12. The volume flow rate of
fuel is shown in Fig. 13. The linear velocity is
shown in Fig. 14.
Thermal-Neutron Flux Depression. — The de-
pression of the thermal-neutron flux has been
7w. H. McAdams, Heat Transmission, p 118, 2d ed.,
McGraw-Hill, New York, 1942.
18y . Lewis, A Semi-Empirical Method of Esti-
mating Flux Depression, MTRL.-54-27, March 11, 1954.
TABLE 3. CALCULATED MAXIMUM AIR
TEMPERATURE IN MARK VIII LOOP
Po = 540 w/cc
Maximum Air Temperature
(Ref) {Rea) Q)
1,500 19,000 450
3,000 16,500 500
6,000 15,500 540
10,000 15,500 550
Py = 2700 w/cc
1,500 368,000 30
3,000 167,000 200
6,000 118,000 290
10,000 118,000 300
{Ref) = 3000
Maximum Air Temperature
PO (w/cc) (Rea) €C)
540 16,500 500
1,540 70,000 360
2,700 167,000 200
F
estimated by using the trectment of Lewis,'®
This is based on a correlation of some experi-
mental data obtained by various workers at the
MTR. A ‘“‘grayness’’ factor of 0.73 was found
which is due to fuel alone. The Inconel walls of
the fuel tube and of the cooling annulus contribute
UNCLASSIFIED
550-B-1056
8-10
ORNL-LR-DWG-3776
100
LOOP + CONNECTING TUBES
LOCP PROPER
10
aAp, (psi)
0.1
Refl
Fig. 12. Fuel Pressure Drop in Mark VIl Loop.
UNCLASSIFIED
S50-B-108%
ORNL-LR-DWG-3775
100
¥, VOLUMETRIC FLOW (cc/sec)
10° 10° wt
ref
Fig. 13. Volumetric Flow of Fuel in Mark VIl
Loop.
13
SSD-B-1068
UNCLASSIFIED DRNL-LR-DWG-3788
SS5D-8-1054 250
ORNL -LR-DWG-3774
100 | | .
— 1
160
200 — 140
= 120 ¢
& w
.
: 5 100 3
E o 80 &
S 2 PRESENT PUMP S
o 3 - >
O o
> > — 60
™ é
3 _ |
© 100— -~ a0
PROPQOSED 30-cc PUMP —
— 20
ANP LOOP 10-cc PUMP —
/ 0
1 50— / : VOLUME OF CONNECTING TUBES
10° 10 10°
reaf
VOLUME OF LOOP
Fig. 14. Linear Velocity of Fuel in Mark VIII o |
Loop. 0 5 10 15
DILUTION FACTOR
another factor of 0.95. The over-ali grayness
ig. 15. Dilution Factor in Mark Vill Loop.
factor is estimated to be about 0.67; that is, Fig. 15. Dilution Factor in Mar oop
two-thirds of the thermal flux is available to the Volume of loop proper, em3 41.6
sample, Volume of connecting tubes, cm3 14.4
‘Dilution Factor,”” — A quantity known as the
*!dilution factor’’ is of some interest in studying Volume of pump, cm® 76.2
in-pile corrosion. This quontity is defined as the Total 132.2
ratio of the maximum value of the fission power p
generated in unit volume of fuel to the average 0 P 0.08
value, In a loop of the type being considered, s ’
this definition may be stated as J‘
y ds
Po(nrfS + Vp) 0
(23) dilution factor = — Dilution factor 10.6
f y ds The effect of changing the pump volume is
0 indicated in Fig. 15,
where V is the volume of the pump and of any Total Power, — The total fission power generated
connecting tubes. The present situation in regard in the loop is calculated from Eq. 12, The results
to the dilution factor is indicated as follows: are shown in Fig. 16.
14
TOTAL POWER (kw)
50
45
40
35
O
o
N
o
n
o
15
$SD-B-1065
ORNL-LR-DWG-3785
/)
/
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
PO' MAXIMUM SPECIFIC POWER GENERATION (kw/cc)
Fig. 16. Total Power of Mark YiIl Loop.
15
v
Appendix A
IMPORTANCE OF REYNOLDS NUMBER IN MASS TRANSFER
The frequent argument on the relative importance
of linear velocity and Reynolds number in mass
transfer between the container and fuel would seem
to be settled by appeal to chemical engineering
experience.'? It is found that excellent results
are obtained in correlating mass transfer coef-
ficients for fluids flowing in pipes by means of
relations of the type
2r .k’
(24)
= f(Scf), (Ref)] ,
/
where &£’ is the mass transfer coefficient, D/ is
the diffusion coefficient of the transferred species
in the fuel phase, and (Scf) is the Schmidt number
for the fuel:
quee, for example, G. G. Brown, et al.,, Unit Oper-
ations, p 517 #, Wiley, New York, 1950.
ke
(25) (Scf) = ——,
PRy
The similarity of Eq. 24 to the analogous heat
transfer equation is evident, the Schmidt number
replacing the Prandtl number, and the nameless
group on the left side replacing the Nusselt
number., Even more striking is the fact that
for many cases the Dittus-Boelter equation (16)
applies to mass transfer with the same coefficients
and exponents as those used in heat transfer. It
is quite clear, therefore, that the Reynolds number
is fully as significant in mass transfer as in heat
transfer, and is definitely the relevant fuel-flow
variable. Furthermore, an additional statement of
the surface-to-volume ratio is unnecessary, since
this information is also given in the Reynolds
number.
Appendix B
ANALOG SIMULATION IN AN IN-PILE LOOP-LIQUID.SALTS FUEL STUDY
E. R. Mann
F. P. Green
R. S. Stone
Reactor Controls Department, Instrumentation and Controis Division
An analog simulation of an in-pile loop experi-
ment for ART fluoride fuel was performed by
means of a portion of the Reactor Controls Com-
puter to determine the optimum fuel and coolant
flow rates and the temperature variation of the
fuel as it traverses the loop for each of 16 flow
combinations. Constant coefficients for the de-
scriptive equations and a linear plot of the forcing
function, flux, Fig. 17, were provided by D. F.
Weekes. This simulation has provided a series
of 16 curves of fuel temperature vs instantaneous
position of the fuel within the 205-cm ‘‘active’’
portion of the loop and 16 corresponding curves
displaying the radial heat transfer vs fuel position.
The specifications given state that the proposed
fuel loop will conform to the following equations
(the quantities 6, and 6, may be defined as
Tf - T9 and T - Tg, respectively; all other
symbols are defined in the nomenclature list in
16
“‘Derivation of Heat Transfer Equations'’):
40,
(26) -—;::B‘gb-—azy , 0<s<205¢em ,
dy ™
(27) = + ayy = a,p
and > 0<s<102.5em ,
df,
(28) o = % P
(29) % - ay = a2¢\
and } 102.5 < s £ 205 em
df,
(30 ds =Tt J
The analog was established on this basis.
(x10™) 5SD-A-1066
3.5 ORNL-LR~DWG-3786
3.0
™~
¢
[
r
o
¢, NEUTRON FLUX (neutrons - cm - sec" )
O
e
o
————
g
/ \
\
0
o
0 T
n o n o [y 0 o wn o
o @© Te} ™ - ;N M~ iy o
o~ — -— - -—
s, DISTANCE FROM iNLET (cm)
Fig. 17. Neutron Flux Distribution.
The differential equations for electrical po-
tentials in the circuit, Fig. 18, are analegous to
Eqs. 26 through 30, because by Kirchhoff’s law
for currents,
g E, dE, E, )
______CI——---+—=0,a’rnode”A ,
Rl R2 dt R3
E3 = GEI , at node ““B'’' ,
e E £, .
. _ 2——=O,m‘nodeC .
R4 R5 dt
Combination of these equations gives
dE 1 G 1
1
(31) . - E, =——E,
dt RzC]
UNCLASSIFIED
SSD-A-1060
ORNL-LR-DWG-3780
o
e
—
P
o
4 2
-£,
—D—b—oz
m
T
Fig. 18. Block Diagram of Electronic Analog
of the In-Pile Loop.
dEI Gz 1 1
(32) — - - E,= — E,
dt R,C, R,C, R,C,
dEz 1 1
(33) —_ = - E] ,
dt R . C R _C
472 572
where G, and G, are the values of G appropriate
to the two regions of the loop,
If in the loop equations the substitution s = ¢
is made and the function ¢(s) is suitably trans-
formed to ¢(t), the loop equations and electrical
network equations are analogous. This implies
that, in the process of solving the problem, the
quantity ¢(s) is introduced in accordance with
the condition s = v-t, where v is a velocity,
selected here as 1 cm/sec. The various constants
were transformed accordingly, since for the elec-
trical analog, time is the independent variable.
Figure 19 is a block diagram of the system
simulator. Operational amplifiers Nos. 1 and 2
generate the electrical equivalents of the radial
heat transfer, y, and the fuel temperature, 9/.
The remaining five amplifiers are required to
perform special functions relating to simulator
operation, such as signal polarity inversion, iso-
lation, and time-constant modification.
The reactor flux function required in this simu-
lation could not be generated readily by means of
17
the standard electronic components on hand. It
was expedient, therefore, to employ a hitherto unre-
ported scheme to generate the flux function shown
in Fig. 17. This system makes use of a Brown
recorder with an auxiliary 10,000-ohm, ten-turn,
Helipot ‘‘slide-wire,’’ the rotor being coupled to
the recording pen drive. The predrawn, calibrated
flux distribution curve was placed on the strip
chart, and a dry pen was inserted in place of the
normal ink-filled pen. During a run the recorder
pen was made to follow the curve by manually
changing the recorder input to amplifier No. 5
(Fig. 19) by means of a 50,000 ohm Helipot. The
slide-wire output represents the flux *‘seen’’ by
the fuel as it traverses the loop. Since the chart
speed is 1 cm/sec, the fuel position in centimeters
measured from the input corresponds to time in
seconds. This arrangement provides a reasonable
time scale so that the function generator operator
can closely follow the calibrated flux curve.
In the interest of convenience, the free pa-
rameters of the system were fixed as follows:
#flmfi'&l = C2 = 10 ut
R, = Ry =175 megohms ,
R, = 750,000 ohms .
This requires that
(34) C] = ] - 75a3 = A3 ’
(35) G, =1+ 75,=(0+4)),
(36) a = 7.5, ,
10°
B
10°
(38) Rg = — .
%q
The values G, and G, are set by means of
previously calibrated 25,600-ohm potentiometers.
The use of two cperational amplifiers, for example,
Nos. 1 and 3 in Fig. 18, to provide accurately
UNCLASSIFIED
$5D-8-1070
ORNL- LR-DWG-3790
MANUAL
-50v
/
/M
50k
L RECORDER
M
-50v BIAS
|
voLT Box| RN
+23v SET &
25k
25k
25k
(1+4,) iV
A 25k M M [TO y
4 4
~ MV VN > RECORDER
M
-50v BIAS
Fig. 19.
18
Block Diagram for In-Pile Loop Simulation.
calibrated time constants over wide ranges has
not been previously reported, although it has been
conventional for some time with the ORNL simu-
lator. The method is described in Appendix C.
The values of 2, and a, are such that G, varies
between 0 and 1, and G, varies between 1 and 2.
The quantity a is set to 7.52, and may vary
between 0 and 1 (fraction of full-scale). The
addition of potentiometer P makes it possible to
vary the effective value of R, from 750,000 ohms
to infinity with a 25,000-ohm potentiometer.
Each of the resistances R, and R have four
different values and are put into the circuit in
the form of lumped constants, chosen by a four-
position switch. Pertinent values, both given and
calculated, are presented in Table 4. The quantity
20.,/a,0, is a correction coefficient needed in
connection with boundary conditions.
The simulation of y required the breaking up of
each run into two equal parts. An exponentially
decaying function describes y in the first half,
the time constant in each case set by potenti-
ometer A, in amplifier No. 3, whose gain is less
than 1. An exponentially increasing function
describes y in the second half, the time constant
in each case set by potentiometer (1 + A.) in
amplifier No. 4, whose gain is greater than 1 and
less than 2. Because of a small discontinuity in
y at the mid-point, the simulation must be stopped
and a new initial condition for y set on potenti-
ometer Y. This control is necessitated by the
boundary conditions imposed, which are:
1. coolant to enter at each end at the same
temperature (Gao = 0,505) and leave at the
center,
2. fuel to enter at one end and leave at the other,
with no net change in temperature (Ofo = 0[205).
Since there are two distinct and separate coolant
streams, entering at either end and merging to exit
at the mid-point of the loop, there is a discon-
tinvity in 6_ (and hence in y, since 0, is a
continuous function) at s = 102.5 em. This
requires a correction at mid-point, which may be
calculated as follows. Let Y102.5 Ad ¥1g,.5 be
the values of y just before and just after the
discontinuity, respectively. From the differential
equations, it can be shown by integration that
az + aa 0
Yio2.s = Yo 2, 1102
a .8, 102.5
- [ b ds ,
aa, 0
) CL2 - aa 0
Yio2.5 = Yo a,a /102.5
2
a
33‘ 205
- ¢pds .
&1%9 Yy02.5
Since the flux distribution is symmetrical about
s = 102.5 cm, it follows that
2a3
0
(39) ¥io2.s = Y1025 ~ 55 %025 -
1%2
The volt-box input to the 6, generator was used
only at the initiation of the second half of the
run in order to start the fuel temperature at the
tinal value which is attained during the first half.
Amplifier No. 5 was used for isolation, and ampli-
fiers numbered 6 and 7 were used to invert the y
and 8, signals and to transpose these outputs to
the center of the recorder scales.
At the start of a run, switches 2 and 3 were
closed. This set 6 =0 and allowed y, to be
set at some tentative value by means of potenti-
ometer Y. Switches 2 and 3 were then opened
and the run started. At mid-point the calculation
was stopped, and switches 3 and 4 were closed.
Potentiometers T and Y were then used to set
6/102.5 = 0/102,5+ and
Yi02.5 = Y102.5 aa, /102.5 °*
19
0¢
TABLE 4. NUMERICAL DATA USED IN ANALOG SIMULATION OF IN-PILE LOOP
G G a 2a
1 2 R R 3
Case Ref Rea a a a B a a a 4 5 -
3 2 1 1 1 2 3 3 5
(1= 75a;) (1 +75a,) (7.5a)) (10’8, (0°)/a, a,a,
1 1,500 10,000 0.0109 0.0379 0.0080 2.4 0.221 0.190 1.660 0.278 0.860
2 20,000 0.0092 0.0589 0.0060 0.85 144 | o 1yay 0.110 0.310 1.450 0.442 118K 4.35M 0.665
3 30,000 0.0089 0.0744 0.0049 11.4 0.079 0.332 1.368 0.558 0.603
4 60,000 0.0075 0.1060 0.00175 8.0 0.037 0.438 1.131 0.795 0.402
5 3,000 10,000 0.0114 0.0208 0.0102 20.4 0.221 0.145 1.765 0.156 1.86
6 20,000 0.0098 0.0342 0.0079 12.4 0.110 0.265 1.592 0.256 1.53
0.42 0.011 .
7 30,000 0.0097 0.0451 0.0071 9.4 116 0.079 0.272 1.532 0.338 238K 8.61M 1.45
8 60,000 0.0082 0.0707 0.0042 6.0 0.037 0.385 1.315 0.530 1.06
9 6,000 10,000 0.0127 0.0108 0.0110 19.6 0.221 0.048 1.825 0.081 3.88
. . . . . . . . .28
10 20,000 0.0100 0.0183 0.0090 0.212 11.6 0.0053 0.110 0.250 1.675 0.137 471K - 17.2M 3
11 30,000 0.0099 0.0246 0.0085 8.6 0.079 0.258 1.638 0.184 3.16
12 60,000 0.0083 0.0408 0.0060 5.2 0.037 0.378 1.450 0.306 2.47
13 10,000 10,000 0.0115 0.0065 0.0112 19.4 0.221 0.138 1.840 0.049 6.50
14 20,000 0.0099 0.0111 0.0094 0.127 114 0.0035 0.110 0.258 1.705 0.083 186K 285M 5:50
15 30,000 0.0098 0.0151 0.0091 8.4 0.079 0.265 1.682 0.113 5.45
16 60,000 0.0080 0.0258 0.0068 5.0 0.037 0.400 1.510 0.194 4.11
Once this adjustment was made, switches 3 and 4
were ned, switch 1 was thrown from ¢, to Lo
and the second half of the calculation was run.
This procedure was continued for successive runs
until a value of y, was found such that 0]205 = 0]0
and Y205 = Yo
The results of this procedure are illustrated in
the offf®d curves (Figs. 20—-27) and in Table 5,
which lists the initial heat transfer and the
maximum temperature excursion for each case,
along with an estimated statement of accuracy
for these two parameters.
TABLE 5. RESULTS OF THE ANALOG CALCULATIONS
R R Yo A9, Curve Number*
ef ea (w/em # 3&) ©c + 1) urve Number
1,500 10,000 54.5 101.7 1
20,000 49.0 99.5 2
30,000 47.0 98.4 3
60,000 43.0 91.5 4
3,000 10,000 55.0 51.5 5
20,000 52.0 48.3 6
30,000 50.8 48.3 7
60,000 47.0 46.7 8
6,000 10,000 57.0 255 9
20,000 53.0 25.5 10
30,000 52.5 25.1 11
60,000 49.0 24.3 12
10,000 10,000 58.0 26.0 13
20,000 53.5 24.9 14
30,000 54.0 23.9 15
60,000 51.0 23.8 16
*See Figs. 20-27,
21
SSD-B-!E?Z SSD-B-1363
ORNL-LR-DWG-3783
ORNL-LR- DWG -3792
55 :
N
CURVE Ref /
No.
45—+
1 1,500
5 3000
0T 9 6000
3 10,000
. /
(1IN
o
-]
/N o
10 N @
(& <l
9 o
g ® f \ 3
e \ =
2 ~
& © _;
w w
a 2
= N\ L
- &
= AN /
E’F -10 \ 9,13/
o \
-15
¢
o
Q
-
P
T
]
\ et
-25
Ref CURVE
i 1,500 2 —
30 3,000 6 =————
6,000 0y
-35 \ / 10,000 14
-40
us 20 40 60 BO 1025 125 145 165 185 205
\1 / s, DISTANCE FROM INLET (cm)
-50
i
ocooQ o o i iati
°R8922R888859583L8%8 8 Fig. 21. Fuel Temperature VYoriotion for
5, DISTANCE FROM INLET (cm) Rea = 20,000.
Fig. 20. Fuel Tempercture Variation for
ROGEIO,OOO.
22
Sk
o
$SD-B-1075
55 ORNL-LR -DWG -3795
/’5
50
CURVE Ref
as | No.
3 41500 \
? 3,000 j
40T 6000
15 40,000 /
35 \
. \
i
25 /
20 ,/ \
»—"/ \
t5 { i
10 / /é 15 :b\ \
17T TN\
£ // NN
w
g / A\
E o N
@
&
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F
| i
Y w0 \\\ a
w N5 /
cn"\
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IREANANERY
N 7Y
\l | N
-25 ‘
Wl
-35 \
3
-45
-50 -
°C22R39232928330L8eL8LE88
—————— - - = - &
5, DISTANCE FROM INLET (cm)
Fig. 22. Fuel Temperature Variction for
Rea = 30,000.
weapil
ss‘o-la*- 74
ORNL-LR-DWG-3794
50
A\
a5 \
“ \
CURVE Ref
No.
/ BT 4 1500
8 3,000
30T 42 6,000
16 10000 \
25
0
N\
{
L TEN N
e {/ N\
=5
@
2
2 0
L
a
v
Bo-s
-
D 0 \\ 12,16 _/
b ‘\ 4
Cb\
-15 \ /
-20 \ / I
/||
-25 /
-35 /
-40 -
-45 L
°283982°888 2080882885
$,DISTANCE FROM INLET {cm)
Fig. 23. Fuel Temperature Variation for
Rea = §0,000.
23
‘ 062
$SD-A-1076 SSD-A-1
Wk ORNL-LR-DWG-3796 ORNL-LR-DWG-3782
55 60
T 1
SO CURVE Rea 7 55
R\ — — 1 10,000 /
~ a5 AN\ —-— 2 20,000 ;/' 50
£ W\ ————— 3 30,000 ~ T 30,0000 7 mee——e
< \ \ ’ / ] E
5 AN s 80000 | L § 4
z
z | TN\ 7 2
& \\ A A x
= 35 N\ 7 LJ 40
=
: / /’/ & 35
30 A 1A
o vy -
T T // / 5
- Y, w
X925 A T 30
\\ _/' -
\\I/V x5
20 3 25
5 20
0 20 40 60 B8O 1025 125 145 165 185 205
s, DISTANCE FROM INLET (cm) 15
O 20 40 60 80 1025 125 145 165 {85 205
. . .. 5, DISTANCE FROM INLET (em)
Fig. 24. Heat Transfer Variation for Ref = 1500,
Fig. 25. Heat Transfer Variation for Ref = 3000.
L
SSD-A-1077
5 ORNL-LR-DWG-3797
SSD-A-1079
ORNL-LR-DWG-3799
60 / 60 '
55\ ] S5 \ | /'
\ / /‘
=50 CURVE Rea / 50 \\ CURVE Rea /
§ \l\ / \ \ —_— 13 10 //
L \ —_— g 10,000 / = \\ 000 /)
245 4 —~-— 10 20,000 W/ 345 || ————- 14 20,000 4
e N | - 11 30,000 M / 3 — 7= 15 30,000 4
u 12 60,000 4 x 16 60,000 /
£ 40 / ki 40 l /A
! \ 0 I/}r
2 2 /
E 35 ‘ E 35 4
w v — /
I /7 < /)
X 30 z T30 \ v
\\ N /// x % ////
25 N s / 25 N } - / ,'{/
l/ /7/ \ \\///,
N\ .
NV \,'//'/
'°5"20 46 60 80 1025 125 145 165 185 205 2020 36 0 8o 1025 125 145 165 185 205
s, DISTANCE FROM INLET {cm) s, DISTANCE FROM INLET (cm)
Fig. 26, Heat Transfer Variation for Ref = §000. Fig. 27. Heat Transfer Variation for Ref = 10,000.
24
Appendix C
TIME-CONSTANT MODIFICATION THROUGH POSITIVE FEEDBACK
E. R. Mann
F. P. Green
R. S. Stone
Reactor Controls Department, Instrumentation and Controls Division
If Fig. 18 is considered without the amplifier
No. 3 feedback loop, the equation of state for
amplifier No. 1 becomes
R2 dE
40 E, =——F -
(40) %
1
R,C .
271 di
Here, the steady-state gain = R,/R,, and the
time constant = R,C,. As we have seen in
Appendix B, with the feedback loop in place,
C dE
@41 E, =— Fo— :
(I G) (I G dt
V(a2 (&)
R2 R3 R2 R3
In practice, R, = R,, so that
R R,C, dE,
1-G6) a
2
E = ——
R,(1 - G)
o
(42)
Reference to Eqs. 40 and 42 shows that where
R, = R, the effect of such positive feedback is
to divide both time constant and gain by (1 ~ G).
Where G is less than unity, both the time constant
and the gain are effectively increased, and
enormously long time constants are made possible
without the need for excessively large values of
R, or C,. Where G exceeds unity, negative time
constants are created, corresponding to regenerative
equations. The absolute value may be larger or
smaller than in the case without feedback, de-
pending upon whether G is less than or greater
than 2.
Another case of interest is that in which
resistor R, is replaced by a capacitance C; =Gy
When the Kirchhoff current balance is set up for
such a circuit, the equations are
a E E, . dE, . dE .
—_— . ——— - —_— 4 —_ =
( ) R] R2 1 dt 1 ’
(44) E, = GE, ,
R, dE,
(45) E, = R E - R,C,(1-G) -
Comparison of Eq. 40 with Eq. 45 demonstrates
that in the case of capacitive position feedback,
the sole effect is to multiply the time constant
by (1 — G). Where G is less than unity, this
effectively decreases R,C, and gives access to a
range of very short time constants. Where G
exceeds unity, the analog again enters the domain
of negative time constants, whose absolute value
may be larger or smaller than in the case without
feedback, depending upon whether G is above or
below 2.
The potentiometer Gr may be calibrated in such
a circuit, and a linear plot of potentiometer setting
vs A (or 7} can be drawn for resistive or capacitive
feedback. For positive time consfants this is
done for two or more settings of Gr by applying a
constant potential E, allowing the circuit to reach
equilibrium, and then disconnecting the driving
voltage and measuring the e-folding time, 7, of
the decaying function E,.
For negative time constants this could be done
for two or more settings of Gr by applying a
constant potential E, differentiating the output
E,, and measuring the rise time constant, 7, of
the expanding derivative. A typical plot of A vs
potentiometer setting is shown in Fig. 28.
UNCLASSIFIED
SSD-A-1064
ORNL -L R-DWG- 3781
0.015
mn
~ 0.010
0.005
-0.005
= -0.010
-0.015
0 0.5 1.0 1.5 2.0
GAIN
Fig. 28, Time-Constant Calibration Technique,
A s