I
PAUL R. KASTEN and RUSSELL E. AVEN’ Oak Ridge National Laboratory, Oak Ridge, Tenn.
Fuel Costs in Batch- and Continuous-Processed Homogeneous Reactors As inexpensive fuel processing facilities become available, continuous-processed reactors will have a lower fuel cost
1
N AQUEOUS homogeneous reactor systems, radiation damage to the fuel does not appear to limit fuel exposure time. This, in combination with the system flexibility (2) which permits large changes in fuel concentration and removal of fission gases such as xenon-135, indicates that homogeneous reactors may be operated for long times without fuel processing. Under these circumstances nuclear power stations could be built and operated many years without a fuelprocessing facility. Centralized processing facilities to recover the fuel a t the end of the operating period could be designed and built as required. Previous results (3) for homogeneous uranium-235 burner reactors indicated that nuclear difficulties should not limit the operating period, and that the economic penalty associated with batch operation could be small. One purpose of the work reported was to extend the study to reactor systems initially loaded with a fuel of relatively low enrichment, and to determine whether fuel costs in these systems could be appreciably lower than in pure burner reactors. The other purpose was to determine whether it would be necessary from an economic-power viewpoint to process the fuel continuously.
studied for both batch and continuous fuel processing. Case 1B refers to Case 1, batch processing; Case I C refers to Case 1, continuous processing. The terms “batch” and “continuous” refer only to the fuel processing itselfthe reactor fuel is considered processed only a t the end of the period of reactor operation when batch processing is assumed, but is processed throughout the period of reactor operation when continuous processing is assumed. The fuel feed to the reactor is assumed to be continuous throughout reactor operation for both fuel-processing methods.
Cost Factors and Assumptions In making economic calculations the following assumptions were made : 1. Initial inventory of Dz0, 50,000 kg. (All other initial inventories were based on this number and initial concentrations) 2. Cost of DzO, $28/lb. 3. Cost of natural U as UOS, $40/kg. Z T I C \
Present address, Department of Chemical Engineering, University of Mississippi, Oxford, Miss.
-1
CURVE
OPERATING COST OF PERIOD ENRICHED U-235 I O yrs
25 yro
I
1
I
I
OPERATING TIME = I O YEARS COST OF 90% ENRICHED U: $ 3 0 / g m
1
I
I
1
1
I
I
1
I
7
: ----- \ -- I .
-
.-
$ 6
m
6
5
I
I
E
- 5
c 0 v? 1 4 3
2 3
? 2
“ 0
200 400 ’ 600 800 1000 U - 2 3 8 CONCENTRATION (grns/lIler)
I200
Figure 1. Total fuel costs vs. uranium238 concentration for batch operation
0
0
I
200 400 600 800 1000 1200 FUEL PROCESSING CYCLE TIME, 0 ( d a y e l
Figure 2. Total fuel costs for continuous fuel processing VOL. 50, NO. 2
FEBRUARY 1958
171
Table 1. G. U238/
Specific Cases Investigated
G.
U238/
Resonance Escape Probability,
Case
Kg. DzO
Liter
~(28)
1
250 500 750 1000
220
0.770 0.680
2 3 4
440 660 880
0.615
0.562
Slow Diffusion Constant, D ,Cm. 0.966 0.923 0.896 0.870
P(49)
General specifications: Reactor temperature, 250' C. Core diameter 12 feet. Core volume, 25,500 liters. Reactor fuel system volume, 51,000 liters. Moderator-coolant, D20, power, 500 Mw. Av. power density in reactor core, 20 kw./l. Equivalent removal-ponwdensity in fuel system external t o core volume, 20 kw./l. Uranium as UOa.
8. Cost of shipping fuel, 0 for continuous processing; assumed to equal $0.63/g. of 25 and 49 for batch operation-Le., cost of shipping and processing was $l/g. fissionable material for "batch" cases 9. Plant load factor, 0.80 10. Net thermal efficiency, 25%, corresponding to 125 Mw. of electrical power
Fuel costs were estimated for reactor operating times of 10 and 25 years. The total fuel cost is the average over the specified operating period. In estimating the fuel cost, the value of fuel feed was assumed to be that a t the end of the operating period. This treatment was adequate, because the value of uranium per gram of uranium-235 does not vary significantly over the range of enrichments required.
Batch Operation Studies
Mathematical Formulation. The fuel costs are a function of the required fuel concentrations. The time behavior of the fuel concentration is controlled by the rate material balances in combination with the critical equation. -4ssuming that the uranium-238 concentration was independent of time, that the fuel fluid was exposed to an average neutron flux while within the reactor system, that fluid transport times were negligible in comparison to times of interest, that uranium-239 decayed virtually instantly to plutonium-239, and that uranium-236 and plutonium-240 could be treated mathematically as if they had zero neutron absorption cross sections, the appropriate equations were:
-
dN(25)ldt = F ( 2 5 ) u(25)N(25) 4 dN(49)/dt = Z(28) 4
+
(1)
term of the critical equation, while plutonium-240 was effectively considered as a fertile material. Although under steady-state conditions the poisoning effects due to uranium-236 would be appreciable (-15% fraction poisons), under the nonsteady-state conditions considered here the fraction poisoning due to uranium-236 was only about 0.02 after 10 years of operation. The fraction poisoning due to plutonium-240 would be high if it were not a fertile material. However, as plutonium-241 is a nuclear fuel and, in fact, has a higher value of 9 than uranium-235 (2.23 compared with 2.08), the plutonium-240 is a better fertile material than uranium-238. Thus, assuming mathematically that the plutonium-240 acts like uranium-238 would tend to give conservative results. AS the plutonium-240 concentration is low relative to the uranium-238 concentration, the optimum uranium-238 concentration obtained in this study is sufficiently accurate. The value used for (fp)~in this study was 0.02 for most cases; however, fuel costs obtained for Case 2B with (f#)o = 0.05 were virtually the same as those with (f,b),, = 0.02. Equations 1 and 2 are rate material balances on uranium-235 and plutonium-239, Equation 3 is the critical equation, Equation 4 relates the total neutron poison cross section in terms of two effective poison cross sections, as approximated by Robb and others ( 8 ) ; and Equations 5 and 6 are rate material balances on these effective neutron poisons. Defining dT = $dt, Equations 1 to 6 can be written (after manipulating the critical equation) as dN(25) - F(25) - 4 2 5 ) N(25) dT d
=
ml = B2D -j- Z(mod)
+ Z(28)
(f#)o = fraction poisoning due to higher isotopes. xenon, and samarium, based on fuel-absorption cross section
Equations 7 to 11 were solved simultaneously and the results written in the following form: N(49) = A1 er:*
+ AtertT + A1er3T
1
N 2 5 ) = ~(25)[mi
D1 eriT
+ A*
(12)
+ D4 +
+ DzerzT + DD3eraT] (13)
+
+
1 / 8 ( 2 5 ) ((Dd m i ) ~ ( 2 5 ) F(25)/d DleriT [u(25) r l ] DzeT2T [u(25)
+ +
r21
f DaersT
where r l , r2, and equation
r3
[&5)
+ rsl)
(14)
are the roots of the
+ Err2 + Eis + E3
$3
+
=
0
(15)
and A,, Di, and E, are functions of input parameter values (defined in nomenclature). As z(28) was assumed constant, the feed rate of uranium-238 required to satisfy this condition was obtained from Ehe equation
Id25) z(25)
+ d 4 9 ) X49)I
(16)
The enrichment of the uranium fed to the system was then obtained from the relation
t and T could be separated, because Z,I#I is a constant for constant power operation. Thus,
(7)
where b is a constant, and Z f ( T ) = uj(49)N(49) f uf(25)'V(25). The value of
Fdt was found analytically from the
relation
172
INDUSTRIAL AND ENGINEERING CHEMISTRY
NUCLEAR TECHNOLOGY
OPERATING TIME = 10 YEARS 90% ENRICHED U AT $15/grn 90% ENRICHED U AT $30/gm
Table II. Total Fuel Costs for Batch Reactor" Operating Period, Years 10 25 Cost of 90% Enriched U,$/G.
---
15
30
15
Case*
30
Total Fuel Cost, Mills/Kw.-Hr.
2.26 4.23 4.04 2.21 3.27 3.17 1.79 1.80 1.72 2.92 1.92 3.31 2.21 3.66 2.70 4.11 Based on reactor temperature of 250' C.; 500 Mw. thermal power: 125 Mw. electrical power; 1B 2B 3B 4B
a
0.8 load factor, 4%/year inventory charges: and shipping and processing costs = $l/g. fission-
able fuel. b B specifies batch fuel processing.
" uranium-238 0
200 400 600 800 f000 U - 2 3 8 CONCENTRATION Igms/litert
1200
Figure 4. Minimum fuel costs for continuous fuel processing vs. U-238 concentration
Results and Discussion. Equations 12 to 19 were used to obtain the concentration behavior and feed requirements as a function of operating time for the various cases. These results were used in conjunction with the cost factors given previously to obtain fuel costs in terms of mills per kilowatt-hour. The total fuel costs for the four cases under study are summarized in Table I1 and plotted in Figure 1 as fuel costs us.
concentration. The results can be compared to a minimum fuel cost of about 4 or 8 mills per kw.-hr. for pure burner reactors ( 3 ) , if highly enriched fuel feed cost $15 or $30 per gram, respectively (the same cost factors were used in the "burner" calculations). Thus, an appreciable savings in fuel costs could be achieved by addition of fertile material to the reactor. Because of the conversion of uranium-238 to plutonium-239, the fuel costs are not so sensitive to the cost of enriched uranium as in the case of a burner. Little savings in fuel costs appear to be effected by operating for 25 rather than 10 years. The shape of the curves given in Figure 1 can be easily explained on a physical basis., At the lower uranium concentrations the initial inventory
Table 111.
charge is relatively low but the feed cost is high. As the initial uranium concentration is increased, fuel inventory requirements increase but feed requirements decrease. The inventory charge eventually becomes predominant as the initial uranium concentration is increased; a t the lower uranium concentrations the feed charge is predominant. A breakdown of the total fuel costs for Case 2B (Table 111) illustrates the relative importance of the various costs. Feed cost was the most important item contributing to the total fuel cost. T h e inventory charge for the initial fuel loading was also important, and increased with case number, as the initial fuel enrichment and the total fuel mass were higher for reactors with higher fertile material concentrations. This charge
Breakdown of Fuel Costs for Case 28" Operating Period, Years 10 Cost, Mills per Kw.-Hr.
25
0.32 0.21 DzO inventory Ghg. 0.07 0.07 D20 operating chg. W/g. $30/g. $15/g. %30/g. Cost of 90% enriched U, $/g. 0.46 0.72 0.18 0.29 Value of initial fuel loading 0.18 0.29 0.18 0.29 Inventory charge for initial loading 1.55 3.10 1.78 3.56 Feed cost 0.86 1.41 0.69 1.20 Value of recovered fuel Cost of fuel recovery plus cost of fuel 0.07 0.05 shippingb 0.009 0.015 0.007 Fuel losseso 0.012 1.80 3.17 1.79 3.27 Total fuel costs Reactor temperature 250' C.; 500 Mw. thermal power; 125 Mw. electrical power; 0.8 load factor; 4%/yr. inventory chg.: 440 g. U2*8/liter. Assumed to equal $l/g. fuel (25 49). This compares with 10.37/g. assumed for continuous processing charge. Assumed to be 1% of final fuel value.
+
Table IV. Initial critical concn., g. UQa/liter U?sB concn., g./liter Initial fuel enrichment, % U2a6 Operating time b, yr. N(25), g./liter N(49), g./liter N(40) 4- N(26) (approx.), g./liter Feed enrichment, y , % UZs6feed rate, g./dag Operating time F(25)dt, g. UZs6X
s, *
Summary of Concentration Data for Batch-Operated Reactors" Case 1B Case 2B Case 3B 2.68 220 1.20 10 4.13 1.93 9 48.7 403
7.40 440 1.66 25 10 8.21 7.03 3.21 5.56 30 12 52.9 37.3 429 335
1.30
3.60
1.02
Case 4B
17.64 660 2.60 25 10 15.4 13.2 9.06 13.3 40 20 40.2 35.7 354 354 2.92
1.07
49.22 880 5.3 25 26.3 22.2 50 37.2 367 3.06
10
71.8 28.3 20 53.0 753 2.85
25 112.4 56.4 50 52.4 735 6.92
Reactor temperature, 250' C.; 500 Mw. thermal power. Values at indicated operating time.
VOL. 50, NO. 2
FEBRUARY 1958
1'73
was partially offset by the value of the uranium-235 recovered a t the end of operation. The variation obtained indicated that recovery of the fuel would become more important as the concentrations of fertile material were increased, and that fuel processing costs would also increase. However, the fuel shipping and processing charges did not become an important item; appreciably higher charges could be permitted for these items than assumed here, without significantly affecting the total fuel costs. Concentration and feed data for Cases 1B to 4B are summarized in Table IV. The feed rate can be compared to a minimum of 660 grams of uranium-235 per day required in a pure burner reactor. An effective conversion ratio, (C,R.)e,can be considered: (C.R.), = 1
assumptions given prior to Equations 1 to 6; however, it was assumed that steady-state conditions apply during the period of operation (deviations from this condition are considered later), that fission-product poisons were removed in the fuel-processing cycle, and that the fuel and fertile material were returned to the reactor a t the same rate they were removed (as the reactor had a conversion ratio less than 1, no difficulty was encountered because of this assumption). The mathematical system was then given by F ( 2 8 ) = Z(28) $
+
Z@)O = effective macroscopic fission prod-
uct cross section for batch operation The value for uJt)
For the above conditions, 2 ( p ) b has been approximated by Robb and others ( 8 )as Z(p)b =
X(P) =
Thus, a feed rate of 400 grams of uranium-235 per day would correspond to (C.R.),= 0.4. The fuel feed rate for Case 4B is much greater than for Case 3B, because of build-up of fission-product poisons which increases the critical mass requirements for Case 4B more than for Case 3C. Although the atoms of fuel produced per atom of fuel burned is higher a t the higher uranium concentration, it takes much longer to reach equilibrium conditions. This results in a negative value for the effective conversion ratio for Case 4B; feed requirements for this case are greater than for a burner reactor. This “extra” feed is required to keep the reactor critical and accumulates within the reactor system.
cuZ:/+
Case
1c
2c
3c
4c
W.P.), % 0 14.14 21.00 27.73 34.33
wherec = constant, atoms of fission product uoisons uroduced vo1.-time-fission power P = reactor thermal power A’(@) = concentration of fission product oisons Z(P) = N & ) u(p) = effective macroscopic cross section of fission prdduct poisons e = fuel processing cycle time All other symbols have their previous meaning. In solving the above system, the effect of fuel processing should be considered in evaluating Z(p). This was done on the following basis: If a reactor were operated at a constant fuel loading and constant flux, the value for Z ( p ) a t time 1 after startup would be given by
0
2.92 11.53 14.35 21.28 0 0.74 2.20 4.40 7.29 11.58 0
0.74 1.18 1.62 2.94
174
Steady-State Results for Continuous Fuel Processing 0, F(25), Ar(25), X(49), N(49) Days y, % G./Day G./L. G./L. Ar(25),G./L.
+
0
211.4 397.3 676.5 1115 0 91.56 535.6 771.6 1752 0 58.94 194.4 455.2 959.9 2407 0
347.7 632.7 967.6 2861
23.62 30.57 34.05 37.53 41.01 17.56 19.13 23.84 25.40 29.32 15.68 16.08 16.90 18.13 19.76 22 + 22 14.84 15.26 15.55 15.76 16.51
(1
rt
J-m
Mathematical System. The equations used here were also based on the
Table V.
GI
189.4 240.4 265.2 289.5 313.3 143.4 155.5 191.1 202.7 231.4 128.7 131.9 138.3 147.8 160.3 178.9 122.1 125.4 127.7 129.3 135.2
INDUSTRIAL AND ENGINEERING CHEMISTRY
0.648 1.070 1.383 1.816 2.452 1.423 1.672 2.725 3.237 5.236 3.508 3.731 4.230 5.146 6.832 11.41 17.41 20.88 24.05 26.94 45.07
0.874 1.014 1.118 1.261 1.472 2.787 2 * 949 3.634 3.967 5.268 7.876 8.124 8.679 9.698 11.58 16.67 41.70 48.40 54.51 60.10 95.11
- e-dh)&) + c 2 (1 - e - u ( P d + t )
(28)
continuously processed reactor, the effective poison cross section will be lowered, as the fuel processing cycle would remove fissiop product poisons, Under these circumstances the Z(p) of Equation 24 or 23 would be given by
(20)
Continuously Processed Reactors
would then be
so that uP(t) can be evaluated. For a
-
feed requirements feed requirements in burner
where CJ = atoms of fission product poisons produced per fission and
1.522 2.084 2.501 3.077 3.924 4.210 4.621 6.359 7.204 10.504 11.384 11.855 12.909 14.844 18.412 28.08 59.11 69.28 78.56 87.04 140.18
o,(t
- w)e
t--w -~
8
dw
(29)
By combining Equations 27 to 29, 2(p) was obtained as
84)=
The mathematical system then consisted of Equations 21 to 24 and Equation 30, and was solved to give feed rates and element concentrations for various process cycle times, 8. This was most easily done by assuming that 2(p) was some fraction of 2(25) 2(49), and solving Equations 21 to 24 for feed rates and fuel concentrations. -4s X f r $ would be a constant for constant power operation, the above results were used to With r$ known, the value evaluate of 6’ which produced the assumed value of Z ( p ) was obtained from Equation 30, The procedure was repeated until 8 covered the desired range. Results a n d Discussion. The steadystate results, summarized in Table V? include feed enrichment, feed rate, and fuel concentration as a function of fuelprocessing cycle time. These results were combined with previous cost factors to give the total fuel costs. Table VI gives the total fuel cost as a funcrion of enriched uranium cost, fuel-processing cycle time. fertile material concentration, and operating period. These results are plotted in Figures 2 and 3 for a 10-year operating period. The minimum fuel cost for each case was dependent on the processing cycle time. For 0 greater than this optimum, the fuel inventory charge became excessive because of the increased poison level; for smaller values of 8 the fuel-processing cost became excessive. The results given in Figures 2 and 3 were combined and are presented in Figure 4, which plots minimum fuel cost against concen-
+
+.
NUCLEAR TECHNOLOGY Table VI.
Operating Time, Years
Total Fuel Cost for Continuous Fuel Processing, Mills/Kw.-Hr.” Case6 IC Case 2C Case 3C Cost of 90% Enriched U, $/G. 15 30 15 30 15 30
Case 4C 15
30
10.58 6.86 5.02 4.10 3.66 3.34 9.41 6.68 4.83 3.91 3.45 3.11
12.60 8.89
T,
Processing Cycle Time, Days 50 100 200 400 800 1200
Total Fuel Cost, Mills/Kw.-Hr. 2.97 3.02 3.19 3.47 3.80 3.98
1.81 1.78 1.84 1.97 2.13 2.22
2.91 2.71 2.70 2.81 2.99 3.14
1.98 1.74 1.67 1.69 1.76 1.83
4.25 3.55 3.24 3.15 3.18 3.24
3.17 2.47 2.13 2.00 1.97 1.99
7.05 6.14 5.72 5.41 12.43 4.13 2.80 3.05 2.86 1.87 50 1.70 8.71 3.43 2.60 2.35 2.91 1.63 1.67 100 6.86 3.12 2.59 2.01 3.08 1.56 1.73 200 25 5.95 3.03 2.70 1.88 3.36 1.58 400 1.86 5.51 3.05 2.87 1.84 3.69 1.64 800 2.02 5.18 3.11 3.87 1.71 3.02 1.86 2.11 I200 Z , Based on reactor temperature of 250’ C.; 500 Mw. thermal power; 125 Mw. electrical power, 0.8 load factor, and 4%/year inventory charges. Reactor diameter was 12 feet, system volume 51,000liters, and cost of DzO taken as $28/lb. * C specifies continuous fuel processing.
Table VII.
Fuel processing cycle time 8, days “Continuous” processing cost, mill/ kw.-hr. Final processing cost, mill/kw.-hr. Cost of 90% enriched
u,$/g.
Breakdown of Total Fuel Costs for Case 2C, for 10-Year Operating Time” DzO Inventory Chg., 0.316 Mill/Kw.-Hr. Dz0 Operating Chg., 0.070 Mill/Kw.-Hr. 100
200
400
800
1200
0.56
0.293
0.159
0.092
0.057
0.045
0.009
0.010
0.011
0.012
0.016
0.019
50
15
30
Steady-state U feed cost, mill/kw.-hr. 0.835 1.67 Inventory chg. for initial loading of U,mill/kw.-hi. 0.185 0.287 Total fuel cost, mill/ kw.-hr. 1.98 2.91 Based on reactor temperature of 250’ C.; steady-state calculations.
tration of fertile material. Cycle time a t specific points is also indicated. As the uranium-238 concentration increased, the optimum value of 0 also increased; this behavior was due to the decreasing effect of fission product poisons on fuel concentration. As the cost of feed increased, the optimum 0 decreased, as a shortek 0 increased the conversion ratio and tended to decrease fuel costs. However, if the processing costs were increased (with a given feed cost), the optimum 8 would be greater than given here. A breakdown of the total fuel costs is given in Table VI1 for Case 2C, based on steady-state calculations for a 10-year operating period. This case corresponds to about the optimum economic system. The fuel-processing cost decreased with increasing 8, but the fuel-feed cost increased. Changing the operating period from 10 to 25 years lowered the fuel costs only slightly. In calculating the fuel costs, the initial fuel loading was assumed to be of low-
15
15
30
15
30
15
0.868
1.736
0.927
1.854
1.015
2.03
l.13
2.26
1.21
2.42
0.185
0.287
0.185
0.287
0.185
0.287
0,185
0.287
0.185
0.287
30
15
30
30
3.14 2.99 ‘1.83 1.67 1.74 2.71 2.70 1.69 1.76 2.81 500 Mw. thermal power; 125 Mw. electrical power; 0.8 load factor, 4%/yr. inventory charges, and
enrichment uranium, corresponding to that for the “batch” reactors, Under this condition, the fuel feed rate would not be given by the steady-state result; rather, it would be determined from the time-dependent equations, taking into consideration the. poison-removal rate. However, if the operating period were long, the final fuel concentrations would be adequately given by steady-state results. To indicate the approach to steady-state conditions, the time-dependent equations were solved for Case 2, assuming that no fission-product poisons were present (e = 0). The results given in Figure 5 consist of the uranium-235 feed rate, total uranium-235 feed, feed enrichment, and fissionable-fuel concentration as a function of reactor operating time. They indicate that steady-state conditions would exist for times greater than 2 years after start-up for this particular case. Thus, steady-state calculations should adequately give the final fuel concentrations after 10 and 25 years’ operating time.
With all poisons removed, the total uranium-235 feed given by the steadystate results would be slightly higher than the actual feed required, as during the initial 2 years of operation the actual feed rate would be less than the steadystate value. This is a result of the conversion of uranium-238 to plutonium239, and is due to the lower critical mass of a plutonium-239-fueled reactor relative to a uranium-235-fueled reactor. The steady-state results for F(25) would be about 10% high after 10 years and 4% high after 25 years (Figure 5). If not all the fission-product poisons were removed (e > 0), the uranium-235 feed-rate curve would vary as indicated in Figure 6. Under such conditions the overestimate of feed costs given by steady-state results would not be as great as indicated above. [With 0 = 0, the overestimate of the feed cost would be equivalent to about 0.08 mill per kw.hr. after 10 years’ operation for Case 2C ($15 per gram for highly enriched uranium), while with 0 = 200 days the overVOL. 50, NO. 2
*
FEBRUARY 1958
175
440
7
120
6
400
5
80
4
60
3
40
2
tI
1
t
Y
‘PARTIAL
POISON REMOVAL
FISSION PRODUCT
\ I TIME
0
0 20
5
25
TIME, years
Figure 5. Feed rate, total feed, amount of fissionable material present and enrichment vs. time for case 2, with no poisons present
estimate would be equivalent to about 0.04 mill per kw.-hr. ($15 per gram for highly enriched uranium).] With finite 8 (some poison build-up), the actual feed rate could be appreciably greater than the steady-state value. However, it was assumed that this “extra” fuel would be recovered by fuel processing a t the end of the operating period. The fuel-processing costs were based on the steady-state fuel concentrations; fuel processing was assumed to be carried out throughout the operating period. However, it would not be necessary to operate the fuel-processing plant during initial reactor operation. for times less than about 8 ; the steady-state fuel concentrations would not be applicable over the entire period of operation, and so it might be argued that the “continuous” fuel-processing costs were overestimated. However, because of the relatively rapid approach to steady-state conditions, and the short processing cycle time relative to the assumed operating period, this overestimate would be small; it could exist only if the amortization costs of the processing plant were small in comparison to operating costs. The previous discussion indicates that
Table VIII.
the curves in Figure 4 are not exactly the correct shape; however, the minimum fuel cost would still occur near the conditions of Case 2, but be lower than indicated by about 0.05 or 0.10 mill per kw.hr. if highly enriched uranium cost $15 or $30 per gram, respectively. The cost of fuel processing itself has not as yet been firmly established. Lewis (6) arbitrarily assigned $10 per gram of fissionable material as the cost for processing low enriched fuel; this is much higher than the estimates of Ohlgren, Lewis, and Weech (7). If the fuel-processing cost assumed here were increased by a factor of 10 (from $0.37 per gram of fissionable fuel to $3.70 per gram), the fuel costs would be increased. The results given in Table VI11 are less accurate for the higher fuel-processing charge, because the fuel cycle time increases with processing cost; this decrease in accuracy is associated with the inadequacy of the mathematical model as the cycle time increases. Comparison of Results I n comparing total fuel costs over a 10-year period in reactors operating on either a continuous or a batch fuel-
Effect of Fuel Processing Costs on Minimum Fuel Costs‘ Case 1C
Case 2C
state conditions are considered.
INDUSTRIAL AND ENGINEERING CHEMISTRY
processing cycle, the minimum fuel costs would be 0.2 to 0.5 mill per kw.-hr. lower for continuous processed reactors if fuel-processing costs were $0.37 per gram of fissionable material; however, if processing costs were a factor of 10 higher, the optimum cycle time associated with continuous processing would approach the period of reactor operation. Thus, the fuel costs for these two processing schemes depend upon fuelprocessing charges in a different way, the cost of processing having a greater effect upon the continuous-processed reactor, This indicates that batchoperated reactors have an economic advantage, from the viewpoint of generating nuclear power some time before relatively inexpensive fuel processing is available. However, it would be economically necessary to recover the fuel a t the end of the operating period, so that processing facilities would eventually be needed. As inexpensive fuelprocessing facilities become available, reactors processed on a continuous basis would have the lower fu.el cost. Comparing these results with burner fuel costs ( 3 ) .it is seen that the addition of a fertile material to a burner-type system could halve the total fuel costs, without causing nuclear difficulties during a 25-year. operating period. However, the technology associated with a burner-type reactor has been developed to a greater degree than that associated with reactor systems involving slurries or highly concentrated solutions. Nomenclature
Case 3C
Cost of fuel processing, $/g. fissionable material 0.37 3.70 0.37 3.70 0.37 3.70 Cost of enriched uranium,$/g. 15 30 15 30 15 30 15 30 15 30 15 30 Approx. 8 associated with min. fuelcost,days 100 50 200 200 200 200 1200 800 800 400 >I200 >1200 Min. fuel costb, mil1sjkw.-hr. 1 . 8 3.0 2 . 4 3.6 1.7 2.7 2.2 3.4 2.0 3.2 2.9 4.1 0 Based on results of steady-state calculations and 10-year operating period. Costs are high by about 0.05 to 0.20 mill/kw.-hr. (for Case 2C) if deviations from steady-
176
-
Figure 6. Effect of poison-removal upon U235feed rate
20
0
POISONS R E M O V E D
Values are for average temperature of
250’ C.
AI
-
A2
--
b,
=
+ + -
P1rl2
rI(r1
rlJ1
pln2
rzJ1
YI) (r2
-clql
+ -4 +-JZ J2
- r2) (YI
73)
NUCLEAR TECH N0 LOGY = fraction poisons,
20) Z(fue1)
= absorption
cross section, barns; u(25) refers to absorption cross section of U216;4 4 9 ) refers to absorption cross section of PuZ39; u(p1) refers to absorption cross section of effective fission product poison p l ; etc.; u(25) = 435; ~ ( 2 8 )= 1.82; U(#I) = 132; u(,b2) = 13.9; u(49) = 1060 = microscopic fission cross section of fuel, barns. ~ ~ ( 2 5 ) 368, ~ f ( 4 9 ) = 685 = macroscopic absorption cross section, cm.-l Z = Nu = macroscouic fission cross section, im.-l. Zf(25) = ~ f ( 2 5 ) N(25); Zf(49) = ~ ~ ( 4 N(49); 9) 8, = Z f ( 2 5 )
U
= fraction poisons due to high-
cross-section poisons and higher isotopes = feed rate of uranium to reactor system, atoms/barn cm.-sec. or grams per day for system. F(25) = U236 feed rate; F(28) = U%* feed rate = base of natural logarithm
= buckling of reactor; B2 =
(i)2 where R is the phys-
icai reactor radius plus a n effective extrapolation distance (taken as 3 inches), cm.-2 = ratio of thermal neutron leakage to absorption, dimensionless = ratio of fast neutron leakage to moderation; dimensionless
UI
1- g(28) Bar v ( x ) u(x)
+
e;
2:
z,
+ 2,(49)
Z(mod) = macroscopic absorption cross-section of moderator, cm.-‘ Z(D20) = 4.49 X 10-5 (99.7595 D,O) Z(p) = fission-products macroscopic absorption cross section, cm.-‘ Zr =: total macroscopic absorption cross section, cm.-l ZT = Z(28) Z(49) Z(25) I U
atoms of fission product poisons produced unit vo1.-unit time-fission power = fraction of atoms produced having effective cross section a(pJ per fission = fraction of atoms produced having effective cross section 4 4 per fission = c1 c2 = atoms of fission product poisons produced per fission
= Concentration Of
-
=p
+
articular element; atomsfbarn-cm. or grams per liter. N(25) refers to U236 concentration ; N(28), to U238 concentration; N(49). to PuZsS concentration; N(40), to Pu24o concentration; N(26), to U236 concentration = resonance escape probability as determined by U238 concentration
= 1-
6 w
I
+
+
+ ZdP) + Fermi age sq. cm. ~ ( D z 0 )= Z(P)
187 = neutron flux; neutrons/barnsec. = integration variable
Acknowledgment Acknowledgment is given to T. B. Fowler for his assistance in the calculations,
= -c2q1 = -q3u(inJ/m1
(C.R.),
=
+
I
feed requirements feed requirements in burner
literature Cited
~
t
= roots of cubic equation: s3 Elsa f E ~ s E3 = 0 = time, seconds, or years
T
=
vz
- -4 2 6 ) p(25) ‘(Pa)
P(25)
=
r l , 72, r3
+
+
+dt, barn-’
[a
- 1 - (fP)o]
X
u(25), barns
1 Y
=
e
11
= =
0
=
+ Bar
u(49), barns fraction enrichment of feed uranium fast effect, taken to be 1 fast neutrons per thermal neutron absorbed in fuel, ~ ( 2 5 ) = 2.08; ~ ( 4 9 ) = 1.95 fuel processing cycle time, days
S. E., Winters, C. E., Chem. Eng. Progr. 5D,256 (1954). ( 2 ) Brims, -- R . B., Swartout, J. A., (
