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Construction of nuclear reactors to obtain desired electricity-generating capacity
Eneryu. Vol. I, pp. 45-51
PerpamonPress 1976. Printed in Great htam
CONSTRUCTION OF NUCLEAR REACTORS TO OBTAIN DESIRED ELECTRICITY-GENERATING CAPACITYt S.
S.
PENNER
Department of Applied Mechanics and Engineering Sciences and Energy Center, University of California. San Diego,LaJoIla,CA 92037, U.S.A. (Received 31 July 1975) Abstract-Schedules for nuclear-reactor development may be determined for any desired electricitygenerating capacity. Net-energy ratios for nuclear-energy production from LWRs (light-water-moderated reactors) using high-grade ore are sufficiently favorably to allow construction of reactor scenarios that imply large savings in fossil-fuel resources during predetermined periods of time.
I. INTRODUCTION
The development of any new resource requires an energy investment of existing resources. Depending on the new energy technology involved, the annual and total energy returns may be negative or positive. A continuous time-dependent energy-investment schedule is shown schematically in Fig. 1 for a successful new resource development. We have defined the following three times in this resource development schedule: the time A? corresponding to the minimum completion time for the new resource, the time 6, after which there is a net annual return of energy, and the time t, after which the time-integrated energy investment first becomes positive. Figure 1 represents an idealized energy-development schedule. In practice, both the annual rates of energy investment and the annual rates of energy return may be step-functions of the time because new energy-resource developments are initiated and completed in incremental units. A schematic diagram depicting this more realistic situation is shown in Fig. 2. The times At, fh and t, are defined as before. The ratio of the height of the first positive increment of the annual return of energy to that for the first negative annual investment of energy depends critically on the net energy ratio, R, for the technology under development and on the life of the new energy-producing plant (see Section 3 for details). For a sufficiently large value of R, t, will generally occur within a year or two of fh. 2. SIMPLIFIED
SCHEMES FOR DETERMINING REACTOR-DEVELOPMENT SCENARIOS; EVALUATIONS OF rh AND r,
We make no distinction between the times in starting a resource development within a given year, i.e. we assume in effect that all energy-resource development is initiated at the beginning of the year and completed at the end of the year. Let N(i) be the number of reactors started during the i th year after the current year; the current year is identified by the label 0 and the number of reactors started during the current year is N(0). The parameter EC is defined to be the total energy cost for the construction of a new reactor which requires At years for completion. Assuming that uniform energy expenditures are made during capital construction and capital investment, the average yearly energy cost is &/At for each new reactor. The total net energy return from the reactor (with proper allowance for all operating charges) is designated as E,, during the plant life of rY years. Again assuming a uniform rate of energy return, the yearly energy return for each new reactor after completion becomes E,/t, The net energy ratio is defined as
(2.1)
R = E,/Ec tThis paper is based on a study performed for Physical Dynamics, 4.5
Inc.. of La Jolla, California (April 1975).
S. S. PENNER
46
ANNUAL
t
RATE
OF ENERGY
RETURN
2 2 ?i
NET
ANNUAL
RETURN
OF ENERGY
5 20 =, z
.
a
I
{ANNUAL RATE /‘ 1 I OF ENERGY INVESTMENT’
2 a
L At
1
d \
I I I
tb
t,
I
TIME+
I
Fig. 1. Schematic diagram showing continuous annual rates of energy investment, return and net energy available as functions of time for a successful new resource development. The minimum construction time for the new resource is At, the break-even time f,, corresponds to the time when the net annual return of energy reaches zero, and the time t, is the time required to achieve a time-integrated energy return that equals zero. The entire enterprise becomes a positive contribution to energy availability after t, years when the total area of the shaded region just equals zero.
t
+I
5
1
ANNUAL
K z Ly
RATE OF ENERGY
RETURN-
5
I
-NET
ANNUAL
._L........\ ?..........,. ... ......,.,l,l,.jj, OF ENERGY RiiiiB:~#:i:i:ri:~:~~:~
~~~~~1
I
, ,
a
I ,ANNLm ,OF ENERGY
RATE RETURN
“FWL,
INVESTMENT
;
;I At
tb
TIME
-
6
Fig. 2. Schematic diagram showing incremental annual rates of energy investment, return and net energy available as functions of time for a successful new resource development. The minimum construction time for the new resource is At, the break-even time fs corresponds to the time when the net annual return of energy reaches zero, and the time t, is the time required to achieve a time-integrated energy return that equals zero. The entire enterprise becomes a positive contribution to energy availability after t, years when the total area of the shaded region just equals zero.
for the construction of a new reactor. As already noted, EC is invested uniformly over At years while IL is returned at a uniform rate during tre years after At years have elapsed from initiation of construction. (a) Net annual rate of return of energy With the specified definitions, the total energy investment years is seen to be (Ec/At)[N(t) while the energy returned
per year in construction
+ N(t - 1) + N(t - 2) +. . . + N(t -At. + I)],
per year is given by the expression
after t
Electricity-generating
capacity
of nuclear
41
reactors
(E,/trp)[N(O) + N(1) + . . . + N(t - At )I, where all time indeces must be positive for physically meaningful definitions of terms. In view of the preceding relations, the net energy that becomes available per year as the result of reactor development is
(2.2) or, in terms of the net energy ratio R, f -‘it = Rthtltv) 2 N(t’)I’=0
~,(t)/(&/At)
A, -I
2 N(t -t’). I =_(,
(2.3)
As already noted, t’, t and At only take on integral values in the present simplified treatment. It is apparent from eqn (2.3) that the energy break-even year, fh, corresponds to
(2.4) The shortest possible break-even time occurs after f,, = At years when the initial reactor development has been completed and useful energy generation first takes place; for th = At. eqn (2.4) leads to the result N(0) = (tv/RA\t) 2 N(At - t’) ,‘=O = (b/RAt)[N(At)+N(At For an accelerated nuclear-reactor t = 25 y, R = 30; hence
development
- l)+.
using high-grade
. .+ N(l)]. fuel, we choose
(2.5) At = Sy,
N(0) = (1/6)[N(S) + N(4) + N(3) + N(2) + N(l)]. i.e. a positive return becomes first possible after 5 years provided the number of reactors started during the initial year equals one sixth of the total number of reactors started during the following 5 years. It is apparent from eqn (2.5) that smaller values of R dictate heavier initial program scheduling. For example, for R = 10, N(0) = (l/2) i I’-
N(f) I
if the annual energy investment is to be returned after f,, = At = 5 years following the year of start-up of construction. In practice, we shall generally be concerned with the evaluation of ea for values of t appreciably larger than At but normally shorter than t(-25~). It is convenient to use eqns (2.2) or (2.3) for a parametric representation of eO. We shall now examine several representative reactor-development schedules. (i) Uniform scheduling of reactor development. For uniform scheduling of reactor development, all of the N(t’)=No are equal. Hence eqn (2.3) becomes ~,(f)/(Ec/At) and c,(t)/(&/At)
= N,,[R(At/ts)(l+
t -At)-
At]
(2.6)
will be positive for R(At/tr)(l+t
-At)>At;
thus, for R = 30, At = 5y, tie = 25y, a positive energy return will first be realized during the fifth EGY Vol. I, No. I-D
S. S.
48
PENNER
year after the present year; for R = 5, At = Sy, t2 = 25y, t will be longer than 9y before a positive energy return occurs. (ii) Sequential doubling of reactor development. We assume that No reactors are developed during the current (t’ = 0) year, 2N0 during the next (t’ = 1) year, 22N0during the t’ = 2 year, etc. Equation (2.3) becomes
l,(t)/(Ec/At)=
[R(At/cie)(l +2+2*+. _ (2’ + 2”
. .+2’-*‘)
+ . . . + 2’-+‘)]&
(2.7)
The dimensionless parameter ~.(t)/(&lAt)N,, will now assume positive values only for large ratios of R At/f,. In particular, for R = 5, At = Sy, tZ = 25y, a positive energy return will never occur. The minimum value of R(At/t,) for which a positive energy return becomes possible for At = 5y and t = 1Oy is easily found to be R(At/t,) = 31.49 from eqn (2.7), i.e. R = 157.46 for tie = 25~ and At = 5y ; this value of R is unrealistically large. Even for t = 2Oy,t, = 25y, R must be at least 155 before a positive energy return occurs. We conclude that continuous doubling of the rate of reactor development is not a realistically defensible posture for a program that is designed to increase the total U.S. energy availability. (iii) Sequential doubling of reactor development for t, years, followed by termination of construction. For the specified model, eqn (2.3) becomes ~o(t)/(Ec/At)= No[R(At/tu)(l +2+22+...+2’-A’) -
@‘I+
p-
+ . . . + 2”C”“‘)]
(2.8)
for t, 5 t 5 tl + At; for t, + At < t < tie + At, reactor development is complete and maximum energy output occurs. As an example, we choose a 10-y reactor-development schedule, i.e. t, = 10~. Setting again At = 5y, tip = 25y, R = 30, R(At/t,) = 6, we may determine the dimensionless ratio ~,(f)/(E~/dt)N~ as a function of t from eqn (2.8). Some representative numerical results are listed in the following table. Table 1. Nmericalvaluesof c(t)/(E,/At)N,asafunctionof
t,
Y
t forR = 30, t, =
IOY,AI = sy, ~rp= 25~
c,(t)/(EC/At)No
t
5.114
Reference to Table 1 shows that the yearly energy output is positive three years after completion of initiation of construction for the assumed model. If we choose N, to be 10, then the annual energy output equals 15,300 times the annual energy charge for reactor development during the year with index 0 at a time corresponding to 13 years after program initiation and 3 years after initiation of construction has been terminated. Maximum output is reached 15 years after the year during which the program was initiated when the entire reactor-development program has been completed. Thereafter, the energy output remains constant until the year t+At is reached, when the first completed reactors have attained their useful life span. (iv) Other models and concluding remarks. The preceding calculations are easily generalized to arbitrary construction schedules. We conclude by noting the dominant importance of the net energy ratio R, of the reactor construction time At, and of the reactor life time TVin determining the magnitudes of available positive or negative energy charges that are incurred or released during the expansionary phase of an energy-resource development schedule. For fixed values of these parameters, a construction program may be invented that yields the desired energy return on a predetermined schedule provided the ratio R(Atl&) is sufficiently large. In the following Section 3, we comment briefly on applicable values of R.
Electricity-generating
(b) Integrated net-energy returns The integrated energy investment (Ec/At){N(t
capacity of nuclear reactors
49
of the year t is
at the beginning
. *+
- 1) + 2N(t - 2) t 3N(t - 3) +. ..+AtN(t-At)tAt[N(t-At-l)t.
N(O)lI, (2.9)
while the integrated
energy return is
(E,/trp)[N(O)(t Hence the total or integrated
-At)+N(l)(t
-At
- l)t..
.+N(t
-At
- l)].
(2.10)
net energy return is
ET = (E,/tip)[N(O)(t -At) t N(l)(t
- 1) + 2N(t - 2) t 3N(t - 3) +. . . + AtN(t - At)
- (EJAt){N(t t At[N(t
- 1) +. . . + iV(r - At - l)]
-At
(2.11)
- At - 1) +. . . t N(O)]}.
Thus the time t, after which the entire enterprise becomes a net producer of energy is determined by the relation (RAt/tu)[N(O)(t,
-At)
t N(l)(t,
-At
- 1) t
= N(t, - 1) t 2N(t, - 2) t 3N(t, - 3) t + At[N(t,
-At
. . + N(t, -At
- l)]
. . + A\tN(t, - Ar)
- 1) t . . . t N(O)].
(2.12)
We shall now work out analytically simple but instructive examples referring to well-defined construction schedules. (i) Uniform scheduling of reactor deuelopment. For uniform scheduling of reactor development, ail of the N(t ‘) = N0 are equal. Hence’ eqn (2.11) becomes Er/Ec=No[(R/&)(t-At)+(t-At-l)+..,+11 - (l/At)[l
t 2 +. . . t At + At(r -At
- l)].
(2.13)
As an example, we use the values t, = 7y, At = 5y. In this case, the ratio R At/t, is determined by the relation Rhtlt, = 6.67 or R = 33.4 for tip = 25~. Similarly, for t, = 8y, At = Sy, R At/& = 4.17 and R = 21.8 for tu = 25~; for t, = lOy, At = 5y, RAt/& = 2.33 and R = 11.7 for tY = 25~; for t, = 15y, At = Sy, RAtIt, = 1.09 and R = 5.45 for ty = 25~. Thus, the net integrated energy break-even time that can be reasonably expected for a nuclear reactor with R = 30, a 5-year construction time and a 25-year life, for the special case of a uniform construction schedule, is a little longer than 7 years. Comparison with the analysis of section 2a(i) shows that t, exceeds fh by a little more than 1 year. For large values of R, this type of result is generally expected. (ii) Sequential doubling of reactor development for t, years, followed by termination of construction. We now use again a model in which construction of No reactors is initiated during the current year, 2N0 during the following year, 2*No during the year thereafter, etc. Equation (2.12) then takes the form (RAt/t,)[(t,
-At)t2(t,
-At
- 1)+2’,-‘“1
={[2+‘(t,
-r,)+.
. . +2’r-A’(At -I)]
+ At(2’rY”’ ’ + 2’r-“-* + . . . + I)}
(2.14)
for t, 5 t, I t, + Af; for t, t At < t < ty + At, reactor development has been completed and maximum energy output occurs. For t, = 1Oy and At = 5y, we may then calculate from eqn (2.14) the applicable values of R At/t, for selected values of t,. The results of these computations are listed in Table 2, together with the required values of R for At = 5y, t, = 10~. For a value of R = 10 referring to total energy production, or a value of R = 30 referring to electricity production, we find from Table 2 that the integrated energy return equals the
50
S.
Table 2. Numerical values of RAt/t,
S. PENNER
and of R, as computed from eqn (2.14) for selected values of t, - t, with t, = 10y. and I, < 1, < ty +Af
r
--
Rat/t ,
R (for
c t=5y,
t,=ZSy,
10
34.72
173.6
11
33.03
165.
12
23.82
119.1
13
15.29
14
9.09
45.45
76.
tl=lOy)
2
45
15
5.11
25.55
16
2.51
12.
17
1.67
8. 35
18
1.25
b. 25
55
integrated invested energy during the sixteenth or fourteenth year, respectively. Thus, for sequential doubling of nuclear-energy plants, the total energy investment will be returned during the sixth or fourth year after termination of construction, depending on the applicable value of R. For R = 30, the integrated energy return is positive before the last generation of nuclear reactors has been completed. 3. NET
ENERGY
RATIOS
We have considered methodological aspects in the calculation of net energy ratios elsewhere.’ A review of these considerations may be found in Refs. 1, 2. Here we content ourselves with the statement that we shall mean the ratio RI [see Refs. 1 or 2 for precise definition of this term] whenever we use the term R in the following discussion. Representative values of R for shale-oil recovery have been given by Clark and Variscoj and have been found to have values around 8. Chapman* has estimated net energy ratios for a number of nuclear reactors. Energy charges for fuel processing (but not the intrinsic energy content of the fuel) were explicitly included for ores with uranium mass fractions of 3 x lo-’ and 7 x lo-‘, both for refueling and for initial fuel recovery. In Chapman’s analysis, a 10’ Mw, plant produces a net output, after consideration of distribution and other losses, of 255.8 Mw, or 5.602 x 10’” kwh, over a life of 25~ at a 62% average load factor when the lower grade ore is used; the corresponding values for the higher grade ore are 523.4 Mw, and 1.14625 x 10” kwh,, respectively. The total energy costs for construction of the 10’ Mw, reactor are 2.2229 x IO’” kwh, and 1.0162 x IO”’ kwh, for fuel processed from low-grade and high-grade ores, respectively. Thus, the energy ratios, defined as usable electrical energy produced divided by the thermal energy input, lie between 2.5 for low-grade ore and 11.4 for high-grade ore, respectively. Chapman has generally obtained energy ratios around 2 or 3 for low-grade ore and 10 for high-grade ore. Chapman notes4 that other workers have given ratios as low as 3 even for high-grade ore. There is a basic criticism that may be raised against the conclusions that are implied by Chapman’s numerical estimates, which is to some extent corrected by Chapman when he notes that the development of nuclear reactors may be viewed as an efficient procedure for converting fossil-fuel energy to electricity. We note that fossil fuel is used extensively today for the generation of electrical energy at an average conversion efficiency of about 33%. For this segment of the fossil-fuel industries, the applicable energy ratio is therefore the ratio of electrical energy produced by the nuclear reactor to the electrical energy which would be generated through application of the fossil fuels used in the construction of nuclear reactors, i.e. the applicable energy ratios are effectively tripled for that portion of nuclear-power generation which replaces electrical energy use that is normally produced from fossil-fuel plants. There is an obvious corollary to the preceding critique, which refers to the idea expressed by some that nuclear energy will ultimately become a primary source of electricity and a secondary source of portable fuels. For applications of this type, the energy ratios of Chapman must be reduced to allow for the loss of energy in the conversion of electricity to portable fuels. For such sequential processes as electrolysis followed by fuel-cell utilization for transportation, the overall energy loss would be expected to be perhaps 25% (if we assume that efficiencies for implementation of transportation are comparable once we begin with either fossil fuels or portable fuels in fuel cells).
Electricity-generating
capacity of nuclear reactors
51
Davi? has presented the following energy charges for an 1100 Mw, nuclear power plant operating at a 60% load factor: for plant construction, 0.74 x lo9 kwh, ; for the initial fuel load, 1.00 x 10’ kwh,; for replacement fuel, 050 x lo9 kwh,. Assuming a twenty-year plant life, the net energy ratio is seen to be 0.60 x 964 x lo9 kwh, R= (1.74 x 109/20) kwh.. + 0.50 x lo9 kwh, = 9’9’ This calculation involves pessimistic assumptions concerning the fuel-processing costs (i.e. no reduction in power consumption for diffusion plants and plutonium recycle nor allowance for anticipated fuel enrichment by centrifuging). The fuel is recovered from conventionally used, high-grade ore (2000-2500 ppm of UIOa in rock). For recovery from low-grade ore (e.g. Chattanooga shales) with 50-70 ppm of U308, the ore mining and milling costs have been estimated by A. Weinberg to be about 25 kwh,/ton of ore.” For this low-grade ore, Davis then calculates an incremental energy charge of 0.32 x 10’ kwh, for the initial core and 0.14 x lo9 kwh, for annual fuel loading. The corresponding net energy ratio becomes 0.60 x 9.64 x 10’ kwh, R = (2.06 x 109/20) kwh,. + 064 x 10’ kwh, = 7’7’ Davis’ net energy ratios are seen to be appreciably lower than the values which we derived from Chapman’s data. However, Davis states that his numbers are uncertain by perhaps as much as a factor of 2. (a) Application of Chapman’s R to a reactor-development schedule We conclude this brief commentary on net energy ratios by noting that the rather large value of R = 30 may be considered to be consistent with the estimate for the ratio of electrical energy derived from fission reactors to the electrical energy which would have been derived from the fossil fuels used in the construction of nuclear reactors if they had instead been used in fossil-fuel power plants. Thus, some of our optimistic examples of energy return for nuclear-reactor development scenarios represent reasonable quantitative descriptions of possible implementation schedules. In particular, a lo-year development schedule with doubling each year of the installed number of reactors [see Section 2a(iii) and Table 21 will produce within 15 years a total supplementary annual electrical energy output of 2.5 x 10” kwh, if the following number of 10’ Mw,. reactors No were installed this year: 2.5 x 10” kwh,/y = 1.0235 x 10” x [lo” kw, x 8.76 x 103(h/y)] x [(l/30) for the energy charge per reactor] x N, or N,, = 0.836, i.e. 836 Mw, must be constructed during 1975 with a 5-y completion schedule and doubling of the capacity installed each year during the succeeding 10 years. It should be noted that the preceding statements refer to the supplementary electrical energy that can be generated in excess of the electrical energy which would be produced if the fossil-fuels required for nuclear reactor development were instead diverted for the direct production of electricity in fossil-fuel generating plants. An extensive analysis of net energy in nuclear reactors has been published recently.6 REFERENCES
I. UCSDINSF(RANN) 2. 3. 4. 5. 6.
Workshop on Net Energy in Shale-Oil Producfion, report prepared by S. S. Penner, published as Section VIII in UC.SD/NSF(RANN) Workshop on In Situ Recovery of Shale Oil, S. S. Penner. ed.. U.S. Government Printing Office, Washington, D.C., 1975. See. also, S. S. Penner and L. Icerman, Energy: V&me 11, Non-nuclear Energy Technologies, Section 9.14. Addison-Wesley. Reading, Massachusetts (1975). See the Appendix to Ref. 1 by C. E. Clark, Jr. and D. C. Varisco. P. Chapman, The ins and outs of nuclear power. New Scientist 866-869 (19 Dec. 1974). W. K. Davis, Nuclear power’s contribution to energy growth. Paper presented at an Atomic Industrial Forum Conference on Accelerating Nuclear Power Plant Construction, 3 March 1975, New Orleans, Louisiana. R. M. Rotty, A. M. Perry and D. B. Reister, “Net Energy from Nuclear Power,” I.E.4 Rept-75-3, Institute for Energy Analysis, Oak Ridge Associated Universities, Oak Ridge, Tennessee (Nov. 1975).
PerpamonPress 1976. Printed in Great htam
CONSTRUCTION OF NUCLEAR REACTORS TO OBTAIN DESIRED ELECTRICITY-GENERATING CAPACITYt S.
S.
PENNER
Department of Applied Mechanics and Engineering Sciences and Energy Center, University of California. San Diego,LaJoIla,CA 92037, U.S.A. (Received 31 July 1975) Abstract-Schedules for nuclear-reactor development may be determined for any desired electricitygenerating capacity. Net-energy ratios for nuclear-energy production from LWRs (light-water-moderated reactors) using high-grade ore are sufficiently favorably to allow construction of reactor scenarios that imply large savings in fossil-fuel resources during predetermined periods of time.
I. INTRODUCTION
The development of any new resource requires an energy investment of existing resources. Depending on the new energy technology involved, the annual and total energy returns may be negative or positive. A continuous time-dependent energy-investment schedule is shown schematically in Fig. 1 for a successful new resource development. We have defined the following three times in this resource development schedule: the time A? corresponding to the minimum completion time for the new resource, the time 6, after which there is a net annual return of energy, and the time t, after which the time-integrated energy investment first becomes positive. Figure 1 represents an idealized energy-development schedule. In practice, both the annual rates of energy investment and the annual rates of energy return may be step-functions of the time because new energy-resource developments are initiated and completed in incremental units. A schematic diagram depicting this more realistic situation is shown in Fig. 2. The times At, fh and t, are defined as before. The ratio of the height of the first positive increment of the annual return of energy to that for the first negative annual investment of energy depends critically on the net energy ratio, R, for the technology under development and on the life of the new energy-producing plant (see Section 3 for details). For a sufficiently large value of R, t, will generally occur within a year or two of fh. 2. SIMPLIFIED
SCHEMES FOR DETERMINING REACTOR-DEVELOPMENT SCENARIOS; EVALUATIONS OF rh AND r,
We make no distinction between the times in starting a resource development within a given year, i.e. we assume in effect that all energy-resource development is initiated at the beginning of the year and completed at the end of the year. Let N(i) be the number of reactors started during the i th year after the current year; the current year is identified by the label 0 and the number of reactors started during the current year is N(0). The parameter EC is defined to be the total energy cost for the construction of a new reactor which requires At years for completion. Assuming that uniform energy expenditures are made during capital construction and capital investment, the average yearly energy cost is &/At for each new reactor. The total net energy return from the reactor (with proper allowance for all operating charges) is designated as E,, during the plant life of rY years. Again assuming a uniform rate of energy return, the yearly energy return for each new reactor after completion becomes E,/t, The net energy ratio is defined as
(2.1)
R = E,/Ec tThis paper is based on a study performed for Physical Dynamics, 4.5
Inc.. of La Jolla, California (April 1975).
S. S. PENNER
46
ANNUAL
t
RATE
OF ENERGY
RETURN
2 2 ?i
NET
ANNUAL
RETURN
OF ENERGY
5 20 =, z
.
a
I
{ANNUAL RATE /‘ 1 I OF ENERGY INVESTMENT’
2 a
L At
1
d \
I I I
tb
t,
I
TIME+
I
Fig. 1. Schematic diagram showing continuous annual rates of energy investment, return and net energy available as functions of time for a successful new resource development. The minimum construction time for the new resource is At, the break-even time f,, corresponds to the time when the net annual return of energy reaches zero, and the time t, is the time required to achieve a time-integrated energy return that equals zero. The entire enterprise becomes a positive contribution to energy availability after t, years when the total area of the shaded region just equals zero.
t
+I
5
1
ANNUAL
K z Ly
RATE OF ENERGY
RETURN-
5
I
-NET
ANNUAL
._L........\ ?..........,. ... ......,.,l,l,.jj, OF ENERGY RiiiiB:~#:i:i:ri:~:~~:~
~~~~~1
I
, ,
a
I ,ANNLm ,OF ENERGY
RATE RETURN
“FWL,
INVESTMENT
;
;I At
tb
TIME
-
6
Fig. 2. Schematic diagram showing incremental annual rates of energy investment, return and net energy available as functions of time for a successful new resource development. The minimum construction time for the new resource is At, the break-even time fs corresponds to the time when the net annual return of energy reaches zero, and the time t, is the time required to achieve a time-integrated energy return that equals zero. The entire enterprise becomes a positive contribution to energy availability after t, years when the total area of the shaded region just equals zero.
for the construction of a new reactor. As already noted, EC is invested uniformly over At years while IL is returned at a uniform rate during tre years after At years have elapsed from initiation of construction. (a) Net annual rate of return of energy With the specified definitions, the total energy investment years is seen to be (Ec/At)[N(t) while the energy returned
per year in construction
+ N(t - 1) + N(t - 2) +. . . + N(t -At. + I)],
per year is given by the expression
after t
Electricity-generating
capacity
of nuclear
41
reactors
(E,/trp)[N(O) + N(1) + . . . + N(t - At )I, where all time indeces must be positive for physically meaningful definitions of terms. In view of the preceding relations, the net energy that becomes available per year as the result of reactor development is
(2.2) or, in terms of the net energy ratio R, f -‘it = Rthtltv) 2 N(t’)I’=0
~,(t)/(&/At)
A, -I
2 N(t -t’). I =_(,
(2.3)
As already noted, t’, t and At only take on integral values in the present simplified treatment. It is apparent from eqn (2.3) that the energy break-even year, fh, corresponds to
(2.4) The shortest possible break-even time occurs after f,, = At years when the initial reactor development has been completed and useful energy generation first takes place; for th = At. eqn (2.4) leads to the result N(0) = (tv/RA\t) 2 N(At - t’) ,‘=O = (b/RAt)[N(At)+N(At For an accelerated nuclear-reactor t = 25 y, R = 30; hence
development
- l)+.
using high-grade
. .+ N(l)]. fuel, we choose
(2.5) At = Sy,
N(0) = (1/6)[N(S) + N(4) + N(3) + N(2) + N(l)]. i.e. a positive return becomes first possible after 5 years provided the number of reactors started during the initial year equals one sixth of the total number of reactors started during the following 5 years. It is apparent from eqn (2.5) that smaller values of R dictate heavier initial program scheduling. For example, for R = 10, N(0) = (l/2) i I’-
N(f) I
if the annual energy investment is to be returned after f,, = At = 5 years following the year of start-up of construction. In practice, we shall generally be concerned with the evaluation of ea for values of t appreciably larger than At but normally shorter than t(-25~). It is convenient to use eqns (2.2) or (2.3) for a parametric representation of eO. We shall now examine several representative reactor-development schedules. (i) Uniform scheduling of reactor development. For uniform scheduling of reactor development, all of the N(t’)=No are equal. Hence eqn (2.3) becomes ~,(f)/(Ec/At) and c,(t)/(&/At)
= N,,[R(At/ts)(l+
t -At)-
At]
(2.6)
will be positive for R(At/tr)(l+t
-At)>At;
thus, for R = 30, At = 5y, tie = 25y, a positive energy return will first be realized during the fifth EGY Vol. I, No. I-D
S. S.
48
PENNER
year after the present year; for R = 5, At = Sy, t2 = 25y, t will be longer than 9y before a positive energy return occurs. (ii) Sequential doubling of reactor development. We assume that No reactors are developed during the current (t’ = 0) year, 2N0 during the next (t’ = 1) year, 22N0during the t’ = 2 year, etc. Equation (2.3) becomes
l,(t)/(Ec/At)=
[R(At/cie)(l +2+2*+. _ (2’ + 2”
. .+2’-*‘)
+ . . . + 2’-+‘)]&
(2.7)
The dimensionless parameter ~.(t)/(&lAt)N,, will now assume positive values only for large ratios of R At/f,. In particular, for R = 5, At = Sy, tZ = 25y, a positive energy return will never occur. The minimum value of R(At/t,) for which a positive energy return becomes possible for At = 5y and t = 1Oy is easily found to be R(At/t,) = 31.49 from eqn (2.7), i.e. R = 157.46 for tie = 25~ and At = 5y ; this value of R is unrealistically large. Even for t = 2Oy,t, = 25y, R must be at least 155 before a positive energy return occurs. We conclude that continuous doubling of the rate of reactor development is not a realistically defensible posture for a program that is designed to increase the total U.S. energy availability. (iii) Sequential doubling of reactor development for t, years, followed by termination of construction. For the specified model, eqn (2.3) becomes ~o(t)/(Ec/At)= No[R(At/tu)(l +2+22+...+2’-A’) -
@‘I+
p-
+ . . . + 2”C”“‘)]
(2.8)
for t, 5 t 5 tl + At; for t, + At < t < tie + At, reactor development is complete and maximum energy output occurs. As an example, we choose a 10-y reactor-development schedule, i.e. t, = 10~. Setting again At = 5y, tip = 25y, R = 30, R(At/t,) = 6, we may determine the dimensionless ratio ~,(f)/(E~/dt)N~ as a function of t from eqn (2.8). Some representative numerical results are listed in the following table. Table 1. Nmericalvaluesof c(t)/(E,/At)N,asafunctionof
t,
Y
t forR = 30, t, =
IOY,AI = sy, ~rp= 25~
c,(t)/(EC/At)No
t
5.114
Reference to Table 1 shows that the yearly energy output is positive three years after completion of initiation of construction for the assumed model. If we choose N, to be 10, then the annual energy output equals 15,300 times the annual energy charge for reactor development during the year with index 0 at a time corresponding to 13 years after program initiation and 3 years after initiation of construction has been terminated. Maximum output is reached 15 years after the year during which the program was initiated when the entire reactor-development program has been completed. Thereafter, the energy output remains constant until the year t+At is reached, when the first completed reactors have attained their useful life span. (iv) Other models and concluding remarks. The preceding calculations are easily generalized to arbitrary construction schedules. We conclude by noting the dominant importance of the net energy ratio R, of the reactor construction time At, and of the reactor life time TVin determining the magnitudes of available positive or negative energy charges that are incurred or released during the expansionary phase of an energy-resource development schedule. For fixed values of these parameters, a construction program may be invented that yields the desired energy return on a predetermined schedule provided the ratio R(Atl&) is sufficiently large. In the following Section 3, we comment briefly on applicable values of R.
Electricity-generating
(b) Integrated net-energy returns The integrated energy investment (Ec/At){N(t
capacity of nuclear reactors
49
of the year t is
at the beginning
. *+
- 1) + 2N(t - 2) t 3N(t - 3) +. ..+AtN(t-At)tAt[N(t-At-l)t.
N(O)lI, (2.9)
while the integrated
energy return is
(E,/trp)[N(O)(t Hence the total or integrated
-At)+N(l)(t
-At
- l)t..
.+N(t
-At
- l)].
(2.10)
net energy return is
ET = (E,/tip)[N(O)(t -At) t N(l)(t
- 1) + 2N(t - 2) t 3N(t - 3) +. . . + AtN(t - At)
- (EJAt){N(t t At[N(t
- 1) +. . . + iV(r - At - l)]
-At
(2.11)
- At - 1) +. . . t N(O)]}.
Thus the time t, after which the entire enterprise becomes a net producer of energy is determined by the relation (RAt/tu)[N(O)(t,
-At)
t N(l)(t,
-At
- 1) t
= N(t, - 1) t 2N(t, - 2) t 3N(t, - 3) t + At[N(t,
-At
. . + N(t, -At
- l)]
. . + A\tN(t, - Ar)
- 1) t . . . t N(O)].
(2.12)
We shall now work out analytically simple but instructive examples referring to well-defined construction schedules. (i) Uniform scheduling of reactor deuelopment. For uniform scheduling of reactor development, ail of the N(t ‘) = N0 are equal. Hence’ eqn (2.11) becomes Er/Ec=No[(R/&)(t-At)+(t-At-l)+..,+11 - (l/At)[l
t 2 +. . . t At + At(r -At
- l)].
(2.13)
As an example, we use the values t, = 7y, At = 5y. In this case, the ratio R At/t, is determined by the relation Rhtlt, = 6.67 or R = 33.4 for tip = 25~. Similarly, for t, = 8y, At = Sy, R At/& = 4.17 and R = 21.8 for tu = 25~; for t, = lOy, At = 5y, RAt/& = 2.33 and R = 11.7 for tY = 25~; for t, = 15y, At = Sy, RAtIt, = 1.09 and R = 5.45 for ty = 25~. Thus, the net integrated energy break-even time that can be reasonably expected for a nuclear reactor with R = 30, a 5-year construction time and a 25-year life, for the special case of a uniform construction schedule, is a little longer than 7 years. Comparison with the analysis of section 2a(i) shows that t, exceeds fh by a little more than 1 year. For large values of R, this type of result is generally expected. (ii) Sequential doubling of reactor development for t, years, followed by termination of construction. We now use again a model in which construction of No reactors is initiated during the current year, 2N0 during the following year, 2*No during the year thereafter, etc. Equation (2.12) then takes the form (RAt/t,)[(t,
-At)t2(t,
-At
- 1)+2’,-‘“1
={[2+‘(t,
-r,)+.
. . +2’r-A’(At -I)]
+ At(2’rY”’ ’ + 2’r-“-* + . . . + I)}
(2.14)
for t, 5 t, I t, + Af; for t, t At < t < ty + At, reactor development has been completed and maximum energy output occurs. For t, = 1Oy and At = 5y, we may then calculate from eqn (2.14) the applicable values of R At/t, for selected values of t,. The results of these computations are listed in Table 2, together with the required values of R for At = 5y, t, = 10~. For a value of R = 10 referring to total energy production, or a value of R = 30 referring to electricity production, we find from Table 2 that the integrated energy return equals the
50
S.
Table 2. Numerical values of RAt/t,
S. PENNER
and of R, as computed from eqn (2.14) for selected values of t, - t, with t, = 10y. and I, < 1, < ty +Af
r
--
Rat/t ,
R (for
c t=5y,
t,=ZSy,
10
34.72
173.6
11
33.03
165.
12
23.82
119.1
13
15.29
14
9.09
45.45
76.
tl=lOy)
2
45
15
5.11
25.55
16
2.51
12.
17
1.67
8. 35
18
1.25
b. 25
55
integrated invested energy during the sixteenth or fourteenth year, respectively. Thus, for sequential doubling of nuclear-energy plants, the total energy investment will be returned during the sixth or fourth year after termination of construction, depending on the applicable value of R. For R = 30, the integrated energy return is positive before the last generation of nuclear reactors has been completed. 3. NET
ENERGY
RATIOS
We have considered methodological aspects in the calculation of net energy ratios elsewhere.’ A review of these considerations may be found in Refs. 1, 2. Here we content ourselves with the statement that we shall mean the ratio RI [see Refs. 1 or 2 for precise definition of this term] whenever we use the term R in the following discussion. Representative values of R for shale-oil recovery have been given by Clark and Variscoj and have been found to have values around 8. Chapman* has estimated net energy ratios for a number of nuclear reactors. Energy charges for fuel processing (but not the intrinsic energy content of the fuel) were explicitly included for ores with uranium mass fractions of 3 x lo-’ and 7 x lo-‘, both for refueling and for initial fuel recovery. In Chapman’s analysis, a 10’ Mw, plant produces a net output, after consideration of distribution and other losses, of 255.8 Mw, or 5.602 x 10’” kwh, over a life of 25~ at a 62% average load factor when the lower grade ore is used; the corresponding values for the higher grade ore are 523.4 Mw, and 1.14625 x 10” kwh,, respectively. The total energy costs for construction of the 10’ Mw, reactor are 2.2229 x IO’” kwh, and 1.0162 x IO”’ kwh, for fuel processed from low-grade and high-grade ores, respectively. Thus, the energy ratios, defined as usable electrical energy produced divided by the thermal energy input, lie between 2.5 for low-grade ore and 11.4 for high-grade ore, respectively. Chapman has generally obtained energy ratios around 2 or 3 for low-grade ore and 10 for high-grade ore. Chapman notes4 that other workers have given ratios as low as 3 even for high-grade ore. There is a basic criticism that may be raised against the conclusions that are implied by Chapman’s numerical estimates, which is to some extent corrected by Chapman when he notes that the development of nuclear reactors may be viewed as an efficient procedure for converting fossil-fuel energy to electricity. We note that fossil fuel is used extensively today for the generation of electrical energy at an average conversion efficiency of about 33%. For this segment of the fossil-fuel industries, the applicable energy ratio is therefore the ratio of electrical energy produced by the nuclear reactor to the electrical energy which would be generated through application of the fossil fuels used in the construction of nuclear reactors, i.e. the applicable energy ratios are effectively tripled for that portion of nuclear-power generation which replaces electrical energy use that is normally produced from fossil-fuel plants. There is an obvious corollary to the preceding critique, which refers to the idea expressed by some that nuclear energy will ultimately become a primary source of electricity and a secondary source of portable fuels. For applications of this type, the energy ratios of Chapman must be reduced to allow for the loss of energy in the conversion of electricity to portable fuels. For such sequential processes as electrolysis followed by fuel-cell utilization for transportation, the overall energy loss would be expected to be perhaps 25% (if we assume that efficiencies for implementation of transportation are comparable once we begin with either fossil fuels or portable fuels in fuel cells).
Electricity-generating
capacity of nuclear reactors
51
Davi? has presented the following energy charges for an 1100 Mw, nuclear power plant operating at a 60% load factor: for plant construction, 0.74 x lo9 kwh, ; for the initial fuel load, 1.00 x 10’ kwh,; for replacement fuel, 050 x lo9 kwh,. Assuming a twenty-year plant life, the net energy ratio is seen to be 0.60 x 964 x lo9 kwh, R= (1.74 x 109/20) kwh.. + 0.50 x lo9 kwh, = 9’9’ This calculation involves pessimistic assumptions concerning the fuel-processing costs (i.e. no reduction in power consumption for diffusion plants and plutonium recycle nor allowance for anticipated fuel enrichment by centrifuging). The fuel is recovered from conventionally used, high-grade ore (2000-2500 ppm of UIOa in rock). For recovery from low-grade ore (e.g. Chattanooga shales) with 50-70 ppm of U308, the ore mining and milling costs have been estimated by A. Weinberg to be about 25 kwh,/ton of ore.” For this low-grade ore, Davis then calculates an incremental energy charge of 0.32 x 10’ kwh, for the initial core and 0.14 x lo9 kwh, for annual fuel loading. The corresponding net energy ratio becomes 0.60 x 9.64 x 10’ kwh, R = (2.06 x 109/20) kwh,. + 064 x 10’ kwh, = 7’7’ Davis’ net energy ratios are seen to be appreciably lower than the values which we derived from Chapman’s data. However, Davis states that his numbers are uncertain by perhaps as much as a factor of 2. (a) Application of Chapman’s R to a reactor-development schedule We conclude this brief commentary on net energy ratios by noting that the rather large value of R = 30 may be considered to be consistent with the estimate for the ratio of electrical energy derived from fission reactors to the electrical energy which would have been derived from the fossil fuels used in the construction of nuclear reactors if they had instead been used in fossil-fuel power plants. Thus, some of our optimistic examples of energy return for nuclear-reactor development scenarios represent reasonable quantitative descriptions of possible implementation schedules. In particular, a lo-year development schedule with doubling each year of the installed number of reactors [see Section 2a(iii) and Table 21 will produce within 15 years a total supplementary annual electrical energy output of 2.5 x 10” kwh, if the following number of 10’ Mw,. reactors No were installed this year: 2.5 x 10” kwh,/y = 1.0235 x 10” x [lo” kw, x 8.76 x 103(h/y)] x [(l/30) for the energy charge per reactor] x N, or N,, = 0.836, i.e. 836 Mw, must be constructed during 1975 with a 5-y completion schedule and doubling of the capacity installed each year during the succeeding 10 years. It should be noted that the preceding statements refer to the supplementary electrical energy that can be generated in excess of the electrical energy which would be produced if the fossil-fuels required for nuclear reactor development were instead diverted for the direct production of electricity in fossil-fuel generating plants. An extensive analysis of net energy in nuclear reactors has been published recently.6 REFERENCES
I. UCSDINSF(RANN) 2. 3. 4. 5. 6.
Workshop on Net Energy in Shale-Oil Producfion, report prepared by S. S. Penner, published as Section VIII in UC.SD/NSF(RANN) Workshop on In Situ Recovery of Shale Oil, S. S. Penner. ed.. U.S. Government Printing Office, Washington, D.C., 1975. See. also, S. S. Penner and L. Icerman, Energy: V&me 11, Non-nuclear Energy Technologies, Section 9.14. Addison-Wesley. Reading, Massachusetts (1975). See the Appendix to Ref. 1 by C. E. Clark, Jr. and D. C. Varisco. P. Chapman, The ins and outs of nuclear power. New Scientist 866-869 (19 Dec. 1974). W. K. Davis, Nuclear power’s contribution to energy growth. Paper presented at an Atomic Industrial Forum Conference on Accelerating Nuclear Power Plant Construction, 3 March 1975, New Orleans, Louisiana. R. M. Rotty, A. M. Perry and D. B. Reister, “Net Energy from Nuclear Power,” I.E.4 Rept-75-3, Institute for Energy Analysis, Oak Ridge Associated Universities, Oak Ridge, Tennessee (Nov. 1975).