When a power reactor is operating in a steady state (constant coolant flow, constant temperatures, etc.), the effective multiplication factor is 1 and the reactivity is zero. If any of the basic parameters, such as coolant flow or temperature, are changed (e. g., to increase or decrease the power level or to compensate for changes in fuel reactivity), then reactivity must be added or subtracted. The most common situations are those in which the reactivity is inserted at a steady rate or as a step function.

Equations 1.7 and 1.8 can be solved for the case where the reactor is taken from delayed critical (p = 5k/k = 0) to prompt critical (p = 5k/k = (3) by inserting reactivity at a constant rate (ramp insertion). Figure 1.5 shows how the relative neutron density, n/n0 = n(t)/n(0), increases with time for several reactivity insertion rates and for several values of the neutron lifetime. Table 1.4 presents similar data in tabular form.

In Fig. 1.6 the effect of inserting a step change in к is shown. The reactor is at delayed critical at t = 0.

1-3.6 Reactivity Changes

For curves, tables, and equations presented in the pre­ceding sections, we assumed that the reactivity was only being altered by some control mechanism that “inserts reac­tivity.” There are other ways that reactivity is altered in an operating power reactor. The most important are: (1) varia­tion of fission-product concentrations, (2) burnup or deple­tion of fuel, and (3) variations in reactor temperatures, pressures, and densities.

An increase in the concentration of fission products reduces reactivity because the fission products absorb some of the neutrons that carry on the chain reaction. The

•The terminology “reactivity insertion," adding or subtracting reactivity, is used here because it is commonly considered proper language of the trade. More exactly, reactivity can be negative, positive, or zero at any operating instant, and adding reactivity could mean, for example, decreasing the negative reactivity toward zero as in startup or in going critical. If, during reactor operation, power is falling and we do not want it to, we say that we add reactivity. If the power is steady but low and we want to increase it, we again say that we add reactivity. If the power is high and we want to lower it, we say that we decrease reactivity, which more exactly means inserting negative reactivity to cause the power to fall. To level the power at a lower point, however, we again say that we add reactivity to compensate or return to a condition of zero reactivity.

Fig 1.1—Reactor stable and transient periods vs reactivity for 2 3 5 U The broken lines are drawn at the delayed neutron mean lives (т, = 1M, ,r2, , r6) for 2 3 5 U The parameter / is the prompt neutron lifetime in seconds (From H C Paxton and G R Keepin, The Technology of Nuclear Reactor Safety, Vol 1, p 262,

The M 1 T Press, Cambridge, Mass, 1964 )

products of fission comprise a large variety of radioactive and stable nuclei whose relative concentrations m a reactor vary with time, power level, and prior operating history Two thermal-neutron absorbing fission products have strong effects on the reactivity of thermal reactors (the pressurized water reactors, the boiling-water reactors, and the gas-cooled reactors of Chaps 15, 16, and 18, respec­tively), namely, 1 3 5 Xe (a 9 2 hr beta cmittci) and l44Sm (a stable nuclide) Because both these nuclides are strong
absorbers of thermal neutrons, they are referred to as fission-product poisons or simply poisons The absorption of thermal neutrons by 135Xe is about 5000 times more probable, on an atom-for-atom basis, than the absorption of thermal neutrons by 2 3 5 U Similarly, 149Sm absorbs thermal neutrons nearly 90 times as easily as 2 3 5 U In almost all present-day power reactors, the chain reaction is propagated almost entirely by thermal-neutron fission processes Consequently, the presence of thermal-neutron

big 12—Stable (asymptotic) period vs reactivity for 2 3 8 U and 2 3 8 U The p irameter / is prompt neutron lifetime Heavy curves are calculated from 1 aplace-transformed prompt burst decay data corresponding points are calculated from dela>cd neutron periods and abundances (I rom H C Paxton and G R Keepin, I hi 7 iibnolnqy of mil Lit Real tin Safety Vol 1 p 263 The Ml 1 Press Cambridge Mass 1964 )

REACTIVITY 0

absorbers in the fuel reduces the rcactnitv of these reactors b absorbing neutrons that otherwise would be available to carry on the chain reaction

In the following paragraphs the basic effects of these two fission products are briefly summarised for details see Rets 3,5, and 7

(a) Xenon-135 [1] In 6 1% of the fissions of 2 3 5 U (or 5 1% of 23 3 U fissions, or 5 5% of 23 9Pu fissions), one of the fission fragments has a mass number of 135 It decays to stable 1 3 5 Ba in the following chain

fission -> 1 35Te(0 5 min)-^ 1 351(6 7 hr)
-|^13sXe(15 3 min)

О 7 1

3 5 Xe(9 2 hr)-£- 1 3 5 Cs(2 6 X 106 yr)

-£• 135Ba(stable)

Because of the short half life of 1 3 51 e, the above decay scheme tan be simplified for most purposes to

fission 1 351(6 7 hr)^- 1 3SXe(9 2 hr) — Д — 135Cs

In addition to being produced via the above chain, 1 35Xe is produced directly in 0 3% of the fissions of 2 3 5 U

The rate of change in the concentration of 1 5Xc is the difference between its production rate (per cm3) and its loss (per cm3) It is produced from the decay of 1 3I and directly from the fission process It is lost b decay to 135Cs and by neutron absorption to 136Xe(stable) The equation is thus

^=(XII+Yx2f0)-XxX-SxX0 (114)

where X = 135Xe concentration (nuclei/em3)

I = 1 3 5 I concentration (nuclei/cm3 )

Xx = l35Xe decay constant (fractional change in concentration attr’butable to beta decay) = 0 693/9 2 hr = 2 1 X 10 5/see

Xj = 1 3 5 I decay constant (fractional change in con­centration attributable to beta decay) = 0.693/6.7 hr = 2.9 X 10 s /sec ф = thermal-neutron flux (neutrons cm 2 sec 1 )

Ox = microscopic thermal-neutron-capture cross sec­tion of 1 3 5Xe = 3.5 X 10“‘ 8 cm2 Zf = macroscopic thermal-neutron-fission cross sec­tion of fuel = concentration of fuel (nuclei/cm3) times the microscopic thermal-neutron-fission cross section

Yx = fractional yield of 135Xe directly from fission

The quantity X] I can be determined by considering the rate of change in the 135I concentration. The 135I is produced from the decay of 13sTe, which, in turn, is produced directly from the fission process. Since the 135Te is so short-lived, it is valid to consider the 135i as produced directly from fission. In this case the equation for the 1 3 51 concentration is

YTe2f0 — X[I — CTjI ф (1.15)

where YTc is the yield of 135Te (6.1% for 2 3 5 U fission, etc.). The final term is the loss of 1 35I because of neutron capture (O] <g ax).

Equations 1.14 and 1.15 can be solved for various initial (t = 0) conditions and for various values of the thermal-neutron flux. One important solution is the equi­librium concentration of 13SXe. The last term of Eq. 1.15 can be neglected; so X|I= Y-pcZf0 at equilibrium condi­tions, and — 0 = YTeZf0 + Yx^f0 — XxX — 0Хф

Solving for X yields

YZf ф

4 + ОхФ where Y is Yx + Y-pe, the total fractional yield of 13sXe per fission (i. e., the yield via the 1 3 5Te chain and the direct yield). Equation 1.16 shows that, as the thermal-neutron flux is reduced, the equilibrium concentration of l3SXe becomes proportional to the flux-, for high thermal-neu-

REACTIVITY, 0 Fig. 1.4 —Stable (asymptotic) period vs reactivity for 2 3 3 U and 2 3 2 Th See caption for Fig 1 2 (From H C Paxton and G R Ktepin The Technology of Nuclear Reactor Safety Vol 1, p 265, The M I T Press, Cambridge, Mass, 1964 )

tron-flux values, the equilibrium concentration of 135Xe becomes independent of the flux

Xcq = Y2f/ax (for ф>Хх/ах)

(1 17)

Xcq s (Y2f/х)ф (for Ф < AX/dX)

The effect of the I35Xe concentration on the control and operation of a nuclear power reactor is determined by how large the absorption of neutrons by 1 5Xe is relative to the absorption of neutrons by the nuclear fuel This determines the degree that the 135Xe interferes with the chain reaction 1 he ratio of macroscopic thermal neutron — absorption cross sections is defined

Poisoning = P(t)

_ macroscopic neut abs cross section of 1 3 5 Xe macroscopic neut abs cross section of the fuel

Xdx

Nuaa

Y(at/aa)0

P(teq)=Xx+ax0

Equation 1 19 is plotted in Pig 1.7 for 23SU fuel Values of Ax and (7X are given following Eq 1.14. The total yield is Y = 0 064 and df/da = 580 barns/685 barns = 0 85 The figure shows the equilibrium poisoning to be linear with the neutron flux when 0^1O12 neutrons cm 2 sec1 (see Pq 1 17) and to approach a constant for high flux values It can be shown (e g, Ref 3, p 334) that the poisoning defined in Eq 1 18 is approximately equal to the reduction in reactivity in a thermal reactor attributable to fission — product poisoning

Change in reactivity = 5k/k s —P(t) (1.20)

To keep a reactor operating at steady state (k = 1), sufficient reactivity must be added, e g, by withdrawing control rods, to compensate for (or override) the reduction in reactivity caused by the fission products in the fuel Thus, for example, in a 23sU-fueled thermal power reactor that is operating at к = 1 with a thermal-neutron flux at the fuel position of 5 X 1013 neutrons cm 2 sec"1, the reac­tivity that must be added to compensate for the effect of the equilibrium concentration of l3SXe is about ak/k = 0 049 (see Fig 1 7)

The effect of 135Xe poisoning is most pronounced when a reactor is shut down after it has been operating at full power for a time sufficiently long that the equilibrium concentration of 13sXe (Eq 1 16) is present In this case the xenon concentration increases considerably above its equilibrium value since it is no longer being removed by thermal neutron capture The 135Xe is being produced by the decay of the equilibrium concentration of 1 3 5 1 (6 3 hr) and being lost by its own 9 2 hr beta decay 1 he net result is shown in big 18, where the poisoning is plotted as a function of time after shutdown from equilibrium for several values of the thermal neutron flux The t = 0 values of Fig 18 are obtained from the equilibrium curve shown in Fig 17 The 1 35Xe poisoning builds up to a maximum after shutdown For low values of the flux, the time to reach maximum poisoning is only a few hours For the higher flux values normally encountered in power-reactor operation, the poisoning reaches a maximum about 10 hr after shutdown The value of the poisoning does not return to its preshutdown value until 30 or 40 hr after shutdown

Table 1 4—Relative Neutron Density (n/n0) as a Function of Time During Ramp Insertions of 10 3, 10 4 , and 10 5 6k/k Per Second* Time, sec (6k/k)/sec ramp ”3U I3SU 2 3«pu /= 10 2 /= 10 6 / = 10 2 /= 10 6 /= 10 2 / = 10 6 0 5 A = 10 3 1 012 1 254 1 011 1 092 1 012 ~1 35 10 4 1 001 1 021 1 001 1 009 1 001 1 026 10 5 1 0001 1 002 1 0001 1 0009 1 0001 1 003 1 0 A = 10 3 1 047 1 723 1 042 1 218 1 048 2 112 10 4 1 005 1 045 1 004 1 018 1 005 1 057 10 5 1 0005 1 004 1 0004 1 002 1 0005 1 005 (1 4= 3 84) 1 5 A = 10 3 1 106 2 768 1 090 1 388 1 108 (16 = 6 16) 10 4 1 010 1 071 1 009 1 029 1 010 1 093 10 5 1 001 1 007 1 0009 1 003 1 001 1 009 2 0 A = 10 3 1 189 6 3 50 1 156 1 626 1 195 62 5 10 4 1 017 1 101 1 014 1 041 1 018 1 133 10 5 1 002 1 009 1 001 1 004 1 002 1 012 5 0 A = 10 3 2 644 3 0 sec = 3 7 x 10” 2 069 15 9 2 792 2 5 sec = 7 7 x 10” 10 4 1 096 1 358 1 069 1 137 1 103 1 517 10 5 1 009 1 028 1 007 1 012 1 010 1 037 10 0 A = 10 3 46 5 OO 15 7 6 8 sec = 1 4 x 103 8 59 0 OO 10 4 1 366 2 401 1 229 1 406 1 411 3 675 10 5 1 031 1 069 1 020 1 031 1 034 1 094 From J M Harrer, Nuclear Reactor Control 1 ngtneermg p 91, D Van Nostrand t ompany, Inc, Princeton, N J 1963

(b)

Fig. 1.6—Relative neutron density, n/n0, vs. time for step insertions of (a) 10~2 6k/k; (b) 10"3 6k/k; and (e) 10"4 6k/k starting at delayed critical in 2 3 3 U, 23SU, and 2 39 Pu. (From J. M. Harrcr, Nuclear Reactor Control engineering, pp. 88 and 89, D. Van Nostrand Company, Inc., Princeton, N. J., 1963.)

Fig. 1.7—Poisoning P(t), vs thermal neutron flux for equi­librium 135Xc concentration in 335U (I rom J VI Ilarrtr Nuclear Reactor Control I ngtneenng p 393, D Van Nos trand Company, Inc, Princeton, N J, 1963 )

Fig. 1 8 — Poisoning vs time tor various thermal neutron flux values, assuming equilibrium concentration of 1 3 3 Xe at t = 0 (I n>m J M Harrer, Nuclcai Reactor Control l ngtneenng p 394, D Van Nostrand Company, Inc Princeton, N J 1963 )

As Fig 1 8 shows, the value of the poisoning at maximum can be many times the equilibrium (before shutdown) value The excess reactivity required to overcome the maximum poisoning may be more than is available in power reactor, particularly if the fuel has been depicted by prior operation If this is the case, then the reactor shutdown time has to be limited to less than a few hours or more than 30 or 40 hr must be allowed

The full shutdown from equilibrium shown in Fig 1 8 is not the only situation of practical interest Often the power level is cut back or increased by some fraction of full power Initially the 13SXe concentration has a value corresponding to the initial power level, after the change in power level, the l35Xe concentration changes until it reaches a new equilibrium value corresponding to the final power level Figures 1 9 and 110 show the time to reach the maximum poisoning following a step decrease or a step increase of the thermal-neutron flux (which is directly proportional to the reactor power level) Figure 1 9 shows, for example, that a 50% cutback from 4 X 1013 neutrons tin 2 see creates a maximum,35Xt poisoning about 23,000 see (6 4 hr) after the cutback In the reverse process, Fig 1 10 shows that when the flux level is doubled from 2 X 1013 neutrons cm 2 sec ‘, the maximum 13,;Xe poisoning effect occurs about 1 1,600 see (3 2 hr) after the increase 1 rom the initial values of the neutron flux, the initial equilibrium concentration of 13sXe and 1 3,I, and the value at the time the maximum effect occurs, the maximum poisoning or maximum reduction in 5k/k tan be calculated

(b) Samarium-149 * In 1 13% of the fissions of 2 3 5 U

(or 0 66% of 2 3 3 U fissions, or 1 9% of 2 3 9 Pu fissions), one

•Numerical data used in this section are from Ref 7
of the fission fragments has a mass number 149 Some of the fissions form 149Pmand others form 149Nd

Fission -> 1 49Pm(54 hr) 1 49Sm(stahle)

1 ission 149Nd(2 hr) 149Pm(54 hr) 149Sm(stable)

Because the ‘49Nd half-life is small compared to the 1 49Pm half life, the first chain above is a good approxinta tion for both chains As noted earlier, 149Sm strongly absorbs theimal neutrons I he other nuclides in the chain are not anomalous m this respect

The rate of change of the 149Sm concentration is just equal to its production rate from 149Pm decay minus its rate of loss from thermal neutron capture (which converts it to stable 1 50Sm)

= ^Pm (Pm) — (Sm)aSm<A (121)

where (Pm) and (Sm) are the concentrations of l49Pm and l49Sm, respectively, Apm is the decay constant of 149Pm = 3 56 X 10 6/scc, a<,m is the thcrnul-ncutron — eipture cross section of 149Sm = 50,000 barns = 5 X 10 20 cm2, and ф is the thermal neutron flux (neutrons cm 2 see *) Note that, unlike 13sXe, the 149Sm is removed only when it captures thermal neutrons The rate of change of 149Pm is its production rate from fission (neglecting the intermediate 149Nd) minus its loss by beta decay (loss by neutron capture is negligible)

YHm0Zf — Apm (Pm) (1 22)

where (Pm) is the concentration (atoms/cm3) of l49Pm, Ypm is the yield of 149Pm in fission, and 2f is the macroscopic thermal-neutron-fission cross section of the nuclear fuel

0 4 8 12 16 20 24 28 32 36 40 TIME, 103 sec Fig. 1.9—Time to reach maximum xenon poisoning after a decrease of thermal-neutron flux from 01 to 02 .n

Fig. 1.10—Time to reach maximum xenon poisoning after an increase of thermal-neutron flux from фj to Ф7 7

When a power reactor has been operating at a steady state (k = 1) for many hours, the equilibrium concentra­tions of 149Pm and 149Sm (from Eqs. 1.21 and 1.22) are

Ypm02f

^Pm

Xp

m (Ptn)eq _ Yp m £f ^Sm 0 ^Sm

Note that the equilibrium concentration of 149Pm is proportional to the thermal-neutron flux, 0, while the equilibrium concentration of 149Sm is independent of the flux.

The equilibrium 149Sm poisoning (Eq 1.18) is

P(teq) = poisoning of 1 9Sm at equilibrium concentration _ (Sm)eqOsm _ Ypm2f<Jgm _

which is also independent of the thermal-neutron flux Substituting into Eq. 1.24 the values of the yields and the fission/absorption cross section ratios gives

	2 35 u	2 3 3 U	2 39Pu
YPm	0.0113	0.0066	0.019
af/aa	580/685	524/593	860/1220
P(tCq) for 1 4 9 Sm	0.0096	0.0058	0.077

These correspond approximately to 5k/k values of —0.96% for 23SU, -0 58% for 23 3 U, and -7.7% for 23 9Pu Comparison of the equilibrium value of 1 4 9 Sm poisoning in 23 5 U-fueled reactors with the equilibrium values of 1 35Xe poisoning (Fig. 1.7) shows the former to be only about one-fourth of the latter.

When a power reactor that has been operating at steady state (k = 1) for some hours is shut down, the concentra­tion of 149Sm increases from its initial (t = 0) value by the creation of 1 4 9 Sm from 1 4 9 Pm decay

1 4 9 Sm cone, after shutdown

= 1 49Sm cone, at shutdown + (1 — e kPm()

X 1 49Pm cone, at shutdown (1 25)

Since the half-life of 14 9Pm is 54 hr, the 1 4 9 Sm concentra­tion is increased by one-half the 149Pm shutdown concen­tration during the first 54 hr after shutdown After a few hundred hours the 149Sm concentration is equal to the sum of the 149Sm concentration at shutdown and the 149Pm concentration at shutdown.

When a power reactor has been operating at a steady state (k = 1) and at constant flux for a few hundred hours, both the 149Pm and 149Sm concentrations have their
equilibrium values (Eq 1.23). If the reactor is then (at t = 0) shut down, the 149Sm concentration builds up according to Eq. 1.25. Substitution of the equilibrium concentrations into Eq. 1.25 gives

149Sm cone, after shutdown from equilibrium

= (Sm)Cq [1 + (0C7Sm/Xpm)(l — e ЛРт’)1 = (Sm)cq [1 + 1.40 X 1O"140

X (1 — e’^Pmt)] (1.26)

where ф is the thermal-neutron flux m neutrons cm 2 sec 1 and Xpm is the disintegration constant of 149Pm (= 3 56 X 10 6/sec = 0.693/54 hr). It is apparent that the poisoning effect of 1 49Sm becomes quite important in high flux operation A shutdown from operation at a flux of 1014 can increase the poisoning effect by a factor of 1 + 1.40, or 2 40 times the equilibrium poisoning (Eq. 1.24), if the shutdown continues for a few hundred hours. Sufficient excess Sk/k must be available to compen­sate for this poisoning when the reactor is started up.

(c) Fuel Burnup. During the operation of a nuclear power reactor, fuel (23 5 U, 2 33 U, or 23 9Pu) is continually being burned up (l e., fissioned) so the remaining fuel becomes depleted in the fissionable nuclide. The effect of this burnup, or depletion, is to reduce the reactivity available to compensate for fission-product poisoning or for other reactivity-reducing effects Eventually the depletion becomes intolerable, and the reactor has to be refueled

At a point in the reactor fuel where the thermal — neutron flux is ф, the absorption of neutrons decreases the concentration of fissile material (23SU, 2 3 3 U, or 2 3 9 Pu) exponentially with time

Fuel cone at time t

= (Fuel cone at t = 0) exp (—ста f0 ф dt) (1 27)

where cra is the neutron absorption cross section of the fissile material The neutron flux is assumed to vary with time (For convenience the integral can be written as 0dv t, where 0dv is the average flux during the time from t = 0 to t = t )

The fractional burnup is defined as the change in fuel concentration divided by the initial concentration. From bq 1.27 it follows that

Fractional burnup = F = 1 — e <7a<*>ivt (1 28)

As an example, the fractional burnup of 2 3 5 U in three months (7.8 X 106 sec) in a powei reactor with an average thermal-neutron flux at the fuel of 5 X It)1 3 neutrons cm 2 sec 1 is 1 — exp I(—685 barns)(5 X It)1 3 )(7 8 X 1 06 cm"2)] = 1 — exp (—0 267), or 0 234, i. e, a fractional burnup of 23.4%.

Figure 1 11 shows the relation between the fractional burnup of fuel and the resulting loss in reactivity 5k/k For a 23 4% burnup, the loss in reactivity is about 2 7% This is to be compared with the reactivity loss of 4.9% (see discussion after 1 q 1 20) attributable to equilibrium xenon poisoning at the same average neutron flux and the 1% (see discussion after fq 1 24) loss in reactivity from samarium poisoning

FRACTIONAL FUEL BURNUP (F)

I ig 1 11 Poisoning as function of fractional fuel burnup (1 rom J Л1 I Ltrr. г Rinleat Re, и tm Contsol / ngmeenng, p 199, D Van Nostraud Company, Inc Princeton, N J, 1963 )

In any actual power reactor, calculations of burnup must take into account many factors not considered in the foregoing. These include the presence of fertile material (23 8 U or 2 3 2lh) in the fuel, the energy and spatial distributions of neutron flux, the geometry and composi­tion of the fuel elements, and the operating history of the reactor. The data presented here are intended to provide a semi-quantitative indication of the effect of fuel depletion on the reactivity that is needed for instrumentation system design.