PHGN590
Introduction to Nuclear Reactor Physics
Reactor Kinetics
J. A. McNeil
Physics Department
Colorado School of Mines
4/2009
Prompt neutron lifetime
vavg = 2.200 µ 10 ^ 5 H* cmês *L;
sScattU = .397; sAbsU238 = .363;
sAbsU235 = 677; sFissU = 577; PercentU235 = .0072;
sAbsNet = H1 - PercentU235L sAbsU238 + PercentU235 sAbsU235;
sFiss = PercentU235 sFissU;
PromptLifetime = 1 ê Hvavg sAbsNetL µ 10 ^ 6 ;H* micro sec*L
Print@" Natural Uranium: tHpromptL = ", PromptLifetime, " ms"D
Natural Uranium: tHpromptL = 0.868317 ms
SaGSTR = .12; tprompt = 1 ê Hvavg SaGSTRL µ 10 ^ 6 ;H* micro sec*L
Print@" GSTR: tHpromptL = ", tprompt, " ms"D
GSTR: tHpromptL = 37.8788 ms
Delayed neutrons - one delayed neutron group
ü Mean lifetime of delayed neutron precursors
»
H* U235 - List format: Hli,biL, Data from Lamarsh & Barrata, p. 88 *L
lDelayedList = 88.0124, .000215<, 8.0305, .001412<,
8.111, .001274<, 8.301, .002568<, 81.14, .000748<, 83.01, .000273<<;
tDelayedList = 881 ê .0124, .000215<, 81 ê .0305, .001412<, 81 ê .111, .001274<,
81 ê .301, .002568<, 81 ê 1.14, .000748<, 81 ê 3.01, .000273<<;
bTotal = Sum@tDelayedList@@i, 2DD , 8i, 1, 6<D;
tAvg = Sum@tDelayedList@@i, 1DD tDelayedList@@i, 2DD , 8i, 1, 6<D ê bTotal;
lAvg = 1 ê tAvg;
Print@"» b HtotalL = ", bTotal, " » t HavgL = ", tAvg, " s » l HavgL = ", lAvg, " s^-1 »"D
b HtotalL = 0.00649 » t HavgL = 13.003 s »
l HavgL = 0.0769051 s^-1 »
2
Reactor_Kinetics.nb
H* Data from Measurement of Reactor Period handout *L
bvalue = .00642;
tDelayedList =
8855.72, .00021<, 814.1, .00141<, 86.22, .00126<, 82.3, .00253<, 8.61, .00074<, 8.05, .00027<<
tAvg = Sum@tDelayedList@@i, 1DD tDelayedList@@i, 2DD , 8i, 1, 6<D ê bvalue ê Log@2D
balt = Sum@tDelayedList@@i, 2DD , 8i, 1, 6<D
lAvg = 1 ê tAvg
ü Laplace transform method
Clear@l, b, LD
n = 2.07; vavg = 2.2 µ 10 ^ 5; SfGSTR = 0.083;
params = 8l Ø .0767, b Ø .007, L Ø 1 ê Hvavg n SfGSTRL<;
sEqn@s_D = HHs ^ 2 + Hl - Hr - bL ê LL s - r l ê LL ê. r Ø d bL ê. params
s2 + s H0.0767 - 37 798.2 H-0.007 + 0.007 dLL - 20.2939 d
s1@d_D = s ê. Part@Solve@sEqn@sD ã 0, sD, 1D
s2@d_D = s ê. Part@Solve@sEqn@sD ã 0, sD, 2D
0.5 -264.664 + 264.587 d - 264.587
1.00058 - 1.99942 d + 1. d2
0.5 -264.664 + 264.587 d + 264.587
1.00058 - 1.99942 d + 1. d2
dvalue = .25;
Period = 1 ê s2@dvalueD
39.1336
ü Direct solution
Clear@l, b, L, rD
eq1 = n '@tD ã Hr - bL n@tD ê L + l c@tD
eq2 = c '@tD ã b n@tD ê L - l c@tD
n£ @tD ã l c@tD +
H-b + rL n@tD
c£ @tD ã -l c@tD +
L
b n@tD
L
Clear@l, b, LD
n = 2.07; vavg = 2.2 µ 10 ^ 5; SfGSTR = 0.083;
params = 8l Ø .0767, b Ø .007, L Ø 1 ê Hvavg n SfGSTRL<
8l Ø 0.0767, b Ø 0.007, L Ø 0.0000264563<
Reactor_Kinetics.nb
soln = Flatten@DSolve@8eq1, eq2, n@0D ã 1, c@0D ã b ê Hl LL<, 8n@tD, c@tD<, tDD;
nsoln@t_D = Simplify@n@tD ê. solnD
csoln@t_D = c@tD ê. soln;
nplot@d_, t_D = Hnsoln@tD ê. r Ø d bL ê. params;
t b+l L-r+
-
‰
Hb+l L-rL2 +4 l L r
2L
t
Hb+l L-rL2 +4 l L r
-b + ‰
Hb + l L - rL2 + 4 l L r + ‰
t
t
b-lL+‰
L
Hb+l L-rL2 +4 l L r
ü Step negative reactivity insertion
L
Hb+l L-rL2 +4 l L r
L
t
Hb+l L-rL2 +4 l L r
lL-r+‰
Hb + l L - rL2 + 4 l L r
ì 2
L
r+
Hb + l L - rL2 + 4 l L r
3
4
Reactor_Kinetics.nb
ü Step positive reactivity insertion (sub prompt critical)
dvalue = +.25; Dt = 10; tvalue = 40;
Period = Dt ê HLog@nplot@dvalue, tvalue + DtDD - Log@nplot@dvalue, tvalueDDL;
Print@" Insertion of +$", dvalue, " of reactivity gives a period of ", Period, " sec"D
horizaxis = StyleForm@"t HsecondsL",
FontFamily Ø "Tahoma", FontColor Ø Blue, FontWeight Ø Bold, FontSize Ø 12D;
vertaxis = StyleForm@"Log@PêP0D", FontFamily Ø "Tahoma",
FontColor Ø Blue, FontWeight Ø Bold, FontSize Ø 12D;
plotname = StyleForm@"Log@PêP0D versus Time ", FontFamily Ø "Tahoma",
FontColor Ø Black, FontWeight Ø Bold, FontSize Ø 14D;
Plot@Log@nplot@dvalue, tDD, 8t, 0, 100<, PlotStyle Ø 88Red, Thickness@.005D<<,
Frame Ø True, GridLines Ø Automatic, PlotLabel Ø plotname,
FrameLabel Ø 8horizaxis, vertaxis<, ImageSize Ø 600, Background Ø LightYellow, PlotRange Ø AllD
Insertion of +$0.25 of reactivity gives a period of 39.133583519 sec
Log@PêP0D versus Time
2.5
Log@PêP0D
2.0
1.5
1.0
0.5
0.0
0
20
40
t HsecondsL
60
80
100
Reactor_Kinetics.nb
5
ü Step positive reactivity insertion (super prompt critical)
dvalue = +2.25; Dt = .10; tvalue = .40;
Period = Dt ê HLog@nplot@dvalue, tvalue + DtDD - Log@nplot@dvalue, tvalueDDL;
Print@" Insertion of +$", dvalue, " of reactivity gives a period of ", Period, " sec"D
horizaxis = StyleForm@"t HsecondsL",
FontFamily Ø "Tahoma", FontColor Ø Blue, FontWeight Ø Bold, FontSize Ø 12D;
vertaxis = StyleForm@"Ln@PêP0D", FontFamily Ø "Tahoma",
FontColor Ø Blue, FontWeight Ø Bold, FontSize Ø 12D;
plotname = StyleForm@"Ln@PêP0D versus Time ", FontFamily Ø "Tahoma",
FontColor Ø Black, FontWeight Ø Bold, FontSize Ø 14D;
Plot@Log@nplot@dvalue, tDD, 8t, 0, 1<, PlotStyle Ø 88Red, Thickness@.005D<<,
Frame Ø True, GridLines Ø Automatic, PlotLabel Ø plotname,
FrameLabel Ø 8horizaxis, vertaxis<, ImageSize Ø 600, Background Ø LightYellow, PlotRange Ø AllD
Insertion of +$2.25 of reactivity gives a period of 0.00302301 sec
Ln@PêP0D versus Time
300
250
Ln@PêP0D
200
150
100
50
0
0.0
0.2
0.4
t HsecondsL
0.6
0.8
1.0
ü Subcritical multiplication
Suppose k < 1; therefore each neutron in the parent generation produces k offspring neutrons. If a source is present that delivers
S neutrons/sec, then in each neutron lifetime Ns = S t neutrons are delivered by the source. If the source is introduced at time
t=0, then the number of neutrons after one lifetime is Ns, after two lifetimes, Ns + k Ns, after three lifetimes, Ns + k (Ns + k Ns).
In other words after one additional lifetime, the number of neutrons is increased by the number delivered by the source plus k x
the number already present. Thus, after many generations: N=Ns(1+k+k^2+k^3+...)=Ns/(1-k). Thus, introducing a neutron
source into a subcritical (k<1) reactor causes the number of neutrons to be multiplied by the factor: 1/(1-k). This can be reexpressed in terms of the reactivity: r = (k-1)/k or k = 1/(1-r), giving the subcritical multiplication factor as (1-r)/r.
Suppose k < 1; therefore each neutron in the parent generation produces k offspring neutrons. If a source is present that delivers
S neutrons/sec, then in each neutron lifetime Ns = S t neutrons are delivered by the source. If the source is introduced at time
t=0, then the number of neutrons after one lifetime is Ns, after two lifetimes, Ns + k Ns, after three lifetimes, Ns + k (Ns + k Ns).
6
Reactor_Kinetics.nb
In other words after one additional lifetime, the number of neutrons is increased by the number delivered by the source plus k x
the number already present. Thus, after many generations: N=Ns(1+k+k^2+k^3+...)=Ns/(1-k). Thus, introducing a neutron
source into a subcritical (k<1) reactor causes the number of neutrons to be multiplied by the factor: 1/(1-k). This can be reexpressed in terms of the reactivity: r = (k-1)/k or k = 1/(1-r), giving the subcritical multiplication factor as (1-r)/r.