Question

Problem 8 A control rod worth 60c of reactivity is half-inserted into a core of arbitrary height with a multiplication factor ($k_{eff}$) of 1. Determine the change in reactivity in pcm and its associated reactor period when: (b) the rod is instantaneously withdrawn to quarter insertion for two minutes. (b) the rod is then instantaneously inserted to 60% of its total length for thirty. (c) After the withdrawal and subsequent insertion, the rod is then returned to half-insertion. By how much has the reactor flux increased or decreased? The formula for differential rod worth with respect to height is: $\frac{dp_r}{dz} = \frac{p_r}{H}[1 - \cos\frac{2\pi z}{H}]$ where $p_r$ is total rod worth. (d) Is there a control rod configuration that could lead to prompt neutrons controlling the reaction? Instructor note: To lower the difficulty, use the non-differential form: $\rho(z) = p_r[1 - \frac{1}{2\pi}\sin\frac{2\pi z}{H}] \quad \epsilon \{0 \le z \le H\}$ Conversion that may come in handy: $\bullet$ 1 eV = 1.602 $\times$ 10$^{-19}$ J $\bullet$ 1 amu = 931.49 MeV $\bullet$ $N_A$ = 6.022 $\times$ 10$^{23}$ atoms/mol $\bullet$ 1 tonne = 1000 kg $\bullet$ 1 Ci = 3.7 $\times$ 10$^{10}$ Bq $\bullet$ 1 Gy = 100 rad $\bullet$ 1 Sv = 100 rem Recoverable energy from 1 fission process: 190 MeV NRC occupational exposure limit: 5 rem/y. Potentially useful atomic masses Radiation Quality Factors Isotope Atomic mass (u) Proton 1.007276466879 Neutron 1.00866491585 H-1 1.00782503223 H-2 2.01410177812 He-4 4.00260325413 O-16 15.99491461957 U-235 235.043930131 U-238 238.050788423 Type of Radiation QF X-rays, gamma rays 1 beta particles 1 alpha particles 20 thermal neutrons ($\le$ 1 keV) 2 fast neutrons ($\ge$ 1 MeV) 11

          Problem 8
A control rod worth 60c of reactivity is half-inserted into a core of arbitrary height with a multiplication factor ($k_{eff}$) of 1.
Determine the change in reactivity in pcm and its associated reactor period when:
(b) the rod is instantaneously withdrawn to quarter insertion for two minutes.
(b) the rod is then instantaneously inserted to 60% of its total length for thirty.
(c) After the withdrawal and subsequent insertion, the rod is then returned to half-insertion. By how much has the reactor flux increased or decreased? The
formula for differential rod worth with respect to height is:
$\frac{dp_r}{dz} = \frac{p_r}{H}[1 - \cos\frac{2\pi z}{H}]$ 
where $p_r$ is total rod worth.
(d) Is there a control rod configuration that could lead to prompt neutrons controlling the reaction?
Instructor note: To lower the difficulty, use the non-differential form:
$\rho(z) = p_r[1 - \frac{1}{2\pi}\sin\frac{2\pi z}{H}] \quad \epsilon \{0 \le z \le H\}$
Conversion that may come in handy:
$\bullet$ 1 eV = 1.602 $\times$ 10$^{-19}$ J
$\bullet$ 1 amu = 931.49 MeV
$\bullet$ $N_A$ = 6.022 $\times$ 10$^{23}$ atoms/mol
$\bullet$ 1 tonne = 1000 kg
$\bullet$ 1 Ci = 3.7 $\times$ 10$^{10}$ Bq
$\bullet$ 1 Gy = 100 rad
$\bullet$ 1 Sv = 100 rem
Recoverable energy from 1 fission process: 190 MeV
NRC occupational exposure limit: 5 rem/y.
Potentially useful atomic masses
Radiation Quality Factors
Isotope	Atomic mass (u)
Proton	1.007276466879
Neutron	1.00866491585
H-1	1.00782503223
H-2	2.01410177812
He-4	4.00260325413
O-16	15.99491461957
U-235	235.043930131
U-238	238.050788423
Type of Radiation	QF
X-rays, gamma rays	1
beta particles	1
alpha particles	20
thermal neutrons ($\le$ 1 keV)	2
fast neutrons ($\ge$ 1 MeV)	11

$Problem 8 A control rod worth 60c of reactivity is half-inserted into a core of arbitrary height with a multiplication factor (keff) of 1. Determine the change in reactivity in pcm and its associated reactor period when: (b) the rod is instantaneously withdrawn to quarter insertion for two minutes. (b) the rod is then instantaneously inserted to 60% of its total length for thirty. (c) After the withdrawal and subsequent insertion, the rod is then returned to half-insertion. By how much has the reactor flux increased or decreased? The formula for differential rod worth with respect to height is: (dpr)/(dz) = (pr)/(H)[1 - cos(2π z)/(H)] where pr is total rod worth. (d) Is there a control rod configuration that could lead to prompt neutrons controlling the reaction? Instructor note: To lower the difficulty, use the non-differential form: ρ(z) = pr[1 - (1)/(2π)sin(2π z)/(H)] ϵ{0 ≤ z ≤ H} Conversion that may come in handy: ∙ 1 eV = 1.602 × 10^-19 J ∙ 1 amu = 931.49 MeV ∙ NA = 6.022 × 10^23 atoms/mol ∙ 1 tonne = 1000 kg ∙ 1 Ci = 3.7 × 10^10 Bq ∙ 1 Gy = 100 rad ∙ 1 Sv = 100 rem Recoverable energy from 1 fission process: 190 MeV NRC occupational exposure limit: 5 rem/y. Potentially useful atomic masses Radiation Quality Factors Isotope Atomic mass (u) Proton 1.007276466879 Neutron 1.00866491585 H-1 1.00782503223 H-2 2.01410177812 He-4 4.00260325413 O-16 15.99491461957 U-235 235.043930131 U-238 238.050788423 Type of Radiation QF X-rays, gamma rays 1 beta particles 1 alpha particles 20 thermal neutrons (≤ 1 keV) 2 fast neutrons (≥ 1 MeV) 11$

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