Exercises — Chapter 26

Reactor Physics and the Multiplication Factor

Problem 26.1 ⭐ A research reactor uses fuel enriched to 20% ${}^{235}\text{U}$. Using the thermal neutron cross sections $\sigma_f({}^{235}\text{U}) = 584$ b, $\sigma_c({}^{235}\text{U}) = 99$ b, $\sigma_a({}^{238}\text{U}) = 2.68$ b, and $\nu = 2.43$:

(a) Calculate the reproduction factor $\eta$ for this fuel.

(b) Compare $\eta$ for 20% enrichment, 3.5% enrichment (typical PWR), and natural uranium (0.72%). What is the physical significance of the trend?

(c) What is the minimum enrichment for which $\eta > 1$ (necessary but not sufficient condition for criticality)?

Problem 26.2 ⭐ Calculate the thermal utilization factor $f$ for a homogeneous mixture of UO$_2$ (3.5% enriched) and light water, given:

• $\Sigma_a^{\text{fuel}} = 0.12\,\text{cm}^{-1}$
• $\Sigma_a^{\text{H}_2\text{O}} = 0.022\,\text{cm}^{-1}$
• $\Sigma_a^{\text{structure}} = 0.003\,\text{cm}^{-1}$ (Zircaloy cladding, etc.)

If control rods add $\Sigma_a^{\text{rods}} = 0.008\,\text{cm}^{-1}$, recalculate $f$. By what fraction does $k_\infty$ change (assuming all other factors remain constant)?

Problem 26.3 ⭐⭐ The resonance escape probability for a homogeneous natural-uranium/graphite mixture is given by:

$$p = \exp\left(-\frac{N_{238} I_{\text{res}}}{\xi \Sigma_s}\right)$$

Using $I_{\text{res}} = 277$ barns for ${}^{238}\text{U}$, $\xi = 0.158$ for carbon, and $\Sigma_s = 0.385\,\text{cm}^{-1}$ for graphite:

(a) Calculate $p$ for an atom ratio of 1 uranium atom per 400 carbon atoms ($N_U/N_C = 1/400$). Take $N_C = 8.03 \times 10^{22}\,\text{cm}^{-3}$ for graphite.

(b) Show that increasing the uranium-to-carbon ratio decreases $p$. Find the ratio at which $p = 0.5$.

(c) Calculate $k_\infty$ for the ratio in part (a), using $\eta = 1.34$ for natural uranium, $f \approx 0.90$, and $\varepsilon = 1.03$. Is this configuration critical?

Problem 26.4 ⭐⭐ The six-factor formula. For a bare spherical reactor with $k_\infty = 1.65$, Fermi age $\tau = 40\,\text{cm}^2$, and thermal diffusion length $L = 2.7\,\text{cm}$:

(a) Calculate the critical geometric buckling $B_g^2$.

(b) Calculate the critical radius $R_c$ for a bare sphere ($B_g^2 = (\pi/R)^2$).

(c) Calculate the critical mass if the core has an average density of $3.5\,\text{g/cm}^3$.

(d) If a 20-cm thick water reflector reduces the critical radius by 30%, estimate the new critical mass. Why does a reflector reduce the critical size so dramatically?

Problem 26.5 ⭐⭐ A PWR has a core containing 80 tonnes of UO$_2$ fuel enriched to 4.2% ${}^{235}\text{U}$.

(a) How many kilograms of ${}^{235}\text{U}$ does the core contain initially? (The mass fraction of U in UO$_2$ is $M_U/(M_U + 2M_O) = 238/270 = 0.881$.)

(b) If the reactor operates at 3,411 MW(th) with a capacity factor of 92%, how much thermal energy (in MWd) does it produce per year?

(c) Assuming 200 MeV per fission and all fissions are ${}^{235}\text{U}$ (approximation), how many kilograms of ${}^{235}\text{U}$ are consumed per year? (Use $1\,\text{MeV} = 1.602 \times 10^{-13}\,\text{J}$, $1\,\text{day} = 86400\,\text{s}$.)

(d) What is the average burnup (MWd/tU) for a 4.5-year fuel cycle?

Moderators and Neutron Slowing-Down

Problem 26.6 ⭐ A neutron is born with energy $E_0 = 2\,\text{MeV}$ from fission and must be thermalized to $E_{\text{th}} = 0.025\,\text{eV}$.

(a) Calculate the number of collisions required to thermalize this neutron in H$_2$O ($\xi = 0.920$), D$_2$O ($\xi = 0.509$), and graphite ($\xi = 0.158$).

(b) Calculate the moderating ratio $\xi \Sigma_s / \Sigma_a$ for each. Which is the best moderator?

(c) Heavy water is expensive (~$300–600/kg). In light of part (b), why would anyone pay for it? What advantage does it provide?

Problem 26.7 ⭐⭐ Show that in an elastic collision between a neutron (mass $m_n = 1\,\text{u}$) and a nucleus of mass $A\,\text{u}$, the minimum energy of the neutron after the collision is:

$$E_{\min} = E_0 \left(\frac{A-1}{A+1}\right)^2$$

(a) Evaluate $E_{\min}/E_0$ for hydrogen ($A = 1$), deuterium ($A = 2$), carbon ($A = 12$), and uranium ($A = 238$).

(b) Explain why hydrogen is such an efficient moderator but uranium is not.

(c) A 2 MeV neutron collides head-on with a hydrogen nucleus. What is the neutron's energy after the collision? How does this relate to why water is used as a moderator despite its neutron absorption?

Delayed Neutrons and Reactor Control

Problem 26.8 ⭐ The prompt neutron lifetime in a PWR is $\ell_p = 2 \times 10^{-4}\,\text{s}$, and the effective delayed neutron fraction is $\beta_{\text{eff}} = 0.0065$.

(a) If only prompt neutrons existed, what would the reactor period be at a reactivity of $\rho = 0.001$? ($T = \ell_p / \rho$.)

(b) With delayed neutrons, the reactor period for $\rho \ll \beta$ is approximately $T \approx \bar{\ell}_d \cdot \beta / \rho$, where $\bar{\ell}_d \approx 12.7\,\text{s}$. Calculate $T$ at $\rho = 0.001$.

(c) Express $\rho = 0.001$ in dollars. Is the reactor below or above prompt critical?

Problem 26.9 ⭐⭐ A control rod is withdrawn, inserting a positive reactivity of $\rho = +0.003$ (about $0.46\$$). (a) Calculate the stable reactor period using the approximate formula $T \approx \bar{\ell}_d \cdot \beta / \rho$. (b) How long does it take for the reactor power to double? ($t_d = T \ln 2$.) (c) If the reactor was initially at 1 MW, what is the power after 5 minutes? (d) Why is the SCRAM system designed to insert more than $1\$$ of negative reactivity within 2 seconds?

Problem 26.10 ⭐⭐⭐ Prompt criticality. Explain using physics (not just handwaving) why a reactivity insertion of $\rho > \beta$ is catastrophically different from $\rho < \beta$.

(a) Calculate the reactor period at $\rho = 1.5\$$ using $T \approx \ell_p / (\rho - \beta)$, with $\ell_p = 2 \times 10^{-4}\,\text{s}$. (b) How long does it take for the power to increase by a factor of $10^6$? (c) In the Chernobyl accident, the reactivity is estimated to have reached ~$100\$$. Calculate the corresponding period and the time for power to increase by a factor of 100.

(d) Explain why the Doppler effect (fuel temperature coefficient) is the only feedback mechanism fast enough to potentially terminate a prompt-critical excursion.

Xenon Poisoning

Problem 26.11 ⭐⭐ At steady-state full power, a PWR operates at a thermal neutron flux of $\phi = 3 \times 10^{13}\,\text{n/cm}^2\text{/s}$ with $\Sigma_f = 0.12\,\text{cm}^{-1}$.

(a) Calculate the equilibrium ${}^{135}\text{I}$ concentration using $I_{\text{eq}} = \gamma_I \Sigma_f \phi / \lambda_I$ with $\gamma_I = 0.061$ and $\lambda_I = 2.93 \times 10^{-5}\,\text{s}^{-1}$.

(b) Calculate the equilibrium ${}^{135}\text{Xe}$ concentration using:

$$X_{\text{eq}} = \frac{(\gamma_I + \gamma_{\text{Xe}}) \Sigma_f \phi}{\lambda_{\text{Xe}} + \sigma_a^{\text{Xe}} \phi}$$

with $\gamma_{\text{Xe}} = 0.003$, $\lambda_{\text{Xe}} = 2.10 \times 10^{-5}\,\text{s}^{-1}$, and $\sigma_a^{\text{Xe}} = 2.65 \times 10^6\,\text{b}$.

(c) Show that at this flux, neutron absorption dominates over decay as the xenon removal mechanism. What fraction of ${}^{135}\text{Xe}$ removal is by neutron capture vs. decay?

Problem 26.12 ⭐⭐ After a reactor operating at $\phi = 3 \times 10^{13}\,\text{n/cm}^2/\text{s}$ shuts down instantaneously:

(a) Explain qualitatively why the ${}^{135}\text{Xe}$ concentration initially increases after shutdown, even though no new fission is occurring.

(b) The xenon concentration after shutdown can be approximated (for the early buildup phase) by noting that $I$ decays into Xe while Xe decays away with no burnup. Set up the differential equations and show that the peak xenon occurs at approximately:

$$t_{\text{peak}} \approx \frac{1}{\lambda_I - \lambda_{\text{Xe}}} \ln\left(\frac{\lambda_I}{\lambda_{\text{Xe}}} \cdot \frac{I_{\text{eq}} \lambda_I}{I_{\text{eq}} \lambda_I + \lambda_{\text{Xe}} X_{\text{eq}}}\right)$$

Estimate $t_{\text{peak}}$ numerically.

(c) A typical PWR has excess reactivity of about 3,000 pcm available from control rods. If the peak xenon reactivity worth exceeds this, the reactor cannot restart. Estimate whether a 3,000 pcm control rod worth is sufficient to override the xenon peak. (Use the equilibrium values from Problem 26.11.)

The Nuclear Fuel Cycle

Problem 26.13 ⭐ The production of 1 kg of 3.5%-enriched uranium from natural feed (0.72% ${}^{235}\text{U}$) with tails assay of 0.3% requires:

(a) How many kg of natural uranium feed? (Use mass balance: $F \cdot x_F = P \cdot x_P + W \cdot x_W$ where $W = F - P$.)

(b) How many SWU? (Use the value function $V(x) = (2x-1)\ln(x/(1-x))$.)

(c) If enrichment costs $150/SWU and natural uranium costs $130/kg U$_3$O$_8$ ($1\,\text{kg U}_3\text{O}_8 = 0.848\,\text{kg U}$), what is the cost of 1 kg of enriched uranium from feed + enrichment alone?

Problem 26.14 ⭐⭐ Plutonium production. During irradiation, ${}^{238}\text{U}$ captures a neutron and, through two beta decays, produces ${}^{239}\text{Pu}$:

$${}^{238}\text{U}(n,\gamma){}^{239}\text{U} \xrightarrow[\beta^-]{23.5\,\text{min}} {}^{239}\text{Np} \xrightarrow[\beta^-]{2.36\,\text{d}} {}^{239}\text{Pu}$$

(a) At a flux of $\phi = 3 \times 10^{13}\,\text{n/cm}^2/\text{s}$ and $\sigma_c({}^{238}\text{U}) = 2.68$ b, calculate the rate of ${}^{239}\text{Pu}$ production per atom of ${}^{238}\text{U}$.

(b) A PWR core contains $\sim$80 tonnes of ${}^{238}\text{U}$. How many kilograms of ${}^{239}\text{Pu}$ are produced per year? (Neglect Pu burnup for this estimate.)

(c) In practice, by end of fuel life, ${}^{239}\text{Pu}$ fission produces roughly 40% of the energy. Explain why: does the Pu fraction increase or decrease during the fuel cycle? What other Pu isotopes accumulate?

Problem 26.15 ⭐⭐ Fuel burnup calculation. A PWR operates at 3,000 MW(th) for 18 months (one fuel cycle) with a capacity factor of 90%. The core contains 100 tonnes of heavy metal (uranium).

(a) Calculate the total energy produced in MWd.

(b) Calculate the average burnup in MWd/tU.

(c) If each fission of ${}^{235}\text{U}$ releases 200 MeV, how many kg of fissile material were consumed? (Include both ${}^{235}\text{U}$ and estimate that 35% of fissions were from Pu.)

(d) If the initial enrichment was 4.5%, what fraction of the initial ${}^{235}\text{U}$ was consumed?

Reactor Safety and Accident Analysis

Problem 26.16 ⭐ Immediately after shutdown, a 3,000 MW(th) reactor produces decay heat at approximately 6% of full power.

(a) What is the decay heat in MW immediately after shutdown?

(b) The decay heat power approximately follows $P(t) = P_0 \cdot 0.066 \cdot [t^{-0.2} - (t + t_s)^{-0.2}]$ where $t$ is time after shutdown (seconds) and $t_s$ is the total operating time (seconds). For $t_s = 3$ years, calculate the decay heat at 1 hour, 1 day, 1 week, and 1 year after shutdown.

(c) This decay heat must be removed or the fuel will melt. If the fuel melting temperature is ~2,800$°$C and the UO$_2$ has a heat capacity of $\sim$300 J/(kg$\cdot$K), roughly how long would it take for the core (100 tonnes) to heat from operating temperature (1,200$°$C) to melting if all cooling is lost at $t = 0$? (Use the immediate decay heat from part (a).)

Problem 26.17 ⭐⭐ TMI analysis. At Three Mile Island Unit 2, the stuck-open PORV valve released primary coolant at a rate sufficient to uncover the core.

(a) If the primary loop contains 250 m$^3$ of water at 155 bar and 300$°$C, and the PORV discharges at a rate of approximately 20 kg/s, estimate how long it takes to lose enough coolant to uncover the top of the fuel ($\sim$50% of coolant volume). Take the water density at operating conditions as $720\,\text{kg/m}^3$.

(b) The operators throttled the emergency core cooling system (ECCS) at about $t = 4\,\text{min}$. Given your answer to (a), was there time for the operators to prevent core damage if they had correctly diagnosed the stuck-open PORV?

Problem 26.18 ⭐⭐⭐ Chernobyl: the void coefficient. In the RBMK, the reactivity can be decomposed as:

$$\rho = \rho_{\text{fuel}} + \rho_{\text{moderator}} + \rho_{\text{coolant}} + \rho_{\text{rods}}$$

The positive void coefficient arises because the light-water coolant acts primarily as a neutron absorber (the graphite provides moderation). When coolant voids increase by $\Delta\alpha_v = 0.1$ (10% increase in void fraction):

(a) Estimate the reactivity insertion if the void coefficient is $\alpha_v = +4\,\text{pcm}/\%$ void (a value characteristic of the RBMK at low power). Express in dollars ($\beta = 0.0048$ for the RBMK fuel mixture, which includes ${}^{239}\text{Pu}$).

(b) The Chernobyl power excursion is estimated to have inserted ~$100\$$ of reactivity in under 1 second. Using $\ell_p \approx 10^{-4}\,\text{s}$, calculate the power doubling time. (c) Explain why the Doppler coefficient (fuel temperature feedback) ultimately terminated the excursion — but not before the fuel had disintegrated. What was the estimated peak power? --- **Problem 26.19** ⭐⭐ **Fukushima: decay heat.** At the time of the earthquake, Fukushima Daiichi Unit 1 (460 MWe, ~1,380 MW(th)) had been operating at full power for many months. (a) Calculate the decay heat 1 hour after shutdown. (b) The reactor core contains ~70 tonnes of UO$_2$ fuel. If no cooling is provided, estimate the rate of temperature increase of the fuel in $°$C/hour at $t = 1$ hour. (Use $c_p \approx 300\,\text{J/(kg}\cdot\text{K)}$ for UO$_2$.) (c) The station batteries (the only remaining power source) could run the emergency cooling system at a flow rate that removed ~15 MW of heat. Was this sufficient at $t = 1$ hour? At $t = 8$ hours (when the batteries were nearly exhausted)? (d) Explain why the zirconium-water reaction ($\text{Zr} + 2\text{H}_2\text{O} \to \text{ZrO}_2 + 2\text{H}_2$), which becomes significant above ~1,200$°$C, is an example of positive feedback that accelerates core damage. --- ## Advanced Reactors and the Nuclear Debate **Problem 26.20** ⭐⭐ **Fast reactor physics.** In a fast reactor (no moderator), neutrons are not thermalized. The average neutron energy is ~100–200 keV. (a) At fast neutron energies, $\sigma_f({}^{238}\text{U}) \approx 0.3\,\text{b}$ and $\sigma_f({}^{239}\text{Pu}) \approx 1.8\,\text{b}$, with $\nu({}^{239}\text{Pu}) \approx 2.95$. Calculate $\eta$ for pure ${}^{239}\text{Pu}$ fuel in a fast spectrum if $\sigma_a({}^{239}\text{Pu}) \approx 2.1\,\text{b}$. (b) Why is $\eta$ higher in a fast spectrum than in a thermal spectrum? What does this imply for the breeding ratio? (c) A fast breeder reactor with breeding ratio $BR = 1.2$ starts with 5,000 kg of fissile Pu. After one fuel cycle (5 years), how much fissile Pu has been produced (net)? How does this compare to the fresh fuel needs of a thermal reactor? --- **Problem 26.21** ⭐⭐ **SMR economics.** An SMR project proposes a 300 MWe reactor with a capital cost of $2.5 billion and a construction time of 5 years. Compare with a conventional 1,200 MWe plant at $10 billion and 8 years construction. (a) Calculate the capital cost per installed kWe for each. (b) Assuming both plants operate at 92% capacity factor, 40-year lifetime, fuel + O&M costs of $15/MWh, and a discount rate of 7%, estimate the LCOE for each using the simplified formula: $$\text{LCOE} \approx \frac{\text{Capital cost} \times \text{CRF}}{\text{Capacity factor} \times 8760} + \text{Fuel + O\&M}$$

where $\text{CRF} = \frac{r(1+r)^n}{(1+r)^n - 1}$ is the capital recovery factor, $r = 0.07$, $n = 40$ years.

(c) If factory learning reduces the SMR capital cost by 20% after the fifth unit, recalculate the LCOE. Is the SMR competitive with the large reactor?

Problem 26.22 ⭐⭐⭐ Waste radiotoxicity. Spent fuel from a PWR contains (per tonne of initial heavy metal at 50 GWd/tU burnup):

(a) Calculate the total activity (in Bq) of each nuclide per tonne at discharge.

(b) Calculate the activity at $t = 300$ years and $t = 10,000$ years. Which nuclides dominate at each time?

(c) The natural uranium ore from which the fuel was mined had an activity of approximately 25 kBq/kg (from U and its decay chain daughters). At 50 GWd/tU burnup, roughly 7.5 tonnes of natural uranium produced 1 tonne of enriched fuel. Calculate the total activity of the original ore. After how many years does the spent fuel activity drop below the original ore activity?

Energy Comparisons and Policy

Problem 26.23 ⭐ A 1 GWe nuclear power plant operating at 92% capacity factor produces how many TWh of electricity per year? Compare with:

(a) A 1 GWe coal plant (85% capacity factor, 30% efficiency, coal energy content 24 MJ/kg). How many tonnes of coal per year?

(b) A wind farm with the same nameplate capacity (1 GWe) but 35% capacity factor. How many TWh per year? How many 3-MW turbines are needed?

(c) A solar PV farm with the same capacity, 25% capacity factor. How many TWh per year? At 5 MW/km$^2$ power density, how many km$^2$ of land?

Problem 26.24 ⭐⭐ Deaths per TWh. Using the data in Section 26.7.2:

(a) The U.S. nuclear fleet has generated approximately 100,000 TWh of electricity since 1957 with zero deaths from radiation to the public. Coal has generated approximately 150,000 TWh with an estimated 370,000 premature deaths from air pollution. Calculate the deaths/TWh for each.

(b) Germany's Energiewende replaced nuclear capacity with renewables and (during the transition) increased fossil fuel use. If shutting down 8 GWe of nuclear (operating at 90% capacity factor) and replacing 30% of the lost generation with lignite coal (at 24.6 deaths/TWh) for 5 years, estimate the excess deaths.

(c) Discuss the ethical implications of the comparison in (b). Is comparing deaths/TWh an adequate metric for nuclear vs. other energy sources? What does it miss?

Computational Problems

Problem 26.25 💻 ⭐⭐ Using reactor_physics.py (or writing your own code):

(a) Reproduce the four-factor formula calculation for PWR fuel at 3.5% enrichment with H$_2$O moderator. Vary the fuel-to-moderator ratio from 0.1 to 2.0 and plot $k_\infty$ vs. this ratio. What is the optimal ratio?

(b) For the optimal ratio found in (a), calculate the critical radius (bare sphere) and the critical mass. Compare with known values for research reactors.

Problem 26.26 💻 ⭐⭐ Model the xenon-135 transient after shutdown from full power ($\phi = 3 \times 10^{13}$ n/cm$^2$/s):

(a) Solve the coupled ODEs for $I(t)$ and $Xe(t)$ numerically. Plot both concentrations (normalized to their equilibrium values) vs. time for 72 hours.

(b) Overlay the xenon reactivity worth ($\rho_{\text{Xe}} = -\sigma_a^{\text{Xe}} X / \Sigma_a$) on a second y-axis.

(c) Identify the time of peak xenon, the peak-to-equilibrium ratio, and the duration of the "xenon dead time" (period during which xenon reactivity exceeds 3,000 pcm).

(d) Repeat for a 50% power reduction (instead of full shutdown). How do the peak xenon and dead time change?

Problem 26.27 💻 ⭐⭐⭐ Burnup and isotopic evolution. Write a simple code that tracks the isotopic concentrations of ${}^{235}\text{U}$, ${}^{238}\text{U}$, ${}^{239}\text{Pu}$, and a lumped fission product category as a function of burnup. Use one-group cross sections:

(a) Start with 3.5% enriched fuel. Evolve the concentrations at constant flux ($\phi = 3 \times 10^{13}$) for 4 years.

(b) Plot the fraction of fission power from ${}^{235}\text{U}$ and ${}^{239}\text{Pu}$ as a function of burnup (MWd/tU).

(c) At what burnup does Pu contribute more than 50% of the fission power? Compare with typical end-of-cycle values (~40%).

Problem 26.28 🔬 Research problem: reactor design comparison. The world's reactor fleet includes PWRs, BWRs, PHWRs, and (historically) RBMKs, each with different safety characteristics determined by their physics.

(a) Read the IAEA's "Nuclear Power Reactors in the World" report (latest edition) and tabulate the number, total capacity, and average age of each reactor type currently operating.

(b) Research the void coefficient for each reactor type. Explain, using the physics of the moderator-coolant interaction, why the void coefficient is negative for PWR/BWR, effectively zero for PHWR, and positive for RBMK at low power.

(c) Select one Gen IV reactor concept (SFR, MSR, HTGR, or LFR). Read two recent technical papers on the concept and write a 2-page summary of its physics advantages, engineering challenges, and timeline to deployment. Include at least one quantitative comparison with a PWR.