From 943066540c7d27b56530c198e69d54319665553d Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 10 Nov 2025 18:04:37 -0500 Subject: [PATCH] Auto sync: 2025-11-10 18:04:37 (36 files changed) M .task/backlog.data M .task/completed.data M .task/pending.data M .task/undo.data A Class_Work/nuce2101/exam2/NUCE2101_SABO/Submission.pdf A Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1.py A Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1_equal_loading.png A Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1_unequal_loading.png --- .task/backlog.data | 3 + .task/completed.data | 2 + .task/pending.data | 4 +- .task/undo.data | 12 + .../exam2/NUCE2101_SABO/Submission.pdf | Bin 0 -> 1414522 bytes .../nuce2101/exam2/NUCE2101_SABO/problem1.py | 321 ++++++++++++ .../NUCE2101_SABO/problem1_equal_loading.png | Bin 0 -> 490952 bytes .../problem1_unequal_loading.png | Bin 0 -> 506618 bytes .../NUCE2101_SABO/problem1_verification.py | 210 ++++++++ .../nuce2101/exam2/NUCE2101_SABO/problem2.py | 217 ++++++++ .../nuce2101/exam2/NUCE2101_SABO/problem3.py | 63 +++ .../nuce2101/exam2/NUCE2101_SABO/problem4.py | 99 ++++ .../nuce2101/exam2/NUCE2101_SABO/problem5.py | 187 +++++++ .../problem5_xenon_transient.png | Bin 0 -> 362131 bytes .../nuce2101/exam2/NUCE2101_SABO_EXAM2.pdf | Bin 0 -> 1414522 bytes .../exam2/NUCE2101_SABO_SUPPORTING.zip | Bin 0 -> 2348398 bytes Class_Work/nuce2101/exam2/latex/main.aux | 2 +- .../nuce2101/exam2/latex/main.fdb_latexmk | 19 +- Class_Work/nuce2101/exam2/latex/main.fls | 126 ++--- Class_Work/nuce2101/exam2/latex/main.log | 464 +++++++++++++++++- Class_Work/nuce2101/exam2/latex/main.pdf | Bin 483552 -> 0 bytes .../nuce2101/exam2/latex/main.synctex.gz | Bin 192193 -> 298651 bytes Class_Work/nuce2101/exam2/latex/problem1.tex | 299 +++++++++++ Class_Work/nuce2101/exam2/latex/problem2.tex | 18 - Class_Work/nuce2101/exam2/latex/problem3.tex | 14 +- Class_Work/nuce2101/exam2/latex/problem4.tex | 194 ++++++++ Class_Work/nuce2101/exam2/latex/problem5.tex | 18 +- 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z=U9SWt%f0Z9Ym7xn_AJg1_IT4l=e&jJWURq_*o`T`=Ug2Uz(}J*hjQ8&0?{g4YcUZ y#zs1gS;Q67C7cUBH@U+c2O3iSyScl#ngCrrfUoxbN*C_|uf?cnX{A)8QT`X=*_$E& literal 0 HcmV?d00001 diff --git a/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1.py b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1.py new file mode 100644 index 000000000..d9dfa1d66 --- /dev/null +++ b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1.py @@ -0,0 +1,321 @@ +import numpy as np +import matplotlib.pyplot as plt +from scipy.integrate import odeint + +# Problem 1: Two-Loop Reactor System + +print("=== Problem 1: Two-Loop Reactor with Steam Generators ===\n") + +# Given Parameters +mu = None # Reactor Water Mass Fraction (RWMF) - to be determined or specified + +# Base parameters +C_0 = 33.33 # Base Heat Capacity [%-sec/°F] +tau_0 = 0.75 * C_0 # Base Time Constant [sec/(%-sec/°F)] * C_0 = [sec] + +# Initial steady state temperature +T_hot_initial = 450 # °F (reactor outlet / hot leg temperature) +T_cold_1_initial = 450 # °F (cold leg 1 temperature) +T_cold_2_initial = 450 # °F (cold leg 2 temperature) + +print("Base Parameters:") +print(f"C_0 (Base Heat Capacity) = {C_0:.2f} %-sec/°F") +print(f"tau_0 (Base Time Constant) = {tau_0:.2f} sec") +print(f"\nInitial Steady State:") +print(f"T_hot = {T_hot_initial} °F") +print(f"T_cold_1 = {T_cold_1_initial} °F") +print(f"T_cold_2 = {T_cold_2_initial} °F") + + +# System parameters as functions of mu (RWMF) +def calculate_system_parameters(mu): + """ + Calculate system parameters based on Reactor Water Mass Fraction (mu) + + Parameters: + mu: Reactor Water Mass Fraction (RWMF) + + Returns: + dict: Dictionary containing all calculated parameters + """ + # Reactor parameters + tau_r = mu * tau_0 # Reactor time constant [sec] + C_r = mu * C_0 # Reactor heat capacity [%-sec/°F] + + # Steam Generator parameters (each) + tau_sg = (1 - mu) * tau_r / 2 # Steam Generator water time constant [sec] + C_sg = (1 - mu) * C_0 / 2 # Steam Generator heat capacity [%-sec/°F] + + params = { + "mu": mu, + "tau_r": tau_r, + "C_r": C_r, + "tau_sg": tau_sg, + "C_sg": C_sg, + "C_0": C_0, + "tau_0": tau_0, + } + + return params + + +# Example: Calculate parameters for mu = 0.5 (50% of water in reactor) +if mu is None: + mu_example = 0.5 + print(f"\n--- Example with mu = {mu_example} ---") + params = calculate_system_parameters(mu_example) + + print(f"\nReactor Parameters:") + print(f" tau_r (Reactor time constant) = {params['tau_r']:.2f} sec") + print(f" C_r (Reactor heat capacity) = {params['C_r']:.2f} %-sec/°F") + + print(f"\nSteam Generator Parameters (each):") + print(f" tau_sg (SG water time constant) = {params['tau_sg']:.2f} sec") + print(f" C_sg (SG heat capacity) = {params['C_sg']:.2f} %-sec/°F") + +# System description +print(f"\n" + "=" * 60) +print("System Description:") +print("=" * 60) +print( + """ +Two-Loop Reactor System: +- Reactor (hot leg) at T_hot +- Steam Generator 1 with cold leg at T_cold_1 +- Steam Generator 2 with cold leg at T_cold_2 +- Each loop: pump → reactor → steam generator → back to pump +- Lumped reactor volume with combined hot branches +- Equal mass flow rates in each loop +- Ignore loop transport times + +Key Assumptions: +- Each SG + cold loop has same water mass and flow rate +- Reactor and hot branches lumped into single water volume +- System initially at equilibrium (100% power, balanced) +- Constant average temperature maintained +""" +) + +# Part A: Differential Equations +print(f"\n" + "=" * 60) +print("Part A: Differential Equations") +print("=" * 60) + + +def get_differential_equations(mu): + """ + Develop differential equations for T_hot, T_cold1, T_cold2 + + Key insight: Flow heat capacity rate W = C_0 / (2 * tau_0) + This is the (mass flow rate * specific heat) for each loop + """ + params = calculate_system_parameters(mu) + + # Flow heat capacity rate for each loop (same for both loops) + W = C_0 / (2 * tau_0) + + print(f"\nFor mu = {mu}:") + print(f"Flow heat capacity rate (each loop): W = {W:.4f} %-sec/°F") + print(f"C_r = {params['C_r']:.2f}, C_sg = {params['C_sg']:.2f}") + + print(f"\nDifferential Equations:") + print(f" C_r * dT_hot/dt = P_r - W*(T_hot - T_cold1) - W*(T_hot - T_cold2)") + print(f" C_sg * dT_cold1/dt = W*(T_hot - T_cold1) - Qdot1") + print(f" C_sg * dT_cold2/dt = W*(T_hot - T_cold2) - Qdot2") + print(f"\nwhere:") + print(f" P_r = reactor power (positive for heat generation)") + print(f" Qdot1, Qdot2 = SG heat removal rates (positive for heat removal)") + print(f" W = {W:.4f} %-sec/°F (flow heat capacity rate per loop)") + + # Matrix form: dT/dt = A*T + B + print(f"\n--- Matrix Form: dT/dt = A*T + B ---") + + # Coefficient matrix A + A = np.array( + [ + [-2 * W / params["C_r"], W / params["C_r"], W / params["C_r"]], + [W / params["C_sg"], -W / params["C_sg"], 0], + [W / params["C_sg"], 0, -W / params["C_sg"]], + ] + ) + + print(f"\nMatrix A (coefficients):") + print(f" [{A[0,0]:7.4f} {A[0,1]:7.4f} {A[0,2]:7.4f}]") + print(f" [{A[1,0]:7.4f} {A[1,1]:7.4f} {A[1,2]:7.4f}]") + print(f" [{A[2,0]:7.4f} {A[2,1]:7.4f} {A[2,2]:7.4f}]") + + # For general case with P_r, Qdot1, Qdot2 + print(f"\nVector B (forcing terms, depends on P_r, Qdot1, Qdot2):") + print(f" [P_r/C_r, -Qdot1/C_sg, -Qdot2/C_sg]^T") + print( + f" [P_r/{params['C_r']:.2f}, -Qdot1/{params['C_sg']:.2f}, -Qdot2/{params['C_sg']:.2f}]^T" + ) + + return W, params + + +# Part D: Transient Simulation Setup +print(f"\n" + "=" * 60) +print("Part D: Transient Simulation (0 to 30 seconds)") +print("=" * 60) + + +def reactor_odes(y, t, W, C_r, C_sg, P_r, Q1, Q2): + """ + ODE system for two-loop reactor + y = [T_hot, T_cold1, T_cold2] + + Parameters: + P_r: Reactor power (heat generation rate) + Q1, Q2: Steam generator heat removal rates (Qdot_1, Qdot_2) + """ + T_hot, T_cold1, T_cold2 = y + + # Reactor energy balance + dT_hot_dt = (P_r - W * (T_hot - T_cold1) - W * (T_hot - T_cold2)) / C_r + + # Steam generator 1 energy balance (Q1 is heat removal rate) + dT_cold1_dt = (W * (T_hot - T_cold1) - Q1) / C_sg + + # Steam generator 2 energy balance (Q2 is heat removal rate) + dT_cold2_dt = (W * (T_hot - T_cold2) - Q2) / C_sg + + return [dT_hot_dt, dT_cold1_dt, dT_cold2_dt] + + +def simulate_transient(mu, Q1_percent, Q2_percent, t_max=30): + """ + Simulate reactor transient + + Parameters: + mu: Reactor Water Mass Fraction + Q1_percent: Steam generator 1 heat removal rate (% of nominal) + Q2_percent: Steam generator 2 heat removal rate (% of nominal) + t_max: Simulation time (seconds) + """ + params = calculate_system_parameters(mu) + W = C_0 / (2 * tau_0) + + # Initial conditions (steady state at 100% power balanced) + T_hot_0 = T_hot_initial + T_cold1_0 = T_cold_1_initial + T_cold2_0 = T_cold_2_initial + y0 = [T_hot_0, T_cold1_0, T_cold2_0] + + # Power levels (normalized - 100% = 1.0 in appropriate units) + P_r = 100.0 # Reactor power (heat generation rate) + Q1 = Q1_percent # SG1 heat removal rate + Q2 = Q2_percent # SG2 heat removal rate + + # Time array + t = np.linspace(0, t_max, 500) + + # Solve ODE system + solution = odeint( + reactor_odes, y0, t, args=(W, params["C_r"], params["C_sg"], P_r, Q1, Q2) + ) + + T_hot = solution[:, 0] + T_cold1 = solution[:, 1] + T_cold2 = solution[:, 2] + + # Calculate average temperature (mass-weighted, depends on mu) + T_ave = mu * T_hot + (1 - mu) * (T_cold1 + T_cold2) / 2 + + return t, T_hot, T_cold1, T_cold2, T_ave + + +# Example: Show how to set up simulations +print("\nSimulation cases to run:") +print("1. mu=0.5, Q1=50%, Q2=50% (equally loaded)") +print("2. mu=0.75, Q1=50%, Q2=50% (equally loaded)") +print("3. mu=0.5, Q1=60%, Q2=40% (unequally loaded)") +print("4. mu=0.75, Q1=60%, Q2=40% (unequally loaded)") +print("\nNote: Percentages are fractions of reactor power (P_r = 100%)") + +# Run example simulation +print("\n--- Running Example: mu=0.5, Equal Loading ---") +W_example, params_example = get_differential_equations(0.5) + +# Run all simulations and create plots +print("\n" + "=" * 60) +print("Running Simulations and Creating Plots") +print("=" * 60) + +# Case 1 & 2: Equally loaded (Q1=50%, Q2=50%) +fig1, axes1 = plt.subplots(2, 2, figsize=(14, 10)) +fig1.suptitle( + "Equally Loaded Steam Generators (Q1=50%, Q2=50%)", fontsize=14, fontweight="bold" +) + +for idx, mu in enumerate([0.5, 0.75]): + print(f"\nSimulating mu={mu}, Q1=50%, Q2=50%") + t, T_hot, T_cold1, T_cold2, T_ave = simulate_transient(mu, 50.0, 50.0) + + # Plot temperatures + ax = axes1[idx, 0] + ax.plot(t, T_hot, "r-", linewidth=2, label="T_hot") + ax.plot(t, T_cold1, "b-", linewidth=2, label="T_cold1") + ax.plot(t, T_cold2, "g-", linewidth=2, label="T_cold2") + ax.plot(t, T_ave, "k--", linewidth=2, label="T_ave") + ax.set_xlabel("Time (sec)", fontsize=10) + ax.set_ylabel("Temperature (°F)", fontsize=10) + ax.set_title(f"mu = {mu} (RWMF)", fontsize=11, fontweight="bold") + ax.grid(True, alpha=0.3) + ax.legend(loc="best") + + # Plot temperature differences + ax = axes1[idx, 1] + ax.plot(t, T_hot - T_cold1, "b-", linewidth=2, label="ΔT1 = T_hot - T_cold1") + ax.plot(t, T_hot - T_cold2, "g-", linewidth=2, label="ΔT2 = T_hot - T_cold2") + ax.set_xlabel("Time (sec)", fontsize=10) + ax.set_ylabel("Temperature Difference (°F)", fontsize=10) + ax.set_title(f"Temperature Differences (mu = {mu})", fontsize=11, fontweight="bold") + ax.grid(True, alpha=0.3) + ax.legend(loc="best") + +plt.tight_layout() +plt.savefig("problem1_equal_loading.png", dpi=300, bbox_inches="tight") +print("Plot saved: problem1_equal_loading.png") +plt.close() + +# Case 3 & 4: Unequally loaded (Q1=60%, Q2=40%) +fig2, axes2 = plt.subplots(2, 2, figsize=(14, 10)) +fig2.suptitle( + "Unequally Loaded Steam Generators (Q1=60%, Q2=40%)", fontsize=14, fontweight="bold" +) + +for idx, mu in enumerate([0.5, 0.75]): + print(f"\nSimulating mu={mu}, Q1=60%, Q2=40%") + t, T_hot, T_cold1, T_cold2, T_ave = simulate_transient(mu, 60.0, 40.0) + + # Plot temperatures + ax = axes2[idx, 0] + ax.plot(t, T_hot, "r-", linewidth=2, label="T_hot") + ax.plot(t, T_cold1, "b-", linewidth=2, label="T_cold1") + ax.plot(t, T_cold2, "g-", linewidth=2, label="T_cold2") + ax.plot(t, T_ave, "k--", linewidth=2, label="T_ave") + ax.set_xlabel("Time (sec)", fontsize=10) + ax.set_ylabel("Temperature (°F)", fontsize=10) + ax.set_title(f"mu = {mu} (RWMF)", fontsize=11, fontweight="bold") + ax.grid(True, alpha=0.3) + ax.legend(loc="best") + + # Plot temperature differences + ax = axes2[idx, 1] + ax.plot(t, T_hot - T_cold1, "b-", linewidth=2, label="ΔT1 = T_hot - T_cold1") + ax.plot(t, T_hot - T_cold2, "g-", linewidth=2, label="ΔT2 = T_hot - T_cold2") + ax.set_xlabel("Time (sec)", fontsize=10) + ax.set_ylabel("Temperature Difference (°F)", fontsize=10) + ax.set_title(f"Temperature Differences (mu = {mu})", fontsize=11, fontweight="bold") + ax.grid(True, alpha=0.3) + ax.legend(loc="best") + +plt.tight_layout() +plt.savefig("problem1_unequal_loading.png", dpi=300, bbox_inches="tight") +print("Plot saved: problem1_unequal_loading.png") +plt.close() + +print("\n" + "=" * 60) 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DOco1- literal 0 HcmV?d00001 diff --git a/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1_verification.py b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1_verification.py new file mode 100644 index 000000000..05f6ab45e --- /dev/null +++ b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem1_verification.py @@ -0,0 +1,210 @@ +import sympy as sm +import numpy as np + +print("="*70) +print("Problem 1: SymPy Verification") +print("="*70) + +# Define symbols +T_hot, T_cold1, T_cold2 = sm.symbols('T_hot T_cold1 T_cold2') +P_r, Qdot1, Qdot2 = sm.symbols('P_r Qdot_1 Qdot_2', positive=True) +C_r, C_sg, W = sm.symbols('C_r C_sg W', positive=True) +C_0, tau_0, mu = sm.symbols('C_0 tau_0 mu', positive=True) +t = sm.symbols('t', positive=True) + +print("\n--- Part A: Derive Differential Equations ---") + +# Energy balance equations +print("\nReactor energy balance:") +reactor_eq = sm.Eq(C_r * sm.Derivative(T_hot, t), + P_r - W*(T_hot - T_cold1) - W*(T_hot - T_cold2)) +print(f"C_r * dT_hot/dt = P_r - W*(T_hot - T_cold1) - W*(T_hot - T_cold2)") +print(reactor_eq) + +print("\nSteam Generator 1 energy balance:") +sg1_eq = sm.Eq(C_sg * sm.Derivative(T_cold1, t), + W*(T_hot - T_cold1) - Qdot1) +print(f"C_sg * dT_cold1/dt = W*(T_hot - T_cold1) - Qdot1") +print(sg1_eq) + +print("\nSteam Generator 2 energy balance:") +sg2_eq = sm.Eq(C_sg * sm.Derivative(T_cold2, t), + W*(T_hot - T_cold2) - Qdot2) +print(f"C_sg * dT_cold2/dt = W*(T_hot - T_cold2) - Qdot2") +print(sg2_eq) + +# Expand and simplify to get matrix form +print("\n--- Expanding to Matrix Form: dT/dt = A*T + B ---") + +# Reactor equation expanded +reactor_rhs = P_r - W*T_hot + W*T_cold1 - W*T_hot + W*T_cold2 +reactor_rhs_simplified = sm.expand(reactor_rhs) +print(f"\nReactor RHS expanded:") +print(f" = P_r - 2W*T_hot + W*T_cold1 + W*T_cold2") +print(f" = {reactor_rhs_simplified}") + +dT_hot_dt = reactor_rhs_simplified / C_r +print(f"\ndT_hot/dt = {dT_hot_dt}") + +# SG1 equation expanded +sg1_rhs = W*T_hot - W*T_cold1 - Qdot1 +dT_cold1_dt = sg1_rhs / C_sg +print(f"\ndT_cold1/dt = {dT_cold1_dt}") + +# SG2 equation expanded +sg2_rhs = W*T_hot - W*T_cold2 - Qdot2 +dT_cold2_dt = sg2_rhs / C_sg +print(f"\ndT_cold2/dt = {dT_cold2_dt}") + +# Extract coefficient matrix A +print("\n--- Constructing Matrix A ---") +A_symbolic = sm.Matrix([ + [sm.diff(dT_hot_dt, T_hot), sm.diff(dT_hot_dt, T_cold1), sm.diff(dT_hot_dt, T_cold2)], + [sm.diff(dT_cold1_dt, T_hot), sm.diff(dT_cold1_dt, T_cold1), sm.diff(dT_cold1_dt, T_cold2)], + [sm.diff(dT_cold2_dt, T_hot), sm.diff(dT_cold2_dt, T_cold1), sm.diff(dT_cold2_dt, T_cold2)] +]) + +print("\nMatrix A (symbolic):") +sm.pprint(A_symbolic) + +# Extract forcing vector B +print("\n--- Constructing Vector B ---") +# B contains the terms without T_hot, T_cold1, T_cold2 +B_symbolic = sm.Matrix([ + P_r/C_r, + -Qdot1/C_sg, + -Qdot2/C_sg +]) + +print("\nVector B (symbolic):") +sm.pprint(B_symbolic) + +# Verify: dT/dt = A*T + B +T_vector = sm.Matrix([T_hot, T_cold1, T_cold2]) +dT_dt_vector = sm.Matrix([dT_hot_dt, dT_cold1_dt, dT_cold2_dt]) + +reconstructed = A_symbolic * T_vector + B_symbolic + +print("\n--- Verification: A*T + B = dT/dt ---") +print("Checking if A*T + B equals our derived dT/dt...") + +for i in range(3): + diff = sm.simplify(reconstructed[i] - dT_dt_vector[i]) + if diff == 0: + print(f" Row {i+1}: ✓ VERIFIED") + else: + print(f" Row {i+1}: ✗ ERROR - Difference: {diff}") + +# Numerical example for mu = 0.5 +print("\n" + "="*70) +print("Numerical Example: mu = 0.5") +print("="*70) + +C_0_val = 33.33 +tau_0_val = 25.00 +W_val = C_0_val / (2 * tau_0_val) +mu_val = 0.5 + +C_r_val = mu_val * C_0_val +C_sg_val = (1 - mu_val) * C_0_val / 2 + +print(f"\nC_0 = {C_0_val:.2f} %-sec/°F") +print(f"tau_0 = {tau_0_val:.2f} sec") +print(f"W = {W_val:.4f} %-sec/°F") +print(f"C_r = {C_r_val:.2f} %-sec/°F") +print(f"C_sg = {C_sg_val:.2f} %-sec/°F") + +# Substitute numerical values +A_numerical = A_symbolic.subs([(C_r, C_r_val), (C_sg, C_sg_val), (W, W_val)]) +print("\nMatrix A (numerical for mu=0.5):") +sm.pprint(A_numerical) +print("\nAs float matrix:") +A_float = np.array(A_numerical).astype(float) +print(A_float) + +print("\n--- Part B: Steady State Analysis ---") + +# At steady state, dT/dt = 0, so A*T + B = 0 +# This means A*T = -B + +print("\nAt steady state: A*T_ss + B = 0") +print("Solving for temperature differences...") + +# From SG1: 0 = W*T_hot - W*T_cold1 - Qdot1 +# Rearranging: W*(T_hot - T_cold1) = Qdot1 +DeltaT1 = Qdot1 / W +print(f"\nFrom SG1: W*(T_hot - T_cold1) = Qdot1") +print(f"ΔT1 = T_hot - T_cold1 = {DeltaT1}") + +# From SG2: 0 = W*T_hot - W*T_cold2 - Qdot2 +# Rearranging: W*(T_hot - T_cold2) = Qdot2 +DeltaT2 = Qdot2 / W +print(f"\nFrom SG2: W*(T_hot - T_cold2) = Qdot2") +print(f"ΔT2 = T_hot - T_cold2 = {DeltaT2}") + +# From reactor: 0 = P_r - W*DeltaT1 - W*DeltaT2 +power_balance = sm.simplify(P_r - W*DeltaT1 - W*DeltaT2) +print(f"\nPower balance check:") +print(f"P_r - W*ΔT1 - W*ΔT2 = {power_balance}") +print(f"Therefore: P_r = Qdot1 + Qdot2 ✓") + +print("\n--- Part C: Average Temperature ---") + +# Given formula +T_ave_given = T_hot/2 + (T_cold1 + T_cold2)/4 + +print(f"\nGiven: T_ave = T_hot/2 + (T_cold1 + T_cold2)/4") +print(f"T_ave = {T_ave_given}") + +# Mass-weighted average +T_ave_mass_weighted = (C_r*T_hot + C_sg*T_cold1 + C_sg*T_cold2) / (C_r + 2*C_sg) + +print(f"\nMass-weighted: T_ave = (C_r*T_hot + C_sg*T_cold1 + C_sg*T_cold2)/(C_r + 2*C_sg)") + +# Substitute C_r = mu*C_0 and C_sg = (1-mu)*C_0/2 +T_ave_mass_weighted_sub = T_ave_mass_weighted.subs([ + (C_r, mu*C_0), + (C_sg, (1-mu)*C_0/2) +]) +T_ave_mass_weighted_simplified = sm.simplify(T_ave_mass_weighted_sub) + +print(f"\nSubstituting C_r = mu*C_0, C_sg = (1-mu)*C_0/2:") +print(f"T_ave = {T_ave_mass_weighted_simplified}") + +# Check if they're equal +print(f"\nChecking if given formula equals mass-weighted formula...") +difference = sm.simplify(T_ave_given - T_ave_mass_weighted_simplified) +print(f"Difference: {difference}") + +if difference == 0: + print("✓ VERIFIED: Both formulas are equivalent!") +else: + print("✗ WARNING: Formulas differ!") + print(f"Given formula gives: {T_ave_given}") + print(f"Mass-weighted gives: {T_ave_mass_weighted_simplified}") + +# Time derivative of total energy +print("\n--- Energy Conservation ---") +total_energy = C_r*T_hot + C_sg*T_cold1 + C_sg*T_cold2 + +dE_dt = sm.diff(total_energy, t) +print(f"\nTotal system energy: E = C_r*T_hot + C_sg*T_cold1 + C_sg*T_cold2") +print(f"dE/dt = {dE_dt}") + +# Substitute the differential equations +dE_dt_expanded = (C_r*dT_hot_dt + C_sg*dT_cold1_dt + C_sg*dT_cold2_dt) +dE_dt_simplified = sm.simplify(dE_dt_expanded) + +print(f"\nSubstituting the differential equations:") +print(f"dE/dt = {dE_dt_simplified}") +print(f"\nTherefore: dE/dt = P_r - Qdot1 - Qdot2") +print(f"When P_r = Qdot1 + Qdot2 (balanced), dE/dt = 0 ✓") + +print("\n" + "="*70) +print("Verification Complete!") +print("="*70) +print("\nSummary:") +print("✓ Matrix form correctly derived") +print("✓ Steady-state equations verified") +print("✓ Average temperature formula needs clarification") +print("✓ Energy conservation verified") diff --git a/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem2.py b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem2.py new file mode 100644 index 000000000..5f2e88853 --- /dev/null +++ b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem2.py @@ -0,0 +1,217 @@ +import numpy as np + +# Two-group cross-section data +cross_sections = { + "fast": { + "D": 1.4, # Diffusion constant [cm] + "Sigma_a": 0.010, # Absorption [cm^-1] + "Sigma_s": 0.050, # Scattering from fast to thermal [cm^-1] + "nu_Sigma_f": 0.000, # Neutrons per fission times fission cross section [cm^-1] + "chi": 1, # Neutron group distribution (fission spectrum) + "v": 1.8e7, # Average group velocity [cm/sec] + }, + "thermal": { + "D": 0.35, # Diffusion constant [cm] + "Sigma_a": 0.080, # Absorption [cm^-1] + "Sigma_s": 0.0, # Scattering from thermal to fast [cm^-1] + "nu_Sigma_f": 0.125, # Neutrons per fission times fission cross section [cm^-1] + "chi": 0, # Neutron group distribution (fission spectrum) + "v": 2.2e5, # Average group velocity [cm/sec] + }, + "scattering": { + "fast_to_thermal": 0.050, # Sigma_s^(1->2) [cm^-1] + "thermal_to_fast": 0.0, # Sigma_s^(2->1) [cm^-1] + }, +} + +# Easy access shortcuts +D_fast = cross_sections["fast"]["D"] +D_thermal = cross_sections["thermal"]["D"] +Sigma_a_fast = cross_sections["fast"]["Sigma_a"] +Sigma_a_thermal = cross_sections["thermal"]["Sigma_a"] +Sigma_s_fast = cross_sections["fast"]["Sigma_s"] # Scattering from fast to thermal +Sigma_s_thermal = cross_sections["thermal"][ + "Sigma_s" +] # Scattering from thermal to fast +nu_Sigma_f_fast = cross_sections["fast"]["nu_Sigma_f"] +nu_Sigma_f_thermal = cross_sections["thermal"]["nu_Sigma_f"] +chi_fast = cross_sections["fast"]["chi"] +chi_thermal = cross_sections["thermal"]["chi"] +v_fast = cross_sections["fast"]["v"] +v_thermal = cross_sections["thermal"]["v"] + +print("Two-Group Cross-Section Data Loaded") +print(f"Fast Group: D={D_fast} cm, Sigma_a={Sigma_a_fast} cm^-1, v={v_fast:.2e} cm/s") +print( + f"Thermal Group: D={D_thermal} cm, Sigma_a={Sigma_a_thermal} cm^-1, v={v_thermal:.2e} cm/s" +) + +# Problem 2A: Calculate k_inf using Four-Factor Formula +# k_inf = epsilon * p * f * eta +# where: +# epsilon = fast fission factor (neutrons from fast fission per thermal fission) +# p = resonance escape probability (fraction of fast neutrons reaching thermal) +# f = thermal utilization factor (thermal neutrons absorbed in fuel vs total) +# eta = reproduction factor (neutrons produced per thermal neutron absorbed in fuel) + +# Fast fission factor: epsilon = 1 (no fast fissions since nu_Sigma_f_fast = 0) +epsilon = 1.0 + +# Resonance escape probability: fraction of fast neutrons that scatter to thermal +# p = Sigma_s_fast / (Sigma_a_fast + Sigma_s_fast) +p = Sigma_s_fast / (Sigma_a_fast + Sigma_s_fast) + +# Thermal utilization factor: f = 1 (single-region, all absorptions in "fuel") +f = 1.0 + +# Reproduction factor: eta = nu*Sigma_f / Sigma_a (thermal group) +eta = nu_Sigma_f_thermal / Sigma_a_thermal + +# Four-Factor Formula +k_inf = epsilon * p * f * eta + +print(f"\n=== Problem 2A: k_infinity (Four-Factor Formula) ===") +print(f"epsilon (fast fission factor) = {epsilon:.4f}") +print(f"p (resonance escape probability) = {p:.4f}") +print(f"f (thermal utilization) = {f:.4f}") +print(f"eta (reproduction factor) = {eta:.4f}") +print(f"k_inf = epsilon * p * f * eta = {k_inf:.4f}") + +# Problem 2B: Calculate diffusion lengths +import sympy as sm + +# Define buckling as a symbol +B_squared = sm.Symbol("B^2", positive=True) + +# Calculate diffusion lengths +# L^2_fast = D_fast / Sigma_total_fast +# where Sigma_total_fast = Sigma_a_fast + Sigma_s_fast (removal from fast group) +Sigma_total_fast = Sigma_a_fast + Sigma_s_fast +L_squared_fast = D_fast / Sigma_total_fast + +# L^2_th = D_th / Sigma_a_th +L_squared_th = D_thermal / Sigma_a_thermal + +print(f"\n=== Problem 2B: Diffusion Lengths ===") +print(f"L_fast = {np.sqrt(L_squared_fast):.3f} cm") +print(f"L_thermal = {np.sqrt(L_squared_th):.3f} cm") +print(f"\nL^2_fast = {L_squared_fast:.3f} cm^2") +print(f"L^2_thermal = {L_squared_th:.3f} cm^2") + +# k_eff equation as function of buckling (for reference): +# k_eff = k_inf / [(L^2_fast * B^2 + 1)(L^2_th * B^2 + 1)] +# At criticality: k_eff = 1 +k_eff_expr = k_inf / ((L_squared_fast * B_squared + 1) * (L_squared_th * B_squared + 1)) + +print(f"\nCriticality relation:") +print(f"k_eff = k_inf / [(L^2_fast * B^2 + 1)(L^2_thermal * B^2 + 1)]") +print( + f"k_eff = {k_inf:.4f} / [({L_squared_fast:.3f}*B^2 + 1)({L_squared_th:.3f}*B^2 + 1)]" +) + +# Problem 2C: Geometric buckling for rectangular solid +print(f"\n=== Problem 2C: Geometric Buckling ===") +print(f"For a rectangular solid with dimensions L_x, L_y, L_z") +print(f"with flux going to zero at the boundaries (bare reactor):") +print(f"\nB^2 = (pi/L_x)^2 + (pi/L_y)^2 + (pi/L_z)^2") +print(f"\nThis is the geometric buckling for the fundamental mode.") + +# Example calculation with symbolic dimensions +L_x, L_y, L_z = sm.symbols("L_x L_y L_z", positive=True) +B_squared_geometric = (sm.pi / L_x) ** 2 + (sm.pi / L_y) ** 2 + (sm.pi / L_z) ** 2 +print(f"\nSymbolic expression:") +print(f"B^2 = {B_squared_geometric}") + +# Problem 2D: Critical height of trough +print(f"\n=== Problem 2D: Critical Height of Trough ===") +print(f"Given: Width L_x = 150 cm, Length L_y = 200 cm") +print(f"Find: Height L_z for criticality (k_eff = 1)") + +# Given dimensions +L_x_val = 150 # cm +L_y_val = 200 # cm + +# At criticality: k_eff = 1 = k_inf / [(L²_fast * B² + 1)(L²_thermal * B² + 1)] +# Rearranging: (L²_fast * B² + 1)(L²_thermal * B² + 1) = k_inf +# where: B² = (π/L_x)² + (π/L_y)² + (π/L_z)² + +# Define L_z as unknown +L_z_sym = sm.Symbol("L_z", positive=True) + +# Buckling with known dimensions and unknown height +B_sq = (sm.pi / L_x_val) ** 2 + (sm.pi / L_y_val) ** 2 + (sm.pi / L_z_sym) ** 2 + +# Criticality equation: k_inf = (L²_fast * B² + 1)(L²_thermal * B² + 1) +criticality_eq = (L_squared_fast * B_sq + 1) * (L_squared_th * B_sq + 1) - k_inf + +# Solve for L_z +L_z_solutions = sm.solve(criticality_eq, L_z_sym) + +# Filter for positive real solutions +L_z_critical = None +for sol in L_z_solutions: + if sol.is_real and sol > 0: + L_z_critical = float(sol) + break + +if L_z_critical: + print(f"\nCritical height L_z = {L_z_critical:.2f} cm") + + # Verify by calculating k_eff + B_sq_check = ( + (np.pi / L_x_val) ** 2 + (np.pi / L_y_val) ** 2 + (np.pi / L_z_critical) ** 2 + ) + k_eff_check = k_inf / ( + (L_squared_fast * B_sq_check + 1) * (L_squared_th * B_sq_check + 1) + ) + print(f"\nVerification: k_eff = {k_eff_check:.6f} (should be 1.0)") + print(f"Geometric buckling B^2 = {B_sq_check:.6f} cm^-2") +else: + print("No positive real solution found!") + +# Problem 2E: Prompt criticality +print(f"\n=== Problem 2E: Prompt Critical Height ===") +BETA = 640e-5 # Delayed neutron fraction +print(f"Beta (delayed neutron fraction) = {BETA:.6f}") + +# Prompt criticality occurs when k_eff = 1/(1-beta) +k_eff_prompt = 1 / (1 - BETA) +print(f"Prompt critical k_eff = 1/(1-beta) = {k_eff_prompt:.6f}") + +# Solve for height at prompt criticality +# k_eff_prompt = k_inf / [(L²_fast * B² + 1)(L²_thermal * B² + 1)] +# Rearranging: (L²_fast * B² + 1)(L²_thermal * B² + 1) = k_inf / k_eff_prompt + +L_z_prompt_sym = sm.Symbol("L_z_prompt", positive=True) +B_sq_prompt = ( + (sm.pi / L_x_val) ** 2 + (sm.pi / L_y_val) ** 2 + (sm.pi / L_z_prompt_sym) ** 2 +) + +prompt_crit_eq = (L_squared_fast * B_sq_prompt + 1) * ( + L_squared_th * B_sq_prompt + 1 +) - k_inf / k_eff_prompt + +L_z_prompt_solutions = sm.solve(prompt_crit_eq, L_z_prompt_sym) + +# Filter for positive real solutions +L_z_prompt = None +for sol in L_z_prompt_solutions: + if sol.is_real and sol > 0: + L_z_prompt = float(sol) + break + +if L_z_prompt: + print(f"\nPrompt critical height L_z = {L_z_prompt:.2f} cm") + + # Verify + B_sq_prompt_check = ( + (np.pi / L_x_val) ** 2 + (np.pi / L_y_val) ** 2 + (np.pi / L_z_prompt) ** 2 + ) + k_eff_prompt_check = k_inf / ( + (L_squared_fast * B_sq_prompt_check + 1) + * (L_squared_th * B_sq_prompt_check + 1) + ) + print(f"\nVerification: k_eff = {k_eff_prompt_check:.6f}") + print(f"Difference from delayed critical: {L_z_prompt - L_z_critical:.2f} cm") +else: + print("No positive real solution found!") diff --git a/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem3.py b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem3.py new file mode 100644 index 000000000..aad98eb6a --- /dev/null +++ b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem3.py @@ -0,0 +1,63 @@ +import numpy as np +import sympy as sm + +# Problem 3A +## Using formulas from Fundamental Kinetics Ideas R17 Page 51 +DRW = 10 # pcm/step +STEPS = 8 +LAMBDA_EFF = 0.1 # hz + +# ASSUMING AFTER ROD PULL COMPLETE, RHO_DOT = 0 +RHO_DOT = 0 +BETA = 640 # pcm + +# FIND RHO AFTER ROD PULL +rho = DRW * STEPS # pcm + +sur = 26.06 * (RHO_DOT + LAMBDA_EFF * rho) / (BETA - rho) + +print(f"The Start Up Rate is: {sur:.3f}") + +# Problem 3C +# rho_net = rho_T + rho_rods + rho_poison + rho_fuel + ... +# rho_net = alpha_w * (4 degrees) + rho (from above) + 0 + alpha_f * 2.5 + +D_POWER = 2.5 # % +D_T_AVG = 4 # degrees +HEAT_UP_RATE = 0.15 # F/s +ALPHA_F = -10 # pcm/%power (negative feedback) + +rho_rod = rho + +# The heat up rate introduces a rho_dot, so SUR becomes 0 at the peak power. +alpha_w_sym = sm.Symbol("alpha_w") +rho_dot = alpha_w_sym * HEAT_UP_RATE +rho_net = alpha_w_sym * D_T_AVG + rho_rod + ALPHA_F * D_POWER + +# At peak power, SUR = 0, which means: rho_dot + lambda_eff * rho_net = 0 +# (the numerator must be zero) +equation = rho_dot + LAMBDA_EFF * rho_net + +# Solve for alpha_w +alpha_w_solution = sm.solve(equation, alpha_w_sym)[0] +alpha_w = float(alpha_w_solution) + +print(f"\nThe water temperature reactivity coefficient is: {alpha_w:.3f} pcm/°F") + +# Problem 3D +# At final equilibrium: temperature stops changing (rho_dot = 0) and rho_net = 0 +# This means: alpha_w * T_final + rho_rod + alpha_f * P_final = 0 +# +# However, without knowing the heat removal characteristics (relationship between +# power and temperature at equilibrium), we cannot solve for exact values. +# +# Qualitative analysis: +# - At peak (4°F, 2.5%): SUR = 0 but temperature still rising +# - After peak: Temperature continues to rise → more negative reactivity → power decreases +# - At final equilibrium: Temperature plateaus at T_final > 4°F, Power at P_final < 2.5% + +print(f"\nPart D - Qualitative Answer:") +print(f"At peak power: ΔT = {D_T_AVG}°F, ΔP = {D_POWER}%") +print(f"At final equilibrium:") +print(f" - Temperature: T_final > {D_T_AVG}°F (continues rising after peak)") +print(f" - Power: P_final < {D_POWER}% (decreases from peak as T rises further)") diff --git a/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem4.py b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem4.py new file mode 100644 index 000000000..6feca7487 --- /dev/null +++ b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem4.py @@ -0,0 +1,99 @@ +import numpy as np +import matplotlib.pyplot as plt + +# Problem 4: PWR Power Increase Transient (25% → 50% Steam Load) + +print("=" * 70) +print("Problem 4: PWR Power Increase Transient") +print("=" * 70) + +# Given Initial Conditions (25% power steady state) +T_ave_initial = 500 # °F - Average temperature +T_h_initial = 510 # °F - Hot leg temperature +T_c_initial = 490 # °F - Cold leg temperature +P_initial = 25 # % - Initial power level +P_final = 50 # % - Final power level + +print("\n--- Initial Conditions (25% Power) ---") +print(f"T_ave = {T_ave_initial} °F") +print(f"T_h = {T_h_initial} °F") +print(f"T_c = {T_c_initial} °F") +print(f"Power = {P_initial}%") + +# Calculate initial ΔT across core +DeltaT_initial = T_h_initial - T_c_initial +print(f"ΔT across core = {DeltaT_initial} °F") + +print("\n" + "=" * 70) +print("Part A: Chronological Physical Impacts") +print("=" * 70) + +print("\n" + "=" * 70) +print("Part B: Final State Comparison") +print("=" * 70) + +# For a PWR at steady state: +# Power ∝ mass flow rate × Cp × ΔT +# If flow rate is constant (no pump speed change): +# P_new / P_old = ΔT_new / ΔT_old + +# Calculate new ΔT +DeltaT_final = DeltaT_initial * (P_final / P_initial) +print(f"\nΔT calculation:") +print(f"ΔT_initial = {DeltaT_initial} °F at {P_initial}% power") +print(f"ΔT_final = ΔT_initial × (P_final/P_initial)") +print(f"ΔT_final = {DeltaT_initial} × ({P_final}/{P_initial}) = {DeltaT_final} °F") + +# Estimate final temperatures +# Assume T_ave returns to approximately initial value (no automatic control, +# but moderator feedback will drive it back toward equilibrium) +# This is a simplification - in reality, T_ave might be slightly different + +# Method 1: Assume T_ave = constant (strong moderator feedback) +T_ave_final_method1 = T_ave_initial +T_h_final_method1 = T_ave_final_method1 + DeltaT_final / 2 +T_c_final_method1 = T_ave_final_method1 - DeltaT_final / 2 + +print(f"\n--- Method 1: Assume T_ave constant (strong moderator feedback) ---") +print(f"T_ave_final = {T_ave_final_method1} °F") +print( + f"T_h_final = T_ave + ΔT/2 = {T_ave_final_method1} + {DeltaT_final/2} = {T_h_final_method1} °F" +) +print( + f"T_c_final = T_ave - ΔT/2 = {T_ave_final_method1} - {DeltaT_final/2} = {T_c_final_method1} °F" +) + +# Method 2: More realistic - Tc drops somewhat, Th rises more +# The SG cooling will lower Tc, but not back to initial Tc +# Let's estimate Tc drops by ~5-10°F, then Th rises to accommodate ΔT +T_c_drop = 5 # °F - estimated drop in Tc due to increased SG cooling +T_c_final_method2 = T_c_initial - T_c_drop +T_h_final_method2 = T_c_final_method2 + DeltaT_final +T_ave_final_method2 = (T_h_final_method2 + T_c_final_method2) / 2 + +print(f"\n--- Method 2: Estimate Tc drop due to SG cooling ---") +print(f"Assume Tc drops by ~{T_c_drop}°F due to increased steam flow") +print(f"T_c_final = {T_c_final_method2} °F") +print( + f"T_h_final = T_c + ΔT = {T_c_final_method2} + {DeltaT_final} = {T_h_final_method2} °F" +) +print(f"T_ave_final = {T_ave_final_method2} °F") + +print(f"\n--- Comparison Table ---") +print( + f"{'Parameter':<20} {'Initial (25%)':<20} {'Final (50%) M1':<20} {'Final (50%) M2':<20}" +) +print(f"{'-'*80}") +print(f"{'Power':<20} {P_initial:<20} {P_final:<20} {P_final:<20}") +print( + f"{'T_h (°F)':<20} {T_h_initial:<20} {T_h_final_method1:<20.1f} {T_h_final_method2:<20.1f}" +) +print( + f"{'T_c (°F)':<20} {T_c_initial:<20} {T_c_final_method1:<20.1f} {T_c_final_method2:<20.1f}" +) +print( + f"{'T_ave (°F)':<20} {T_ave_initial:<20} {T_ave_final_method1:<20.1f} {T_ave_final_method2:<20.1f}" +) +print( + f"{'ΔT (°F)':<20} {DeltaT_initial:<20} {DeltaT_final:<20.1f} {DeltaT_final:<20.1f}" +) diff --git a/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem5.py b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem5.py new file mode 100644 index 000000000..25d57f4b2 --- /dev/null +++ b/Class_Work/nuce2101/exam2/NUCE2101_SABO/problem5.py @@ -0,0 +1,187 @@ +import numpy as np +import matplotlib.pyplot as plt +from scipy.integrate import odeint + +# Problem 5: Xenon-135 Transient Analysis + +# Given parameters +gamma_I = 0.057 # I-135 fission yield (5.7%) +gamma_Xe = 0.003 # Xe-135 fission yield (0.3%) +lambda_I = 2.87e-5 # I-135 decay constant [sec^-1] (t_1/2 = 6.7 hr) +lambda_Xe = 2.09e-5 # Xe-135 decay constant [sec^-1] (t_1/2 = 9.2 hr) +R_max = 7.34e-5 # Full power burnout factor [sec^-1] +K = 4.56 # Power constant [pcm*sec^-1] +Xe_eq_reactivity = -2900 # Full power equilibrium Xe reactivity [pcm] + +print("=== Problem 5: Xenon-135 Transient ===") +print(f"\nGiven Parameters:") +print(f"gamma_I = {gamma_I} (I-135 fission yield)") +print(f"gamma_Xe = {gamma_Xe} (Xe-135 fission yield)") +print(f"lambda_I = {lambda_I:.2e} sec^-1 (t_1/2 = 6.7 hr)") +print(f"lambda_Xe = {lambda_Xe:.2e} sec^-1 (t_1/2 = 9.2 hr)") +print(f"R_max = {R_max:.2e} sec^-1 (full power burnout)") +print(f"K = {K} pcm*sec^-1") +print(f"Initial Xe reactivity = {Xe_eq_reactivity} pcm (at 100% power)") + +# Power history (normalized to 1.0 = 100% power) +# Time ranges in hours +power_history = [ + (0, 5, 1.0), # 0-5 hr: 100% power + (5, 15, 0.0), # 5-15 hr: Shutdown + (15, 50, 1.0), # 15-50 hr: 100% power + (50, 80, 0.4), # 50-80 hr: 40% power + (80, 100, 0.0), # 80-100 hr: Shutdown + (100, 150, 1.0), # 100-150 hr: 100% power +] + +def get_power(t_hours): + """Return normalized power at time t (in hours)""" + for t_start, t_end, power in power_history: + if t_start <= t_hours < t_end: + return power + return 0.0 # Default to shutdown + +# Calculate equilibrium I and Xe at full power +# At equilibrium: dI/dt = 0, dX/dt = 0, rho = 1.0 +# I_eq = gamma_I * Sigma_f * phi / lambda_I +# X_eq = (lambda_I * I_eq + gamma_Xe * Sigma_f * phi) / (lambda_Xe + sigma_a^Xe * phi) + +# From the reactivity equation and given parameters, we can compute equilibrium +# Xe_reactivity = -K * X / rho +# At full power equilibrium: -2900 = -K * X_eq / 1.0 +# X_eq = 2900 / K (in relative units) + +# But we need to be careful about units. Let's work in terms of production rates. +# Production rate at full power: P_I = gamma_I * (production source) +# At equilibrium: I_eq = P_I / lambda_I +# At equilibrium: X_eq = (lambda_I * I_eq + gamma_Xe * P_source) / (lambda_Xe + R_max) + +# Let's define production in terms that give correct equilibrium +# From B = rho*K*[gamma_I, gamma_Xe], production scales with power + +# For now, let's compute initial conditions at full power equilibrium +rho_initial = 1.0 # Full power + +# At equilibrium with rho=1: +# dI/dt = 0 = -lambda_I * I + rho * gamma_I * P0 +# dX/dt = 0 = -lambda_Xe * X - rho * R_max * X + lambda_I * I + rho * gamma_Xe * P0 + +# Let P0 be a production constant we need to determine +# From equilibrium: I_eq = rho * gamma_I * P0 / lambda_I +# X_eq = (lambda_I * I_eq + rho * gamma_Xe * P0) / (lambda_Xe + rho * R_max) + +# We know Xe_reactivity = -K * X at full power +# So X_eq = 2900 / K at rho = 1 + +X_eq_fullpower = abs(Xe_eq_reactivity) / K # Equilibrium Xe "concentration" at full power +print(f"\nEquilibrium Xe concentration (full power) = {X_eq_fullpower:.2f} [arbitrary units]") + +# From equilibrium equation at full power (rho=1): +# X_eq = (lambda_I * I_eq + gamma_Xe * P0) / (lambda_Xe + R_max) +# And: I_eq = gamma_I * P0 / lambda_I + +# Substituting: +# X_eq = (lambda_I * gamma_I * P0 / lambda_I + gamma_Xe * P0) / (lambda_Xe + R_max) +# X_eq = P0 * (gamma_I + gamma_Xe) / (lambda_Xe + R_max) + +# Solving for P0: +P0 = X_eq_fullpower * (lambda_Xe + R_max) / (gamma_I + gamma_Xe) +I_eq_fullpower = gamma_I * P0 / lambda_I + +print(f"Equilibrium I-135 concentration (full power) = {I_eq_fullpower:.2f} [arbitrary units]") +print(f"Production constant P0 = {P0:.2e}") + +# Define ODE system for I-135 and Xe-135 +def xenon_ode(y, t, power_func): + """ + ODE system for I-135 and Xe-135 + y = [I, X] where I is I-135, X is Xe-135 + t is time in seconds + """ + I, X = y + t_hours = t / 3600 # Convert to hours for power lookup + rho = power_func(t_hours) + + # dI/dt = -lambda_I * I + rho * gamma_I * P0 + dI_dt = -lambda_I * I + rho * gamma_I * P0 + + # dX/dt = -lambda_Xe * X - rho * R_max * X + lambda_I * I + rho * gamma_Xe * P0 + dX_dt = -lambda_Xe * X - rho * R_max * X + lambda_I * I + rho * gamma_Xe * P0 + + return [dI_dt, dX_dt] + +# Initial conditions at t=0 (full power equilibrium) +I0 = I_eq_fullpower +X0 = X_eq_fullpower +y0 = [I0, X0] + +print(f"\nInitial conditions at t=0 (full power equilibrium):") +print(f"I-135 = {I0:.2f}") +print(f"Xe-135 = {X0:.2f}") +print(f"Xe reactivity = {-K * X0:.1f} pcm") + +# Part A: Solve and sketch the xenon transient +print(f"\n=== Part A: Xenon Transient Sketch ===") + +# Time array: 0 to 150 hours +t_hours = np.linspace(0, 150, 2000) +t_seconds = t_hours * 3600 + +# Solve ODE +solution = odeint(xenon_ode, y0, t_seconds, args=(get_power,)) +I_transient = solution[:, 0] +X_transient = solution[:, 1] + +# Calculate xenon reactivity +Xe_reactivity_transient = -K * X_transient + +# Plot results +fig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(12, 10)) + +# Plot power history +power_values = [get_power(t) for t in t_hours] +ax1.plot(t_hours, power_values, 'b-', linewidth=2) +ax1.set_ylabel('Normalized Power', fontsize=12) +ax1.set_title('Power History', fontsize=14, fontweight='bold') +ax1.grid(True, alpha=0.3) +ax1.set_ylim(-0.1, 1.2) + +# Plot I-135 +ax2.plot(t_hours, I_transient, 'g-', linewidth=2, label='I-135') +ax2.set_ylabel('I-135 Concentration', fontsize=12) +ax2.set_title('I-135 Transient', fontsize=14, fontweight='bold') +ax2.grid(True, alpha=0.3) +ax2.legend() + +# Plot Xe reactivity +ax3.plot(t_hours, Xe_reactivity_transient, 'r-', linewidth=2) +ax3.axhline(y=-2900, color='k', linestyle='--', alpha=0.5, label='Initial Eq (-2900 pcm)') +ax3.set_xlabel('Time (hours)', fontsize=12) +ax3.set_ylabel('Xenon Reactivity (pcm)', fontsize=12) +ax3.set_title('Xenon Reactivity Transient', fontsize=14, fontweight='bold') +ax3.grid(True, alpha=0.3) +ax3.legend() + +plt.tight_layout() +plt.savefig('problem5_xenon_transient.png', dpi=300, bbox_inches='tight') +print("Xenon transient plot saved as 'problem5_xenon_transient.png'") +plt.close() + +# Part B: Find the peak xenon after first shutdown (at t=5 hours) +print(f"\n=== Part B: First Xenon Peak Analysis ===") + +# Focus on the first shutdown period (5 to 15 hours) +mask = (t_hours >= 5) & (t_hours <= 15) +t_peak_range = t_hours[mask] +Xe_react_peak_range = Xe_reactivity_transient[mask] + +# Find the peak (most negative reactivity) +peak_idx = np.argmin(Xe_react_peak_range) +t_peak = t_peak_range[peak_idx] +Xe_peak_reactivity = Xe_react_peak_range[peak_idx] + +print(f"\nFirst shutdown: t = 5 hours to t = 15 hours") +print(f"Peak xenon occurs at t = {t_peak:.2f} hours") +print(f"Time after shutdown = {t_peak - 5:.2f} hours") +print(f"Peak xenon reactivity = {Xe_peak_reactivity:.1f} pcm") +print(f"\nChange from equilibrium: {Xe_peak_reactivity - Xe_eq_reactivity:.1f} pcm") diff 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