add msrDynamics implementation of the dynamic model

dynamic_model/msrDynamics_implementation/data/ORNL_msre_1MW_U233_insertion.csv

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dynamic_model/msrDynamics_implementation/data/ORNL_msre_1MW_U233_magnitude.csv

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dynamic_model/msrDynamics_implementation/data/ORNL_msre_1MW_U233_magnitude_theoretical.csv

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dynamic_model/msrDynamics_implementation/data/ORNL_msre_1MW_U233_phase.csv

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dynamic_model/msrDynamics_implementation/data/ORNL_msre_1MW_U233_phase_theoretical.csv

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dynamic_model/msrDynamics_implementation/frq_script.py

@@ -0,0 +1,168 @@
+# Imports
+from parameters_U233 import *
+import numpy as np
+import matplotlib.pyplot as plt
+from jitcdde import t
+from msrDynamics.system_objects import Node, System
+import pandas as pd
+import sympy as sp
+from concurrent.futures import ProcessPoolExecutor
+from scipy.signal import find_peaks
+
+f_range = np.logspace(-2, 1, num=100)
+# tau_l = 2 * tau_l
+# tau_c = 2*tau_c
+# tau_c_hx = 2 * tau_c_hx
+# tau_hx_c = 2 * tau_hx_c
+def process_frequency(f):
+
+    MSRE = System()
+
+    # radiator
+    T_out_rc = Node(m = mn_rp, scp = mcp_rpn/mn_rp, W = W_rp, y0 = T0_rp)
+    T_out_air = Node(m = mn_rs, scp = mcp_rsn/mn_rs, W = W_rs, y0 = T0_rs)
+
+    # heat exchanger
+    T_hf1 = Node(m = mn_p, scp = mcp_pn/mn_p, W = W_p, y0 = T0_p1)
+    T_hf2 = Node(m = mn_p, scp = mcp_pn/mn_p, W = W_p, y0 = T0_p2)
+    T_hf3 = Node(m = mn_p, scp = mcp_pn/mn_p, W = W_p, y0 = T0_p3)
+    T_hf4 = Node(m = mn_p, scp = mcp_pn/mn_p, W = W_p, y0 = T0_p4)
+    T_ht1 = Node(m = m_tn, scp = scp_t, y0 = T0_t1)
+    T_ht2 = Node(m = m_tn, scp = scp_t, y0 = T0_t2)
+    T_hc1 = Node(m = mn_s, scp = mcp_sn/mn_s, W = W_s, y0 = T0_s1)
+    T_hc2 = Node(m = mn_s, scp = mcp_sn/mn_s, W = W_s, y0 = T0_s2)
+    T_hc3 = Node(m = mn_s, scp = mcp_sn/mn_s, W = W_s, y0 = T0_s3)
+    T_hc4 = Node(m = mn_s, scp = mcp_sn/mn_s, W = W_s, y0 = T0_s4)
+
+    # core 
+    n = Node(y0 = n_frac0)
+    C1 = Node(y0 = C0[0])
+    C2 = Node(y0 = C0[1])
+    C3 = Node(y0 = C0[2])
+    C4 = Node(y0 = C0[3])
+    C5 = Node(y0 = C0[4])
+    C6 = Node(y0 = C0[5])
+    rho = Node(y0 = 0.0)
+
+    # add reactivity input
+    r = 1e-5
+    def rho_insert(t):
+        return r*sp.sin(f*t)
+
+    rho_ext = MSRE.add_input(rho_insert, T)
+
+    T_cg = Node(m = mcp_g1/scp_g, scp = scp_g, y0 = T0_g1)
+    T_cf1 = Node(m = mn_f, scp = scp_f, W = W_f, y0 = T0_f1)
+    T_cf2 = Node(m = mn_f, scp = scp_f, W = W_f, y0 = T0_f2)
+
+    MSRE.add_nodes([T_out_rc,T_out_air,T_hf1,T_hf2,T_hf3,T_hf4,T_ht1,T_ht2,T_hc1,
+                T_hc2,T_hc3,T_hc4,n,C1,C2,C3,C4,C5,C6,T_cg,T_cf1,T_cf2,rho])
+
+    # dynamics 
+
+    # radiator
+    T_out_rc.set_dTdt_advective(source = T_hc4.y(t-tau_hx_r))
+    T_out_rc.set_dTdt_convective(source = [T_out_air.y()], hA = [hA_rpn])
+
+    T_out_air.set_dTdt_advective(source = Trs_in)
+    T_out_air.set_dTdt_convective(source = [T_out_rc.y()], hA = [hA_rsn])
+
+    # heat exchanger
+    T_hf1.set_dTdt_advective(source = T_cf2.y(t-tau_c_hx))
+    T_hf1.set_dTdt_convective(source = [T_ht1.y()], hA = [hA_pn])
+
+    T_hf2.set_dTdt_advective(source = T_hf1.y())
+    T_hf2.dTdt_convective = T_hf1.dTdt_convective
+
+    T_hf3.set_dTdt_advective(source = T_hf2.y())
+    T_hf3.set_dTdt_convective(source = [T_ht2.y()], hA = [hA_pn])
+
+    T_hf4.set_dTdt_advective(source = T_hf3.y())
+    # T_hf4.set_dTdt_convective(source = [T_ht2.y()], hA = [hA_pn])
+    T_hf4.dTdt_convective = T_hf3.dTdt_convective
+
+    # T_ht1.set_dTdt_convective(source = [T_hf1.y(),T_hf2.y(),T_hc3.y(),T_hc4.y()], hA = [hA_pn,hA_pn,hA_sn,hA_sn])
+    # T_ht2.set_dTdt_convective(source = [T_hf3.y(),T_hf4.y(),T_hc1.y(),T_hc2.y()], hA = [hA_pn,hA_pn,hA_sn,hA_sn])
+    T_ht1.set_dTdt_convective(source = [T_hf1.y(),T_hf1.y(),T_hc3.y(),T_hc3.y()], hA = [hA_pn,hA_pn,hA_sn,hA_sn])
+    T_ht2.set_dTdt_convective(source = [T_hf3.y(),T_hf3.y(),T_hc1.y(),T_hc1.y()], hA = [hA_pn,hA_pn,hA_sn,hA_sn])
+
+    T_hc1.set_dTdt_advective(source = T_out_rc.y(t-tau_r_hx))
+    T_hc1.set_dTdt_convective(source = [T_ht2.y()], hA = [hA_sn])
+
+    T_hc2.set_dTdt_advective(source = T_hc1.y())
+    T_hc2.dTdt_convective = T_hc1.dTdt_convective
+
+    T_hc3.set_dTdt_advective(source = T_hc2.y())
+    T_hc3.set_dTdt_convective(source = [T_ht1.y()], hA = [hA_sn])
+
+    T_hc4.set_dTdt_advective(source = T_hc3.y())
+    T_hc4.dTdt_convective = T_hc3.dTdt_convective
+
+    # core
+    n.set_dndt(r = rho.y()+rho_ext, beta_eff = beta_t, Lambda = Lam, lam = lam, C = [C1.y(),C2.y(),C3.y(),C4.y(),C5.y(),C6.y()])
+    C1.set_dcdt(n.y(),beta[0],Lam,lam[0],tau_c,tau_l)
+    C2.set_dcdt(n.y(),beta[1],Lam,lam[1],tau_c,tau_l)
+    C3.set_dcdt(n.y(),beta[2],Lam,lam[2],tau_c,tau_l)
+    C4.set_dcdt(n.y(),beta[3],Lam,lam[3],tau_c,tau_l)
+    C5.set_dcdt(n.y(),beta[4],Lam,lam[4],tau_c,tau_l)
+    C6.set_dcdt(n.y(),beta[5],Lam,lam[5],tau_c,tau_l)
+
+    T_cg.set_dTdt_convective(source = [T_cf1.y()], hA = [hA_fg])
+    T_cg.set_dTdt_internal(source = n.y(), k = k_g*P)
+
+    T_cf1.set_dTdt_advective(source = T_hf4.y(t-tau_hx_c))
+    T_cf1.set_dTdt_convective(source = [T_cg.y()], hA = [k_1*hA_fg])
+    T_cf1.set_dTdt_internal(source = n.y(), k = k_f1*P)
+
+    T_cf2.set_dTdt_advective(source = T_cf1.y())
+    T_cf2.dTdt_convective = T_cf1.dTdt_convective
+    T_cf2.set_dTdt_internal(source = n.y(), k = k_f2*P)
+
+    rho.set_drdt(sources = [T_cf1.dydt(), T_cf2.dydt(), T_cg.dydt()], coeffs = [a_f/2,a_f/2,a_g])
+
+    MSRE.solve(T)
+
+    i_out = [i for i in range(len(T)) if T[i] >= 500]
+    n0 = n.y_out[i_out[0]-25]
+    n_out = np.array(n.y_out)[i_out]
+
+    # calculate output amplitude
+    peaks, _ = find_peaks(n_out)
+    troughs, _ = find_peaks(-n_out)
+    amplitude = (np.mean(n_out[peaks]) - np.mean(n_out[troughs]))/2
+
+    # calculate Gain
+    input_amplitude = r  
+    gain = amplitude / (input_amplitude*n0)
+
+    # calculate Phase Shift
+    peak_times = T[i_out][peaks]
+    input_period = 2 * np.pi / f
+    input_signal = [i[0] for i in MSRE.input.get_state(T)]
+    input_peaks, _ = find_peaks(input_signal)
+    time_differences = [abs(T[input_peaks[i]] - peak_times[0]) for i in range(len(input_peaks))]
+    closest_peak_index = np.argmin(time_differences)
+    closest_peak_time = T[input_peaks[closest_peak_index]]
+    
+    # calculate Phase Shift
+    phase_shift = 360*(closest_peak_time-peak_times[0])/input_period
+
+    if (phase_shift>180):
+        phase_shift = -360+phase_shift
+    elif (phase_shift<-180):
+        phase_shift = 360-phase_shift
+           
+    return f, gain, phase_shift
+
+with ProcessPoolExecutor() as executor:
+    results = list(executor.map(process_frequency, f_range))
+
+# Process the results
+results_df = pd.DataFrame(results, columns=['Frequency', 'Gain', 'Phase Shift'])
+
+# Write to CSV file
+# csv_filename = f"frequency_response_results_{P}_MW_double_tau.csv"
+csv_filename = f"frequency_response_results_{P}_MW.csv"
+results_df.to_csv(csv_filename, index=False)
+
+print(f"Results written to {csv_filename}")

dynamic_model/msrDynamics_implementation/parameters_U233.py

@@ -0,0 +1,151 @@
+import numpy as np
+
+# domain
+t0 = 0.0
+tf = 5000.00
+T = np.arange(t0,tf,0.01)
+
+# REACTIVITY INSERTION
+inserted = 1.39e-4 # 1MW
+# inserted = 1.96e-4 # 5MW
+# inserted = 2.48e-4 # 8MW
+
+# NEUTRONICS DATA
+tau_l = 16.73
+tau_c = 8.46
+# P = 0.1
+# P = 5
+P = 8    
+n_frac0 = 1  # initial fractional neutron density n/n0
+Lam = 4.0E-04
+lam = np.array([1.260E-02, 3.370E-02, 1.390E-01, 3.250E-01, 1.130E+00, 2.500E+00])
+beta = np.array([0.00023, 0.00079, 0.00067, 0.00073, 0.00013, 0.00009])
+beta_t = np.sum(beta)  # total delayed neutron fraction MSRE
+rho_0 = beta_t-sum(np.divide(beta,1+np.divide(1-np.exp(-lam*tau_l),lam*tau_c))) # reactivity change in going from stationary to circulating fuel
+C0 = beta / Lam * (1.0 / (lam - (np.exp(-lam * tau_l) - 1.0) / tau_c))
+
+# Feedback coefficients
+a_f = -11.034E-5
+a_g = -05.814E-5
+
+# CORE HEAT TRANSFER PARAMETERS
+vdot_f = 7.5708E-02
+rho_f = 2.14647E+03
+W_f = 1.623879934566580e+02
+m_f = W_f * tau_c
+nn_f = 2
+mn_f = m_f / nn_f
+scp_f = 1.9665E-3
+
+# Core Upflow
+v_g = 1.95386
+rho_g = 1.860E3
+m_g = v_g * rho_g
+scp_g = 1.773E-3
+mcp_g1 = m_g * scp_g
+mcp_f1 = mn_f * scp_f
+mcp_f2 = mn_f * scp_f
+hA_fg = 0.02 * 9 / 5
+k_g = 0.07
+k_1 = 0.5
+k_2 = 0.5
+k_f = 0.93
+k_f1 = k_f / nn_f
+k_f2 = k_f / nn_f
+
+# Heat Exchanger
+d_he = 16
+h_he = 72
+od_tube = 0.5
+id_tube = od_tube - 2 * 0.042
+n_tube = 159
+a_tube = 254 * 144
+l_tube = a_tube / n_tube / (np.pi * od_tube)
+v_tube = n_tube * np.pi * (od_tube / 2) ** 2 * l_tube
+v_cool = n_tube * np.pi * (id_tube / 2) ** 2 * l_tube
+v_he = (d_he / 2) ** 2 * np.pi * h_he
+v_he_fuel = v_he - v_tube
+in_m = 1.63871e-5
+W_p = W_f
+m_p = v_he_fuel * in_m * rho_f
+nn_p = 4
+mn_p = m_p / nn_p
+cp_p = scp_f
+vdot_s = 5.36265E-02
+rho_s = 1.922e3
+W_s = 1.005793369810108e+02
+m_s = v_cool * in_m * rho_s
+nn_s = 4
+mn_s = m_s / nn_s
+scp_s = 2.39E-3
+A_phe = 2.359E+01
+ha_p = 6.480E-01
+ha_s = 3.060E-01
+mcp_pn = mn_p * cp_p
+hA_pn = ha_p / nn_s
+nn_t = 2
+rho_tube = 8.7745E+03
+m_tn = (v_tube - v_cool) * in_m * rho_tube / nn_t
+scp_t = 5.778E-04
+mcp_tn = m_tn * scp_t
+mcp_sn = mn_s * scp_s
+hA_sn = ha_s / nn_s
+
+# Initial conditions
+Tf_in = 6.3222E+02
+T0_f2 = 6.5727E+02
+T0_f1 = Tf_in + (T0_f2 - Tf_in) / 2
+T0_g1 = T0_f1 + (k_g * P / hA_fg)
+Tp_in = T0_f2
+T0_p4 = Tf_in
+T0_p1 = Tp_in - (Tp_in - T0_p4) / 4
+T0_p2 = Tp_in - 2 * (Tp_in - T0_p4) / 4
+T0_p3 = Tp_in - 3 * (Tp_in - T0_p4) / 4
+Ts_in = 5.4611E+02
+T0_s4 = 5.7939E+02
+T0_s1 = Ts_in + (T0_s4 - Ts_in) / nn_s
+T0_s2 = Ts_in + 2 * (T0_s4 - Ts_in) / nn_s
+T0_s3 = Ts_in + 3 * (T0_s4 - Ts_in) / nn_s
+T0_t1 = (T0_p1 * hA_pn + T0_s3 * hA_sn) / (hA_pn + hA_sn)
+T0_t2 = (T0_p3 * hA_pn + T0_s1 * hA_sn) / (hA_pn + hA_sn)
+
+# Radiator Parameters
+Trp_in = T0_s4
+T0_rp = Ts_in
+Trs_in = 37.78
+T0_rs = 148.9
+od_rad = 0.01905
+tube_wall_thick = 0.0018288
+id_rad = od_rad - 2 * tube_wall_thick
+n_rtubes = 120
+l_rtube = 9.144
+v_rp = np.pi * (id_rad / 2) ** 2 * l_rtube * n_rtubes
+n_tpr = 12
+n_row = 10
+tube_space = 0.0381
+v_rs = (n_row * od_rad + (n_row - 1) * tube_space) * (n_tpr * od_rad + (n_tpr - 1) * tube_space) * l_rtube
+W_rp = W_s
+m_rp = v_rp * rho_s
+nn_rp = 1
+mn_rp = m_rp / nn_rp
+cp_rp = scp_s
+vdot_rs = 94.389
+rho_rs = 1.1237
+W_rs = vdot_rs * rho_rs
+m_rs = v_rs * rho_rs
+nn_rs = 1
+mn_rs = m_rs / nn_rs
+scp_rs = 1.0085E-3
+A_rad = 6.503E1
+h_roverall = P / A_rad / ((T0_rp + Trp_in) / 2 - (T0_rs + Trs_in) / 2)
+mcp_rpn = mn_rp * cp_rp
+hA_rpn = h_roverall * A_rad / nn_rs
+mcp_rsn = mn_rs * scp_rs
+hA_rsn = h_roverall * A_rad / nn_rs
+
+# Pure time delays between components
+tau_hx_c = 8.67 #+2.145 
+tau_c_hx = 3.77 #+2.145
+tau_hx_r = 4.71
+tau_r_hx = 8.24
+

dynamic_model/msrDynamics_implementation/parameters_U235.py

@@ -0,0 +1,201 @@
+import numpy as np
+import math
+pi = math.pi
+
+# domain
+t0 = 0.0
+tf = 1000.00
+T = np.arange(t0,tf,0.01)
+
+# NEUTRONICS DATA
+tau_l = 16.73  # ORNL-TM-0728 %16.44; % (s)
+tau_c = 8.46  # ORNL-TM-0728 %8.460; % (s)
+P = 8.0  # Thermal Power in MW ORNL-TM-1070, p.2
+n_frac0 = 1.0  # initial fractional neutron density n/n0 (n/cm^3/s)
+Lam = 2.400E-04  # mean generation time ORNL-TM-1070 p.15 U235
+# Lam = 4.0E-04;  # mean generation time ORNL-TM-1070 p.15 U233
+lam = np.array([1.240E-02, 3.05E-02, 1.11E-01, 3.01E-01, 1.140E+00, 3.014E+00])
+beta = np.array([0.000223, 0.001457, 0.001307, 0.002628, 0.000766, 0.00023])  # U235
+# beta = np.array([0.00023, 0.00079, 0.00067, 0.00073, 0.00013, 0.00009])  # U233
+beta_t = np.sum(beta)  # total delayed neutron fraction MSRE
+rho_0 = beta_t-sum(np.divide(beta,1+np.divide(1-np.exp(-lam*tau_l),lam*tau_c))) # reactivity change in going from stationary to circulating fuel
+C0 = beta / Lam * (1.0 / (lam - (np.exp(-lam * tau_l) - 1.0) / tau_c))
+
+# Feedback co-efficients
+a_f = -8.71E-05  # U235 (drho/°C) fuel salt temperature-reactivity feedback coefficient ORNL-TM-1647 p.3 % -5.904E-05; % ORNL-TM-0728 p. 101 %
+a_g = -6.66E-05  # U235 (drho/°C) graphite temperature-reactivity feedback coefficient ORNL-TM-1647 p.3 % -6.624E-05; % ORNL-TM-0728 p.101
+
+# CORE HEAT TRANSFER PARAMETERS
+# FUEL PARAMETERS - DONE
+vdot_f = 7.5708E-02  # ORNL-TM-0728 % 7.571e-2; % vol. flow rate (m^3/s) ORNL-TM-1647 p.3, ORNL-TM-0728 p.12
+rho_f = 2.14647E+03  # (partially enriched U-235)ORNL-TM-0728 p.8 2.243E+03; % (Th-U) density of fuel salt (kg/m^3) ORNL-TM-0728 p.8
+W_f = 1.623879934566580e+02  # 1.83085e+02;%vdot_f*rho_f; % 182.78; % calcd from m_dot*cp*delT=P; vdot_f*rho_f; % fuel flow rate (kg/s)
+# tau_f_c  = tau_c; % ORNL-TM-0728 % 8.45; % transit time of fuel in core (s) ORNL-TM-1070 p.15, TDAMSRE p.5
+m_f = W_f * tau_c  # fuel mass in core (kg)
+nn_f = 2  # number of fuel nodes in core model
+mn_f = m_f / nn_f  # fuel mass per node (kg)
+# cp_f     = 4.2*9/5; % (MJ/deg-C) total fuel heat capacity TDAMSRE p.5
+scp_f = 1.9665E-3  # specific heat capacity of fuel salt (MJ/kg-C) ORNL-TM-0728 p.8
+
+# Core Upflow - DONE
+v_g = 1.95386  # graphite volume(m^3) ORNL-TM-0728 p. 101
+rho_g = 1.860E3  # graphite density (kg/m^3) ORNL-3812 p.77, ORNL-TM-0728 p.87
+m_g = v_g * rho_g  # graphite mass (kg)
+cp_g = 3.6 * 9 / 5  # TDAMSRE p.5 graphite total heat capacity (MW-s/C) ORNL-TM-1647 p.3
+scp_g = 1.773E-3  # cp_g/m_g; % graphite specific heat capacity (MW-s/kg-C) ORNL-TM-1647 p.3
+mcp_g1 = m_g * scp_g  # (mass of material x heat capacity of material) of graphite per lump (MW-s/°C)
+mcp_f1 = mn_f * scp_f  # (mass of material x heat capacity of material) of fuel salt per lump (MW-s/°C)
+mcp_f2 = mn_f * scp_f  # (mass of material x heat capacity of material) of fuel salt per lump (MW-s/°C)
+hA_fg = 0.02 * 9 / 5  # (fuel to graphite heat transfer coeff x heat transfer area) (MW/°C) ORNL-TM-1647 p.3, TDAMSRE p.5
+k_g = 0.07  # fraction of total power generated in the graphite  ORNL-TM-0728 p.9
+k_1 = 0.5  # fraction of heat transferred from graphite which goes to the first fuel lump
+k_2 = 0.5  # fraction of heat transferred from graphite which goes to the second fuel lump
+k_f = 0.93  # fraction of heat generated in fuel - that generated in the external loop ORNL-TM-0728 p.9
+k_f1 = k_f / nn_f  # fraction of total power generated in lump f1
+k_f2 = k_f / nn_f  # fraction of total power generated in lump f2
+
+# New node for power deposited in fuel outside the core
+k_out = 1 - (k_g + k_f)  # fraction of power generated in fuel in external loop ORNL-TM-0728 p.9
+m_out = W_f  # (kg) Mass of node such that resident time is 1 sec (W_f needs to be defined)
+
+# Initial conditions - DONE
+Tf_in = 6.3222E+02  # in °C ORNL-TM-1647 p.2
+T0_f2 = 6.5727E+02  # 6.5444E+02; % in °C 6.461904761904777e+02; ORNL-TM-1647 p.2
+T0_f1 = Tf_in + (T0_f2 - Tf_in) / 2  # 6.405952380952389e+02; in °C
+T0_g1 = T0_f1 + (k_g * P / hA_fg)  # 6.589285714285924e+02; in °C
+# T0_out = k_out * P / m_out / scp_f + T0_f2  # in °C (scp_f needs to be defined)
+
+
+# Heat Exchanger - DONE
+# Geometry
+d_he = 16  # (in) he diameter ORNL-TM-0728 p. 164
+h_he = 72  # (in) active height % 96; %(in) he height ORNL-TM-0728 p. 164
+od_tube = 0.5  # (in) coolant tube OD ORNL-TM-0728 p. 164
+id_tube = od_tube - 2 * 0.042  # (in) coolant tube ID ORNL-TM-0728 p. 164
+n_tube = 159  # number of coolant tubes ORNL-TM-0728 p. 164
+a_tube = 254 * 144  # (in^2) total area of tubes ORNL-TM-0728 p. 164
+l_tube = a_tube / n_tube / (np.pi * od_tube)  # (in) tube length
+v_tube = n_tube * np.pi * (od_tube / 2) ** 2 * l_tube  # (in^3) hx shell volume occupied by tubes
+v_cool = n_tube * np.pi * (id_tube / 2) ** 2 * l_tube  # (in^3) hx volume occupied by coolant
+v_he = (d_he / 2) ** 2 * np.pi * h_he  # (in^3) volume of heat exchanger shell
+v_he_fuel = v_he - v_tube  # (in^3) volume available to fuel in shell
+
+# Unit conversions
+in_m = 1.63871e-5  # 1 cubic inch = 1.63871e-5 cubic meters
+
+# PRIMARY FLOW PARAMETERS - DONE
+W_p = W_f  # fuel flow rate (kg/s)
+
+m_p = v_he_fuel * in_m * rho_f  # fuel mass in PHE (kg)
+nn_p = 4  # number of fuel nodes in PHE
+mn_p = m_p / nn_p  # fuel mass per node (kg)
+cp_p = scp_f  # fuel heat capacity (MJ/(kg-C))
+
+# SECONDARY FLOW PARAMETERS - DONE
+vdot_s = 5.36265E-02  # ORNL-TM-0728 p. 164 % 5.236E-02; % coolant volume flow rate (m^3/s) ORNL-TM-1647 p.3
+rho_s = 1.922e3  # coolant salt density (kg/m^3) ORNL-TM-0728 p.8
+W_s = 1.005793369810108e+02  # vdot_s*rho_s; % calcd from mdot*cp*delT; vdot_s*rho_s; % coolant flow rate (kg/s) ORNL-TM-1647 p.3
+
+m_s = v_cool * in_m * rho_s  # coolant mass in PHE (kg)
+nn_s = 4  # number of coolant nodes in PHE
+mn_s = m_s / nn_s  # coolant mass per node (kg)
+scp_s = 2.39E-3  # cp_s/m_s; % specific heat capacity of coolant (MJ/(kg-C) ORNL-TM-0728 p.8
+
+A_phe = 2.359E+01  # effective area for heat transfer (primary and secondary, m^2) ORNL-TM-0728 p.164
+
+ha_p = 6.480E-01  # heat transfer*area coefficient from primary to tubes (MW/C) ORNL-TM-1647 p.3
+ha_s = 3.060E-01  # heat transfer*area coefficient from tubes to secondary (MW/C) ORNL-TM-1647 p.3
+
+# Primary Side
+mcp_pn = mn_p * cp_p  # (mass of material x heat capacity of material) of fuel salt per lump in MW-s/°C
+hA_pn = ha_p / nn_s  # 3.030; % (primary to tube heat transfer coeff x heat transfer area) in MW/°C
+
+# Tubes - DONE
+nn_t = 2  # number of nodes of tubes in the model
+rho_tube = 8.7745E+03  # (kg/m^3) density of INOR-8 ORNL-TM-0728 p.20
+m_tn = (v_tube - v_cool) * in_m * rho_tube / nn_t  # mass of tubes (kg)
+scp_t = 5.778E-04  # specific heat capacity of tubes (MJ/(kg-C)) ORNL-TM-0728 p.20
+mcp_tn = m_tn * scp_t  # mass*(heat capacity) of tubes per lump in MW-s/°C
+
+# Secondary Side - DONE
+mcp_sn = mn_s * scp_s  # (mass of material x heat capacity of material) of coolant salt per lump in MW-s/°C
+hA_sn = ha_s / nn_s  # (tube to secondary heat transfer coeff x heat transfer area) in MW/°C
+
+# Initial conditions - DONE
+# Primary nodes
+Tp_in = T0_f2  # in °C ORNL-TM-1647 p.2
+T0_p4 = Tf_in  # 6.5444E+02; % in °C 6.461904761904777e+02; ORNL-TM-1647 p.2
+T0_p1 = Tp_in + (T0_p4 - Tp_in) / 4  # in °C
+T0_p2 = Tp_in + 2 * (T0_p4 - Tp_in) / 4  # in °C
+T0_p3 = Tp_in + 3 * (T0_p4 - Tp_in) / 4  # in °C
+
+# Secondary nodes
+Ts_in = 5.4611E+02  # in °C ORNL-TM-1647 p.2
+T0_s4 = 5.7939E+02  # in °C ORNL-TM-1647 p.2
+T0_s1 = Ts_in + (T0_s4 - Ts_in) / nn_s  # in °C
+T0_s2 = Ts_in + 2 * (T0_s4 - Ts_in) / nn_s  # in °C
+T0_s3 = Ts_in + 3 * (T0_s4 - Ts_in) / nn_s  # in °C
+# Tube nodes
+T0_t1 = (T0_p1 * hA_pn + T0_s3 * hA_sn) / (hA_pn + hA_sn)  # in °C
+T0_t2 = (T0_p3 * hA_pn + T0_s1 * hA_sn) / (hA_pn + hA_sn)  # in °C
+
+# Radiator Parameters - DONE
+
+# Initial conditions - DONE
+# Primary nodes
+Trp_in = T0_s4  # 5.933E+02; % in °C ORNL-TM-1647 p.2
+T0_rp = Ts_in  # in °C ORNL-TM-1647 p.2
+
+# Secondary nodes - DONE
+Trs_in = 37.78  # (C) air inlet temperature ORNL-TM-1647 p.2
+T0_rs = 148.9  # (C) air exit temperature ORNL-TM-1647 p.2
+
+# Radiator Geometry
+od_rad = 0.01905  # (m) outer diameter of tubes in the radiator ORNL-TM-0728 p.296
+tube_wall_thick = 0.0018288  # (m) thickness of tubes in the radiator ORNL-TM-0728 p.296
+id_rad = od_rad - 2 * tube_wall_thick
+n_rtubes = 120  # number of tubes in the radiator (rows times tubes per row) ORNL-TM-0728 p.296
+l_rtube = 9.144  # (m) length of tubes in the radiator ORNL-TM-0728 p.296
+v_rp = pi * (id_rad / 2) ** 2 * l_rtube * n_rtubes  # volume available to salt in the radiator
+# v_rtube = pi * (od_rad / 2) ** 2 * l_rtube * n_rtubes - v_rp  # volume of metal in radiator tubes *TUBES NOT MODELED
+
+n_tpr = 12  # number of tubes per row in the radiator matrix
+n_row = 10  # number rows in the radiator matrix
+tube_space = 0.0381  # (m) spacing between tubes and rows of matrix
+v_rs = (n_row * od_rad + (n_row - 1) * tube_space) * (n_tpr * od_rad + (n_tpr - 1) * tube_space) * l_rtube  # volume of air inside radiator
+
+# PRIMARY FLOW PARAMETERS - DONE
+W_rp = W_s  # coolant salt flow rate (kg/s)
+m_rp = v_rp * rho_s  # coolant salt mass in rad (kg)
+nn_rp = 1  # number of coolant salt nodes in the radiator
+mn_rp = m_rp / nn_rp  # coolant mass per node (kg)
+cp_rp = scp_s  # coolant specific heat capacity (MJ/(kg-C))
+
+# SECONDARY FLOW PARAMETERS - DONE
+vdot_rs = 94.389  # ORNL-TM-0728 p. 296; 78.82; % air volume flow rate (m^3/s) ORNL-TM-1647 p.2
+rho_rs = 1.1237  # air density (kg/m^3) REFPROP (310K and 0.1MPa)
+W_rs = vdot_rs * rho_rs  # air flow rate (kg/s)
+
+m_rs = v_rs * rho_rs  # coolant air mass in rad (kg)
+nn_rs = 1  # number of coolant nodes in rad
+mn_rs = m_rs / nn_rs  # coolant mass per node (kg)
+scp_rs = 1.0085E-3  # (MJ/kg-C) specific heat capacity of air at (air_out+air_in)/2 REFPROP
+
+A_rad = 6.503E1  # (m^2) surface area of the radiator ORNL-TM-0728 p.14
+h_roverall = P / A_rad / ((T0_rp + Trp_in) / 2 - (T0_rs + Trs_in) / 2)  # cald as: P/A_rad/((T0_rp+Trp_in)/2-(T0_rs+Trs_in)/2)  3.168E-4; % (MW/m^2-C) polimi thesis
+
+# Primary Side
+mcp_rpn = mn_rp * cp_rp  # (mass of material x heat capacity of material) of fuel salt per lump in MW-s/°C
+hA_rpn = h_roverall * A_rad / nn_rs  # 3.030; % (primary to secondary heat transfer coeff x heat transfer area) in MW/°C
+
+# Secondary Side - DONE
+mcp_rsn = mn_rs * scp_rs  # (mass of material x heat capacity of material) of coolant salt per lump in MW-s/°C
+hA_rsn = h_roverall * A_rad / nn_rs  # (tube to secondary heat transfer coeff x heat transfer area) in MW/°C
+
+# Pure time delays between components - DONE
+tau_hx_c = 8.67  # (sec) delay from hx to core TDAMSRE p.6
+tau_c_hx = 3.77  # (sec) subtracted 1 sec for external loop power generation node resident time; delay from core to fuel hx TDAMSRE p.6
+tau_hx_r = 4.71  # (sec) fertile hx to core TDAMSRE p.6
+tau_r_hx = 8.24  # (sec) core to fertile hx TDAMSRE p.6
+
+first_val = (rho_0 - beta_t) * n_frac0 / Lam + lam[0] * C0[0] + lam[1] * C0[1] + lam[2] * C0[2] + lam[3] * C0[3] + lam[4] * C0[4] + lam[5] * C0[5]