# Simulation doesn't converge with SMA and non-integer characteristic charge **URL:** \ **Category:** CADET Troubleshooting\ **Created:** [January 2, 2024, 9:40pm UTC](https://forum.cadet-web.de/t/simulation-doesnt-converge-with-sma-and-non-integer-characteristic-charge/841 "2024-01-02T21:40:38Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![Stefanos](https://forum.cadet-web.de/letter_avatar_proxy/v4/letter/s/3da27b/32.png) [@Stefanos](https://forum.cadet-web.de/u/Stefanos)\ **Post date:** [January 2, 2024, 9:40pm UTC](https://forum.cadet-web.de/t/simulation-doesnt-converge-with-sma-and-non-integer-characteristic-charge/841/1 "2024-01-02T21:40:38Z")
Hello everyone, I used the Yamamoto method in Cadet Process, resulting in an optimum of approximately 1.69 for the characteristic charge. Unfortunately, the model does not converge afterwards. However, if I then arbitrarily set the characteristic charge to 1, 2 or another natural number, it converges. The model consistently fails to converge when the characteristic charge is not an integer. Could someone help me? ---
**Author:** ![ronald.jaepel](https://forum.cadet-web.de/user_avatar/forum.cadet-web.de/ronald.jaepel/32/986_2.png) [@ronald.jaepel](https://forum.cadet-web.de/u/ronald.jaepel)\ **Post date:** [January 2, 2024, 9:50pm UTC](https://forum.cadet-web.de/t/simulation-doesnt-converge-with-sma-and-non-integer-characteristic-charge/841/2 "2024-01-02T21:50:20Z")
Good evening Stefanos, can you share a minimum reproducible example including your code and input data? ---
**Author:** ![Stefanos](https://forum.cadet-web.de/letter_avatar_proxy/v4/letter/s/3da27b/32.png) [@Stefanos](https://forum.cadet-web.de/u/Stefanos)\ **Post date:** [January 3, 2024, 8:01am UTC](https://forum.cadet-web.de/t/simulation-doesnt-converge-with-sma-and-non-integer-characteristic-charge/841/3 "2024-01-03T08:01:56Z")
Hello Ronald, many thanks for the quick reply and I apologise for not having sent a minimal example before. Here is the example: ```auto from CADETProcess.processModel import ComponentSystem from CADETProcess.processModel import StericMassAction from CADETProcess.processModel import Inlet, GeneralRateModel, Outlet from CADETProcess.processModel import FlowSheet from CADETProcess.processModel import Process import numpy as np Q_ml_min = 0.5 # ml/min Q_m3_s = Q_ml_min/(60*1e6) #m3/s V_tracer = 5e-8 #m3 component_system = ComponentSystem() component_system.add_component('Salt') component_system.add_component('Protein') binding_model = StericMassAction(component_system, name='SMA') binding_model.is_kinetic = False binding_model.adsorption_rate = [1, 1] binding_model.desorption_rate = [1, 1] binding_model.steric_factor = [1, 1] binding_model.capacity = 800 binding_model.characteristic_charge = [0,1.6] # Unit Operations inlet = Inlet(component_system, name='inlet') inlet.flow_rate = Q_m3_s column = GeneralRateModel(component_system, name='column') column.binding_model = binding_model column.length = 0.1 column.diameter = 0.0077 column.bed_porosity = 0.3 column.particle_radius = 3.4e-5 column.particle_porosity = 0.857 column.axial_dispersion =2e-7 column.film_diffusion = [2.0e-3, 2.0e-5] column.pore_diffusion = [7e-3, 7e-5] outlet = Outlet(component_system, name='outlet') # Flow Sheet flow_sheet = FlowSheet(component_system) flow_sheet.add_unit(inlet) flow_sheet.add_unit(column) flow_sheet.add_unit(outlet) flow_sheet.add_connection(inlet, column) flow_sheet.add_connection(column, outlet) # Process process = Process(flow_sheet, 'lwe') wash_start = V_tracer/Q_m3_s gradient_start = (1200 + wash_start) concentration_difference = np.array([1000, 0]) - np.array([0,0]) gradient_volume = 6.11e-5 process.cycle_time = gradient_start + (gradient_volume/Q_m3_s) gradient_duration = process.cycle_time - gradient_start gradient_slope = concentration_difference/gradient_duration _ = process.add_event('Load', 'flow_sheet.inlet.c', [0, 1],0) _ = process.add_event('Wash', 'flow_sheet.inlet.c', [0, 0], wash_start) _ = process.add_event( 'grad_start', 'flow_sheet.inlet.c', [[0, gradient_slope[0]], [0, gradient_slope[1]]], gradient_start ) from CADETProcess.simulator import Cadet simulator = Cadet() sim_res = simulator.simulate(process) ``` As soon as I change the characeristic charge of the protein to a natural number the model converges. Best regards Stefanos ---
**Author:** ![ronald.jaepel](https://forum.cadet-web.de/user_avatar/forum.cadet-web.de/ronald.jaepel/32/986_2.png) [@ronald.jaepel](https://forum.cadet-web.de/u/ronald.jaepel)\ **Post date:** [January 3, 2024, 3:14pm UTC](https://forum.cadet-web.de/t/simulation-doesnt-converge-with-sma-and-non-integer-characteristic-charge/841/4 "2024-01-03T15:14:49Z")
Hey Stefanos, the problem was, that the SMA model doesn’t work well with salt concentrations of exactly 0. My theory is that due to numerical noise concentrations can sometimes dip below 0 and would then get raised to a non-integer, which doesn’t work. To solve this, please set up your model to contain a bit of salt before starting the simulation. The two changes are: adding the salt concentration to your column.c and column.cp initialization states and also initialize the column with all binding sites saturated with salt: ```auto starting_salt_conc = 1 # or any other non-zero number column.c = [starting_salt_conc, 0] column.cp = [starting_salt_conc, 0] column.q = [binding_model.capacity, 0] ``` and changing the inlet concentrations to also use non-zero concentrations: ```auto _ = process.add_event('Load', 'flow_sheet.inlet.c', [starting_salt_conc, 1], 0) _ = process.add_event('Wash', 'flow_sheet.inlet.c', [starting_salt_conc, 0], wash_start) _ = process.add_event( 'grad_start', 'flow_sheet.inlet.c', [[starting_salt_conc, gradient_slope[0]], [0, gradient_slope[1]]], gradient_start ) ``` The complete working code is: ```auto from CADETProcess.processModel import ComponentSystem, LumpedRateModelWithPores from CADETProcess.processModel import StericMassAction from CADETProcess.processModel import Inlet, GeneralRateModel, Outlet from CADETProcess.processModel import FlowSheet from CADETProcess.processModel import Process import numpy as np Q_ml_min = 0.5 # ml/min Q_m3_s = Q_ml_min / (60 * 1e6) # m3/s V_tracer = 5e-8 # m3 starting_salt_conc = 1 component_system = ComponentSystem() component_system.add_component('Salt') component_system.add_component('Protein') binding_model = StericMassAction(component_system, name='SMA') binding_model.is_kinetic = True binding_model.adsorption_rate = [0, 1] binding_model.desorption_rate = [0, 1] binding_model.steric_factor = [0, 1] binding_model.capacity = 800 # binding_model.reference_solid_phase_conc = 800 # binding_model.reference_liquid_phase_conc = 1000 binding_model.characteristic_charge = [0, 1.6] # Unit Operations inlet = Inlet(component_system, name='inlet') inlet.flow_rate = Q_m3_s column = GeneralRateModel(component_system, name='column') column.binding_model = binding_model column.length = 0.1 column.diameter = 0.0077 column.bed_porosity = 0.3 column.particle_radius = 3.4e-5 column.particle_porosity = 0.857 column.axial_dispersion = 2e-7 column.film_diffusion = [2.0e-3, 2.0e-5] column.pore_diffusion = [7e-3, 7e-5] column.c = [starting_salt_conc, 0] column.cp = [starting_salt_conc, 0] column.q = [starting_salt_conc, 0] outlet = Outlet(component_system, name='outlet') # Flow Sheet flow_sheet = FlowSheet(component_system) flow_sheet.add_unit(inlet) flow_sheet.add_unit(column) flow_sheet.add_unit(outlet) flow_sheet.add_connection(inlet, column) flow_sheet.add_connection(column, outlet) # Process process = Process(flow_sheet, 'lwe') wash_start = V_tracer / Q_m3_s gradient_start = (1200 + wash_start) concentration_difference = np.array([1000, 0]) - np.array([0, 0]) gradient_volume = 6.11e-5 process.cycle_time = gradient_start + (gradient_volume / Q_m3_s) gradient_duration = process.cycle_time - gradient_start gradient_slope = concentration_difference / gradient_duration _ = process.add_event('Load', 'flow_sheet.inlet.c', [starting_salt_conc, 1], 0) _ = process.add_event('Wash', 'flow_sheet.inlet.c', [starting_salt_conc, 0], wash_start) _ = process.add_event( 'grad_start', 'flow_sheet.inlet.c', [[starting_salt_conc, gradient_slope[0]], [0, gradient_slope[1]]], gradient_start ) from CADETProcess.simulator import Cadet from CADETProcess.plotting import SecondaryAxis sec = SecondaryAxis() sec.components = ['Salt'] sec.y_label = '$c_{salt}$' simulator = Cadet() sim_res = simulator.simulate(process) sim_res.solution.outlet.outlet.plot(secondary_axis=sec) ``` ---
**Author:** ![Stefanos](https://forum.cadet-web.de/letter_avatar_proxy/v4/letter/s/3da27b/32.png) [@Stefanos](https://forum.cadet-web.de/u/Stefanos)\ **Post date:** [January 4, 2024, 9:06am UTC](https://forum.cadet-web.de/t/simulation-doesnt-converge-with-sma-and-non-integer-characteristic-charge/841/5 "2024-01-04T09:06:38Z")
Hello Ronald, thank you very much for the very quick and detailed explanation! 🙂 ---
**Author:** ![alters](https://forum.cadet-web.de/user_avatar/forum.cadet-web.de/alters/32/294_2.png) [@alters](https://forum.cadet-web.de/u/alters)\ **Post date:** [January 5, 2024, 11:28pm UTC](https://forum.cadet-web.de/t/simulation-doesnt-converge-with-sma-and-non-integer-characteristic-charge/841/6 "2024-01-05T23:28:36Z")
The SMA model also is not really mechanistically valid at salt concentration c\_s = 0. It is based on stoichiometric displacement where salt ions displace adsorbed proteins. There are other models such as Langmuir that can be used with c\_s = 0. ---
**Author:** ![h.lanzrath](https://forum.cadet-web.de/user_avatar/forum.cadet-web.de/h.lanzrath/32/1186_2.png) [@h.lanzrath](https://forum.cadet-web.de/u/h.lanzrath)\ **Post date:** [January 27, 2026, 10:31am UTC](https://forum.cadet-web.de/t/simulation-doesnt-converge-with-sma-and-non-integer-characteristic-charge/841/7 "2026-01-27T10:31:40Z")