3. Moments Analysis
Moments analysis¶
These sets of examples deal with how to calculate the moments (summary statistics of the molecular distribution with respect to time). The moments are often critical in determining kinetic measurements such as the conversion and reactivities.
Note that these methods should only be applied after baseline correcting and calibrating your data.
# imports and reading in the data
import pandas as pd
import tapsap
import plotly
plotly.offline.init_notebook_mode()
data = pd.read_csv('../tapsap/data/irreversible.csv')
times = data['times'].values
inert_flux = data['inert_flux'].values
reactant_flux = data['A_flux'].values
Moments¶
The moments, denoted by M0, M1, and M2, represent the area, mean residence time and the variance residence time (variance = M2 - M1**2). More intuitively, each moment describes a part of the flux: the area is the total number of molecules, the mean residence time is the central location of the flux and the variance residence time describes the spread of the flux (time in reacator). These moments are calculated using the trapezoidal rule and by the function moments in tapsap. The moments function does have an optional input in the time range in which the moments should be taken (integration_time_range).
temp_moments = tapsap.moments(inert_flux, times)
print(temp_moments)
{'M0': 0.9992222700000024, 'M1': 0.49723921110000036, 'M2': 0.4074535171413597}
Diffusion coefficient¶
This function determines the diffusion coefficient using the moments and reactor parameters. An optional input includes the current and reference mass. If the masses are included, then the diffusion coefficient is scaled by Graham's Law. This function allows for determining the diffusion of a reactant species based on the diffusion of the inert.
This function will be important to measuring the gas concentration and rate since each function relies on the diffusion of the reactant.
# setting the reactor parameters
zone_residence_time = {'zone0':0.5, 'zone1':1, 'zone2':0.5}
zone_lengths = {'zone0':0.5, 'zone1':1e-5, 'zone2':0.5}
zone_porosity = {'zone0':0.5, 'zone1':0.5, 'zone2':0.5}
temp_diffusion = tapsap.diffusion_moments(temp_moments, zone_lengths, zone_porosity)
print(temp_diffusion)
{'diffusion': 0.5023951411307574}
Check for reversibility¶
This is a simple function for determining if a reactant gas is reversible. It compares the normalized M1 value of the reactant to the normalized M1 value of the inert. If the inert M1 is less than the reactant M1, then the flux is reversible. This function is helpful in determining the scale of the flux because if the flux is reversible, then the area of the reactant should be equal to the area of the inert flux.
temp_reversible = tapsap.isreversible(reactant_flux, times, inert_flux, 40, 40)
print(temp_reversible)
False
Reactivity coefficients¶
The reactivity coefficients describe the apparent kinetics of the reactant/products based on the inert flux. Depending on the assumed mechanism, these coefficients may describe specific intrinsic kinetic coefficients. This function requires not only the moments, but also the reactor parameters. There are two different functions for reactivities: reactivities for a reactant (reactivities_reactant) and reactivities for the product (reactivities_product). The reactivities for the reactant only require information about the inert, but the product reactivities also requires the moments and the reactivities of the reactant species.
reactant_moments = tapsap.moments(reactant_flux, times)
reactant_reactivities = tapsap.reactivities_reactant(reactant_moments, temp_moments, zone_residence_time)
print(reactant_reactivities)
{'r0': 0.866989719192093, 'r1': 0.035867877284172955, 'r2': -0.0043589577393161605}
Other summary statistics¶
The moments are not the only parameters that may help describe the flux. In some cases, it necessary to check the min, mean, and max of the flux as well as the approximate Gamma distribution parameters. The functions min_mean_max and the rtd_parameters (residence time distribution parameters) determine the min, mean, max, residence time, variance residence time, the Gamma distribution shape, and the Gamma distribution scale
temp_min_mean_max = tapsap.min_mean_max(inert_flux)
temp_rtd = tapsap.rtd_parameters(temp_moments)
print(temp_min_mean_max)
print(temp_rtd)
{'min': 0.0, 'mean': 0.33307409, 'max': 1.84938}
{'mean_residence_time': 0.49762622994781647, 'variance_residence_time': 0.16013878787893676, 'gamma_shape': 1.5463578063253745, 'gamma_scale': 3.1074684437103186}
Application to a Transient object¶
The above examples can be applied in a similar fashion but to all flux within a species using the methods set_min_mean_max, set_moments, set_gas_diffusion, set_rtd_parameters, and set_reactivities. The methods set_gas_diffusion will calculate the diffusion coefficient of the inert species if the reference_gas is set (see set_reference_gas in the Experiment class). Additionally, if the reference_gas is set, the method will automatically apply Graham's Law moving from the inert diffusion to the gas diffusion. If the reference gas is not set, then the diffusion will be calculated based on the current flux/reactor information. When using set_reactivities, there is an option to provide a reactant_obj (the Transient class of the reactant). If supplied, then the reactivities will be calculated as the current object is a product. Otherwise, the reactivities will be calculated as if the species is a reactant.
All moments based values are stored in the Transient object under df_moments.
# reading in the data
data = tapsap.read_tdms('../tapsap/data/argon_100C.tdms')
# set the reactor information
data.reactor.zone_lengths = {'zone0':0.018, 'zone1':1e-5, 'zone2':0.018}
data.reactor.zone_porosity = {'zone0':0.4, 'zone1':0.4, 'zone2':0.4}
data.reactor.reactor_radius = 0.002
# this specifies the reactor parameters to each species
data.set_reactor_params()
# grabbing the transient information
transient_info = data.species_data['AMU_40_1']
# baseline correcting
transient_info.baseline_correct(baseline_time_range = [4.5, 5])
# set the moments
transient_info.set_moments()
# set the min_mean_max
transient_info.set_min_mean_max()
# set the gas diffusion
transient_info.set_gas_diffusion()
# set the residence time distribution parameters
transient_info.set_rtd_parameters()
tapsap.plot_tap(transient_info.df_moments['pulse_number'].values, transient_info.df_moments, legend=True, y_lab='Moments', x_lab='Pulse_number')