Arbitrary RTD Model

Arbitrary RTD model using a user supplied function

class rtdpy.arbitrary.Arbitrary(fun, dt, time_end)

Bases: rtdpy.rtd.RTD

Create an arbitrary Residence Time Distribution (RTD) model from a function.

Parameters

funcallable

Function called with fun(t), where t is an np.ndarray of time. fun(t) should return the RTD at time t as np.ndarray.

dtscalar

Time step for RTD. dt>0

time_endscalar

End time for RTD. time_end>0

Examples

>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> import rtdpy
>>> #
>>> # Create Square RTD function
>>> def fun(t):
...     # Square RTD function between times 10 and 15.
...     return 1 / (15 - 10) * np.greater(t, 10) * np.less(t, 15)
>>> rtdmodel = rtdpy.Arbitrary(fun, dt=.01, time_end=20)
>>> plt.plot(rtdmodel.time, rtdmodel.exitage, label="Square RTD")
>>> print(f"Square RTD mean residence time: {rtdmodel.mrt():.1f}")
Square RTD mean residence time: 12.5
>>> #
>>> # Create a model from experimental data
>>> t_data = np.array([0, 1, 2, 3, 4, 5], dtype=float)
>>> C_data = np.array([0.5, 0.2, 0.3, 0.15, 0.1, 0.0], dtype=float)
>>> def fun_data(t):
...     return np.interp(t, t_data, C_data, left=0, right=0)
>>> rtdmodel_data = rtdpy.Arbitrary(fun_data, dt=.01, time_end=20)
>>> #
>>> # Plot arbitrary models
>>> plt.plot(rtdmodel_data.time, rtdmodel_data.exitage, label="Data RTD")
>>> plt.xlabel('Time')
>>> plt.ylabel('Exit Age Function')
>>> plt.legend()
>>> print(f"Data RTD mean residence time: {rtdmodel_data.mrt():.1f}")
Data RTD mean residence time: 1.7
>>> plt.show()

property dt

Time step for RTD

property exitage

Exit age distribution for RTD

property exitage_norm

Normalized Exit Age Distribtion for RTD

frequencyresponse(omegas)

Parameters

omegasndarray

frequencies at which to evaluate magnitude response

Returns

magnitudendarray

frequency magnitude response at omegas

property fun

Return fun.

funnelplot(times, disturbances)

Return maximum output signal due to square disturbances.

Uses method from [Garcia] . Also returns meshgrid for times and disturbance inputs for ease of plotting.

Parameters

timesarray_like, size m

Times to determine funnelplot

disturbancesarray_like, size n

Disturbance magnitudes

Returns

x2D meshgrid size (mxn)

times

y2D meshgrid size (mxn)

disturbances

response2D meshgrid size (mxn)

maximum response at (x,y)

References

Garcia

Garcia-Munoz S., Butterbaugh A., Leavesley I., Manley L.F., Slade D., Bermingham S. (2018) A flowhseet model for the development of a continuous process for pharmaceutical tablets: An industrial perspective. “AIChE Journal”, 64(2), 511-525.

integral()

Integral of RTD.

mrt()

Mean residence time of RTD.

output(inputtime, inputsignal)

Convolves input signal with RTD

Parameters

inputtimendarray

Times of input signal, which must have same dt as RTD. Size m

inputsignalndarray

Input signal. Size n

Returns

outputsignalndarrary

Output signal at same dt. Size m + n -1

sigma()

Variance of RTD.

property stepresponse

Step respose of RTD

property stepresponse_norm

Normalized step respose of RTD

property time

Time points for exitage function.

property time_end

Last time point for RTD