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APPLICATIONS OF TRANSITIONS AND

MOMENT ANALYSIS

Korben Knudson and Scott Rudge

1 of 25 RMC Pharmaceutical Solutions, Inc.

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OUTLINE

• What’s a Transition, What’s a Moment, What’s aPlate?

• How Do You Use Transition Analysis?• Transition Analysis for Control

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WHAT’S A PLATE?

• The concept of a plate in chromatography comesfrom an analysis by Martin and Synge from 1941

• Series of perfectly mixed CSTR’s passing solventdown a line with a pulse solute in the first tank

• Also analogous with equilibrium plates in distillation

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1 2 3 N

P-1 P-2 P-3


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PLATE RESULT


As N ∞, CN ~ normal error curve


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HETP

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Key concepts:

h = H/dp

v = vdp/D

h = A + B/v + Cv


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WHAT’S A TRANSITION?


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Transitions in Chromatography

These Conductivity transitions are similar to what would be found in an equilibration step


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MULTIPLE TRANSITIONS IN A CHROMATOGRAPHY SEQUENCE

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FIRST DERIVATIVE OF A TRANSITION

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dC

/dV

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WHAT’S A MOMENT?

• The Moment is another statistical concept foranalyzing a distributed data set, without theassumptions of normality

𝜇𝜇0 = �0

𝑉𝑉𝐹𝐹 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑

𝜇𝜇1 = �0

𝑉𝑉𝐹𝐹𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑

𝜇𝜇2 = �0

𝑉𝑉𝐹𝐹𝑑𝑑2

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑


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WHAT’S A MOMENT?

• A moment is a mathematical way to turn atransition into an HETP measurement

• Very useful because an HETP requires a separate“experiment” in order to determine

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MOMENTS HAVE A CHROMATOGRAPHIC

INTERPRETATION

𝜇𝜇0 =𝐿𝐿𝑢𝑢

𝜇𝜇1 =𝐿𝐿𝑢𝑢 +

2𝐷𝐷𝐿𝐿𝑢𝑢2 1 + 𝜀𝜀𝐾𝐾𝑐𝑐 1 + 𝐾𝐾𝑛𝑛

𝜇𝜇2

=2𝐷𝐷𝐿𝐿𝐿𝐿𝑢𝑢3 +

8𝐷𝐷𝐿𝐿2

𝑢𝑢4 1 + 𝜀𝜀𝐾𝐾𝑐𝑐 1 + 𝐾𝐾𝑛𝑛 2

+𝐿𝐿𝑢𝑢 +

2𝐷𝐷𝐿𝐿𝑢𝑢2 𝜀𝜀𝐾𝐾𝑐𝑐

𝑅𝑅2

𝐷𝐷𝑟𝑟1 + 𝐾𝐾𝑛𝑛 2

𝑣𝑣 𝑣𝑣 + 2 +𝜀𝜀 1 + 𝐾𝐾𝑛𝑛 2

𝐻𝐻𝑐𝑐+𝐾𝐾𝑛𝑛𝐻𝐻𝑛𝑛


Molecule specific parameters


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KEY MOMENT EQUATIONS

𝐻𝐻 =𝐿𝐿𝜎𝜎2

𝜇𝜇1𝜇𝜇0

2

𝜎𝜎2 =𝜇𝜇2𝜇𝜇0

−𝜇𝜇1𝜇𝜇0

2

These equations allow us to transform our moment calculations into something that resembles the plate

height that we’re used to


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HOW DO YOU USE MOMENTS?


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dc/dV = 0

dc/dV = 0

In principal, these regions should have moments = 0


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MOMENTS IN PRACTICE

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Volume (mLs)

Baseline Noise

Noise = 0.2

Noise filterAverage +3*noise

This is called a changeband filter14 of 25

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DATA FILTER

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Start calculation here


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WHAT DOES THE CURVE LOOK LIKE?

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Smooth enough to differentiate?We use additional smoothingFive timepoint moving average


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WHAT DO THE RESULTS LOOK LIKE?

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y = 0.0017x + 0.0547R² = 0.309

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H (c

m)

Flowrate (mL/min)

Plate Height vs. Flowrate

The dependence between plate height and flowrate is as expected, but scatter is a little high


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CAN WE CONTROL CHROMATOGRAPHY?

• We know we can use flowrate to affect resolution• We can also use temperature• “” “” “ buffer composition

• We can use transition analysis to understand if resolutionis changing over time

• Our goal is to design methods for the calculation thatallow feedback control• This means measuring noise in real time• Initiating the transition calculation• Referring to a van Deemter plot to determine a new flow rate

set point• Tuning the deadband and proportional response to provide

smooth control


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APPLICATION TO SMB


Seidel-Morgenstern et al., Chem Eng Technol, 31(6) 2008

Zones have boundaries at transitions, theoretically, and there are transitions to measure during each cycle in each zone. Adjusting flow rate to maintain resolution has implications for the operation of the rest of the bed.


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CONCLUSION

• Transition analysis uses moments to monitor the performance of a column in real time.

• Multiple transitions in a chromatography cycle can be monitored, but each is a unique transition with different kinetics

• We have developed some techniques for data filtering and smoothing that increase the repeatability of the method

• Using transition analysis is our best bet for control of chromatography and SMB as columns age or deteriorate


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