**Significance Statement**

Developing a new filtration process to production level in the therapeutic industry is an intricate undertaking with strong consequences on the success of the corresponding drug-development project. The pressure distribution within filters in parallel systems is not homogeneous, and so dimensional analysis alone is unable to provide a full prediction of the capacity of the whole system. The added effect that filters clog at different rates due to the inhomogeneous pressure distribution complicates the whole upscaling process even further. As a result, determining the appropriate filtration capacity for production-level consignment sizes based only on laboratory measurements is fundamentally challenging.

Armin Krupp, Professor Colin Please and Dr Ian Griffiths at the University of Oxford worked with Dr Anil Kumar from Pall Corporation to propose a method for the scaling up of multi-capsule depth filtration systems using only limited small-scale laboratory measurements. This was achieved by deriving a mathematical model for the set-up, and employing well-established constitutive laws to predict the flow and pressure distribution during an entire run. The result significantly reduces both the time and cost normally associated with the scaling-up process. Their work is now published in * Separation and Purification Technology*.

Two full depth-filter devices were built, comprising either five or ten disc-shaped capsules connected in parallel and stacked on top of each other. Inlets and outlets of the capsules were connected in series, and the pressures were then measured at the inlet and outlet. The research team then recorded the total flux through the systems and the height difference between capsules relative to the inlet. A mathematical theory was developed in which the hydrostatic component of the pressure was eliminated from the equations so that only the hydrodynamic component needed to be calculated. The flow was assumed to be laminar and the change in pressure across the flux discharge through a predetermined filter was predicted using Darcy’s law. The mathematical model was then calibrated using the readings obtained from one empirical set-up and validated using the second experiment.

The resulting theory can be used to predict the required number of depth-filter capsules and the appropriate operating conditions to achieve a given processing rate for any given filtration challenge. The model requires the input of only a small amount of data that characterizes the blocking behaviour for that contaminant, which is gained from conducting a simple experiment using only a small amount of filter media. This bypasses the need to conduct more complex large-scale experiments as was traditionally the case.

The method outlined in this paper can be utilized in diverse scaling-up problems in depth-filter devices. The idea of combining a small set of empirical measurements with a mathematical model can also be adapted to any filtration configuration to define its performance during a complete filtration run. The method has the advantage of delivering highly accurate results while freeing the experimentalist of having to estimate the observed clogging characteristic using models.

Corresponding author Ian Griffiths said “*In all of the work we do we place an emphasis on ensuring that the results have an impact in shaping a final product. It is always rewarding to see our mathematical models go from conception to final-stage implementation. Working with industries such as Pall Corporation provides an ideal route to facilitate this and we are excited to continue our tradition of partnership with industries in Oxford Mathematics.*”

**Reference**

A.U. Krupp^{1}, C.P. Please^{1}, A. Kumar^{2}, I.M. Griffiths^{1}. **Scaling-up of multi-capsule depth filtration systems by modeling flow and pressure distribution**. Separation and Purification Technology volume 172 (2017) pages 350–356.

- Mathematical Institute, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK
- Pall Life Sciences, 20 Walkup Drive, Westborough, MA 01581, USA

Go To Separation and Purification Technology