# Attachment rate calculation

Calculating attachment rates relies on the convection-despersion equation used in NanoFASE transport modelling. The attachment rate is calculated for the case of favorable attachment, i.e. attachment only by diffusion or advection of the ENMs, using the single-collector contact efficiency ($$\small \eta_{0}$$). The attachment efficiency is calculated from the attachment rate constant or vice versa. However, the attachment efficiency also takes into account the non-favorable case, where coatings or electrostatic repulsion lower the rates relative to the favorable case.

 1. $$\eta_{0}$$ 1. Calculation of single-collector contact efficiency according to Tufenkji and Elimelech (2004). 2. $$\alpha$$ 2. Determine the attachment efficiency using a batch test or column test. 3.  $$k_{att}=\alpha \frac{3(1-\Theta )v}{2d_{c}}\eta_{0}$$ 3. Calculation of the first-order attachment rate constant. 4. $$\Theta \frac{dC}{dt}=-v\Theta \frac{dC}{dz}+D\Theta \frac{d^{2}C}{dz^2}-k_{att}C$$ 4. Calculation of transport using the convection-dispersion equation.

## Used in

 $$\small \eta_{0}$$ calculation requires knowledge of the effective porosity ($$\Theta$$) and average collector size ($$d_{c}$$). Both can be calculated using pedotransfer functions. The pore velocity is calculated from the rainfall intensity divided by the porosity. The convection-dispersion equation also requires the dispersivity. Visit the NanoFASE Library: Tufenkji N, et al. (2004) Transport of Cryptosporidium oocysts in porous media: Role of straining and physicochemical filtration. Environmental Science & Technology. 20:10818-10828. http://doi.org/10.1021/es049789u Elimelech M, et al. (1995) Particle Deposition and Aggregation: Measurement, Modeling and Simulation. Oxford: Butterworth-Heinemann.

## Contact Karin Norrfors Geert Cornelis

Email: geert.cornelis@slu.se