Roughness features on the walls of a channel wall affect the pressure drop of a fluid flowing through that channel. This roughness effect can be described by (i) flow area constriction and (ii) increase in the wall shear stress. Replotting the Moody’s friction factor chart with the constricted flow diameter results in a simplified plot and yields a single asymptotic value of friction factor for relative roughness values of $\epsilon ∕D>0.03$ in the fully developed turbulent region. After reviewing the literature, three new roughness parameters are proposed (maximum profile peak height $R_p$ , mean spacing of profile irregularities $R_{\mathrm{Sm}}$ , and floor distance to mean line $F_p$ ). Three additional parameters are presented to consider the localized hydraulic diameter variation (maximum, minimum, and average) in future work. The roughness $\epsilon$ is then defined as $R_p+F_p$ . This definition yields the same value of roughness as obtained from the sand-grain roughness [H. Darcy, Recherches Experimentales Relatives au Mouvement de L’Eau dans les Tuyaux (Mallet-Bachelier, Paris, France, 1857); J. T. Fanning, A Practical Treatise on Hydraulic and Water Supply Engineering (Van Nostrand, New York, 1877, revised ed. 1886); J. Nikuradse, “Laws of flow in rough pipes” [“Stromungsgesetze in Rauen Rohren,” VDI-Forschungsheft 361 (1933)]; Beilage zu “Forschung auf dem Gebiete des Ingenieurwesens,” Ausgabe B Band 4, English translation NACA Tech. Mem. 1292 (1937)]. Specific experiments are conducted using parallel sawtooth ridge elements, placed normal to the flow direction, in aligned and offset configurations in a $10.03\mathrm{mm}$ wide rectangular channel with variable gap (resulting hydraulic diameters of $325\mu \mathrm{m}–1819\mu \mathrm{m}$ with Reynolds numbers ranging from 200 to 7200 for air and 200 to 5700 for water). The use of constricted flow diameter extends the applicability of the laminar friction factor equations to relative roughness values (sawtooth height) up to 14%. In the turbulent region, the aligned and offset roughness arrangements yield different results indicating a need to further characterize the surface features. The laminar to turbulent transition is also seen to occur at lower Reynolds numbers with an increase in the relative roughness.

Moving Bed Biofilm Reactors (MBBRs) can efficiently treat wastewater by incorporating suspended biocarriers that provide attachment surfaces for active microorganisms. The performance of MBBRs for wastewater treatment is, among other factors, contingent upon the characteristics of the surface area of the biocarriers. Thus, novel biocarrier topology designs can potentially increase MBBR performance in a significant manner. The goal of this work is to assess the performance of 3-D-printed biofilter media biocarriers with varying surface area designs for use in nitrifying MBBRs for wastewater treatment. Mathematical models, rendering, and 3D printing were used to design and fabricate gyroid-shaped biocarriers with a high degree of complexity at three different levels of specific surface area (SSA), generally providing greater specific surface areas than currently available commercial designs. The biocarriers were inoculated with a nitrifying bacteria community, and tested in a series of batch reactors for ammonia conversion to nitrate, in three different experimental configurations: constant fill ratio, constant total surface area, and constant biocarrier media count. Results showed that large and medium SSA gyroid biocarriers delivered the best ammonia conversion performance of all designs, and significantly better than that of a standard commercial design. The percentage of ammonia nitrogen conversion at 8 hours for the best performing biocarrier design was: 99.33% (large SSA gyroid, constant fill ratio), 94.74% (medium SSA gyroid, constant total surface area), and 92.73% (large SSA gyroid, constant biocarrier media count). Additionally, it is shown that the ammonia conversion performance was correlated to the specific surface area of the biocarrier, with the greatest rates of ammonia conversion (99.33%) and nitrate production (2.7 mg/L) for manufactured gyroid biocarriers with a specific surface area greater than 1980.5 m²/m³. The results suggest that the performance of commercial MBBRs for wastewater treatment can be greatly improved by manipulation of media design through topology optimization.