When the width of channel is varied:Q¼ΔP
8
kXN
i¼ 1Li
WiHiWiHiis the area perpendicular to flow, Liis the length in the direction of flow, and
ΔP is the pressure difference across the length of the channel.
Using parallels between fluidic and electrical resistance, i.e. liquid flow Qi is an
equivalent for current,ΔPi pressure drop along the channel is equivalent to voltage
drop, andμLi/(kWiHi) is an equivalent to fluidic resistance of each individual
channels, one can estimate total flux. If several fluidic elements are connected in
series or in parallel, the total flux through this network will follow analogy of
Ohm’s law, i.e. sum of individual fluxes when connected in series, and reciprocals
when connected in parallel.
3.2 Spreading of Wax and Width of Patterned Channel
Spreading of the molten wax and width of final channels can be predicted using the
Washburn equation:
L¼ffiffiffiffiffiffiffi
γDt
4 ηsWhere L—distance covered by the wax front,η—viscosity (function of time and
temperature to which device was exposed during bake),γ—effective surface
tension, D—average pore diameter, t—time. The same equation can be used to
predict transport of the fluid front in the channel.
The final inner width of the channel formed by wax can be defined as
WC¼WP2L, where
WC—inner width of the hydrophobic channel, WP—inner width of the printed
channel, L—the additional distance that the wax spreads perpendicular to the length
of the channel. L is a function of time, heat and the structural properties of paper.
3.3 Transport Time
Transport time through a multi-segment geometry can be calculated using the
modified Darcy’s law equation:
t¼VReq
ΔP170 E. Vereshchagina