Biomedical Engineering Reference
In-Depth Information
where N is the total number of molecules in the modeled system. The basic steps of an MD-
simulation are:
Determination of initial conditions and geometry parameters,
Determination of interaction force (2.2) , and
Integration of equation of motion (2.10) .
Because of its deterministic nature, MD is extremely expensive in terms of computational
resources. Less resources would be needed if the system is modeled with a statistic method.
Direct simulation Monte Carlo is a statistic method for modeling at the molecular level. In DSMC,
many molecules are modeled as a single particle. The interactions between the molecules of each
particle are determined statistically, while the motion of the particle is modeled deterministically. The
basic steps of DSMC are [3] :
Determination of particle motion,
Indication and cross-referencing of particles,
Simulation of particle collision, and
Sampling of macroscopic properties.
2.1.2 Continuum level
At continuum level, transport phenomena are described with a set of conservation equation. Because
flow in micromixers is laminar, we do not need to deal with turbulent flow, which is impossible to solve
analytically. The three basic conservation equations are:
Conservation of mass: continuity equation,
Conservation of momentum: Newton's second law or Navier-Stokes equation, and
Conservation of energy: first law of thermodynamics or energy equation.
Solving these three equations will result in three basic variables: the velocity field v , the pressure
field p , and the temperature field T . Fluid properties, depending on the thermodynamic state (pressure p
and temperature T ), such as density, viscosity, thermal conductivity, and enthalpy, can be derived from
these variables and fed back into the conservation equations. The above three equations are formulated
for a single phase of homogenous composition. In micromixers, most fluids carry one or more species
other than the carrying fluids. If the mixers are used as microreactors, chemical reactions also need to
be considered. Thus, in addition to the above three conservation equations, two further equations are
needed to describe the transport of species in micromixers:
Conservation of species: convective/diffusive equation and
Laws of chemical reactions.
2.1.2.1 Conservation of mass
The continuity equation has the general form:
D r
D t þ V$
v
¼
0
(2.11)
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