ABSTRACT
By dividing Equation 13.1 by Avogadro’s number, NA, the equivalent expression in terms of the molar flux and the concentration (mol/m3) is obtained:
J J N N
v Cv A A
d d= ′ = =η . (13.2)
Thus, not surprisingly, in addition to a flux due to diffusion, there is also one due to flow because of the applied force, and these fluxes just add to get the total molar flux:
J D dC dx
Cvd= − + . (13.3)
13.3.1 ABSOLUTE MOBILITY Consider a particle (solid, liquid, or gas) settling (or rising) in a liquid under the influence of gravity as shown in Figure 13.2. The particle* will reach a constant drift or terminal velocity, vd, in the (positive or negative) vertical direction when the forces acting on it are balanced. These forces are gravity, Fg, the buoyancy force, Fb, due to the weight of the fluid displaced, and a drag force caused by the viscous forces between the moving particle and the fluid, Fd. As a result, the equation of motion of this particle, F = ma (Newton’s first law, one of the few things that one needs to remember in this world), becomes
ma F F F Fg b d= = + + (13.4)
where the resulting acceleration will be in the –z direction if z is the vertical direction and the gravitational force exceeds the drag and buoyancy force. Viscous drag forces occur in many different areas of physics and engineering and they typically are assumed to be proportional to the velocity of the moving particle: for example, Fd = –γ v. For a particle moving in a fluid in which there is simple laminar flow around the particle, this is indeed the case, and is a good starting point in developing a model for diffusion in liquids. Consider the general result in Equation 13.4 with all the forces
that might be acting on this particle lumped into a single force, F, with the exception of the viscous force. The result is the following simple differential equation for the velocity of the particle as a function of time:
ma m dv dt
F v= = − γ . (13.5)
This too is a very general equation in that F can be any type of net applied force: mechanical, electrical, chemical, and so on. For now, it will be left as a general force. Rearranging Equation 13.5 gives
dv F v m
dt −
= γ
which is easily integrated by substituting u F v= − γ so that dv du− ( )1 γ , resulting in
− −( ) = +1 1 γ
γln F v m
t A
and rearranging
ln F v m
t A−( ) = − + ′γ γ
leading to
F v A m t
− = ′′ −
γ γ
e
where A A A, ,′ ′′ are all integration constants. Solving for the velocity with v = 0 at t = 0, then
v F m t
= −
γ
1 e (13.6)
Equation 13.6 is plotted in Figure 13.3 and at t = ∞, dv/dt = 0 and v = vd = F/γ.* In any event, the drift velocity can be more conveniently written as
v F BFd = =γ (13.7)
where B is called the absolute mobility with units of m/s N and B =1 γ.
