FICK’S LAWS CLASS NOTES
FOR MECHANICAL ENGINEERING
Maxwell-Stefan equations are used for
the mass transfer in multi-component
mixtures. Fick’s laws equations are
limiting equations of Maxwell-Stefan
equations. In these, it has been
assumed that the mixture is extremely
dilute. Further assumption is that every
chemical species is interacting only with
the bulk mixture and not with the other
While dealing with the multiple species non-dilute mixture, several modified Maxwell-Stefan equations are available. A single phase system containing two or more species with different concentrations will result in mass transfer. Further, it minimizes the concentration differences within the system.
Whereas in a multi-phase system mass transfer is due to chemical potential differences between the species. Temperature and pressure are uniform in a single phase system. The difference in chemical potential is due to the variation in concentration of each species. Mass transfer is the basis for many chemical and biological processes. For example, the removal of water by the air conditioner in a rainy season is a chemical process. Further, pushing out unwanted material from the human body is a biological process.
Fick’s First Law of molar diffusion
In a solution having two molecular species A and B, Fick’s first law gives
J*AZ = — DAB dCA/dZ
JAZ is the molar flux of component A in the z direction (( kmol of A) / s.m2)
DAB is the molecular diffusivity (Diffusion coefficient) of the molecule A in B
( m2/s) Or it is the mass transfer coefficient at micro level
CA is the concentration of A (k mol/m3)
z is the distance of diffusion in z direction (m)
Application of Fick’s First Law to a Steady State Diffusion
Steady state condition means no change of concentration with respect to time.
Rearranging and integrating the Fick’s equation, we get
J*AZ ∫Z2Z1dZ = — DAB ∫c1c2 dCA
Molecular diffusion from CA1 To CA2 and from Z1 to Z2
J*AZ (z2—Z1)= DAB(CA1 — CA2)
J*AZ = DAB (CA1 — CA2) / (z2—Z1)
FICK’S SECOND LAW OF DIFFUSION
Applies to non- steady state diffusion. It states that the rate of change of concentration difference (dCx/dt ) is equal to the diffusivity times the rate of the change of the concentration gradient (D ∂2Cx/∂x2) provided diffusion coefficient ‘ D ’ is not a function of x in x direction.)
Non-Steady-State Diffusion is a realty. In this, the concentration at any point changes with time.
Mathematical form of Second law
dCx/dt = D ∂2Cx/∂x2
(This equation tells that the rate of change of concentration difference (dCx/dt ) is equal to the diffusivity times the rate of the change of the concentration gradient (D ∂2Cx/∂x2) provided diffusion coefficient ‘ D ’ is not a function of x in x direction.)
Let the local concentration flux at position “x” be c(x,t) and and diffusion flux (per unit area) as J(x) respectively.
Change in concentration over distance dx and time interval ‘dt’
dc(x,t ) = [ (Jx )—J (x+dx)]dt A /Adx
But J(x+dx) = J(x) + (dJ/dx)dx
dC(x,t) /dt = –dJ/dx
Writing in partial differential form
∂C(x,t) /∂t = –∂J/∂x
From first law
J = –D dCx/dx
∂C (x,t)/∂t = —∂J/∂x = D∂2C/∂2x
This is Fick’s Second Law
Fick’s 2nd law of diffusion is for finding the rate of accumulation (or depletion) of concentration within the volume.
Mathematical form of Fick’s Second Law is
dCx/dt = (D dCx/dx)
Assuming D is not a function of ‘ x ‘,Fick’s law In one dimensional diffusion becomes
dCx/dt = D ∂2Cx/∂x2
This equation tells that the rate of change of concentration difference (dCx/dt ) is equal to the diffusivity times the rate of the change of the concentration gradient (D ∂2Cx/∂x2) provided diffusion coefficient ‘ D ’ is not a function of x in x direction.
In 3D case, equation becomes
dCx/dt = D (∂2Cx/∂x2 + ∂2Cx/∂y2 + ∂2Cx/∂z2)
With specific initial and boundary conditions, the partial differential equation can be solved to give the concentration as function of spatial position and time
Equation dCx/dt = D∂2Cx/∂x2 is of complex nature and can be solved only by Gauss error method i.e. using tabular data and chart only.