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

species.

Single-phase system

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.

Multi-phase 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 = — D_AB dC_A/dZ

J_AZ is the molar flux of component A in the z direction (( kmol of A) / s.m²)

D_AB is the molecular diffusivity (Diffusion coefficient) of the molecule A in B
( m²/s) Or it is the mass transfer coefficient at micro level

C_A 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  ∫^Z2_Z1dZ = — D_AB ∫_c1^c2 dC_A

Molecular diffusion from C_A1 To C_A2 and from Z1 to Z2
J*_AZ (z2—Z1)= D_AB(C_A1 — C_A2)
J*_AZ = D_AB (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 (dC_x/dt ) is equal to the diffusivity times the rate of the change of the concentration gradient (D ∂²C_x/∂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

dC_x/dt = D ∂²C_x/∂x²

(This equation tells that the rate of change of concentration difference (dC_x/dt ) is equal to the diffusivity times the rate of the change of the concentration gradient (D ∂²C_x/∂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 ) = [ (J_x )—J _(x+dx)]dt A /Adx

But J_(x+dx) = J_(x) + (dJ/dx)dx

Substituting

dC(x,t) /dt = –dJ/dx

Writing in partial differential form

∂C(x,t) /∂t = –∂J/∂x

From first law

J = –D dC_x/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 dC_x/dx)

Assuming D is not a function of ‘ x ‘,Fick’s law In one dimensional diffusion becomes

dC_x/dt = D ∂²C_x/∂x²

This equation tells that the rate of change of concentration difference (dC_x/dt ) is equal to the diffusivity times the rate of the change of the concentration gradient (D ∂²C_x/∂x²) provided diffusion coefficient  ‘ D ’ is not a function of x in x direction.

In 3D case, equation becomes

dC_x/dt = D (∂²C_x/∂x² + ∂²C_x/∂y² + ∂²C_x/∂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 dC_x/dt = D∂²C_x/∂x² is of complex nature and can be solved only by Gauss error method i.e. using tabular data and chart only.

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