# FICK’S LAWS CLASS NOTES FOR MECHANICAL ENGINEERING

**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.m^{2})

#### D_{AB} is the molecular diffusivity (Diffusion coefficient) of the molecule A in B

( m^{2}/s) Or it is the mass transfer coefficient at micro level

#### C_{A} is the concentration of A (k mol/m^{3})

#### 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}_{Z1}dZ = — 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 ∂^{2}C_{x}/∂x^{2}) 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 ∂^{2}C_{x}/∂x^{2}

#### (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 ∂^{2}C_{x}/∂x^{2}) 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∂^{2}C/∂^{2}x

#### 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 ∂^{2}C_{x}/∂x^{2}

#### 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 ∂^{2}C_{x}/∂x^{2}) provided diffusion coefficient ‘ D ’ is not a function of x in x direction.

#### In 3D case, equation becomes

#### dC_{x}/dt = D (∂^{2}C_{x}/∂x^{2} + ∂^{2}C_{x}/∂y^{2 }+ ∂^{2}C_{x}/∂z^{2})

#### 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∂^{2}C_{x}/∂x^{2} 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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