## What is mass transfer?

Mass transfer is the transport of a substance (mass) in a fluid (liquid or gas) but it may also be described in solid-phase materials. You may have come across this term while studying the types of systems in thermodynamics. Let us recall it. A system is a region or area of focus. There are three types of systems:

### Open system

Both energy and mass transfer take place in an open system.

Example: Piston cylinder arrangement with valves.

### Closed system

Energy transfer takes place but the mass transfer does not take place in a closed system.

Example: Piston cylinder arrangement without any valves.

### Isolated system

Neither energy nor mass is transferred in isolated system.

Example: A thermos flask for some given amount of time.

The mass transfer may take place in a single phase or over phase boundaries in multiphase systems. Mass transfer occurs when there is a gradient in the concentration of the species. The basic mechanism is the same no matter whether it is a solid, liquid, or gas.

## What are the types of mass transfer?

Depending on the nature, the boundary conditions, and the forces responsible for mass transfer, there are four types of mass transfer, as mentioned below:

### Diffusion in a quiescent medium

The phenomenon of mass transfer that causes the distribution of a chemical species to become more uniform in space as time passes, is called diffusion.

### How diffusion takes place?

The driving force for diffusion is the thermal motion of molecules. Molecules at temperatures above absolute zero are always moving and possess kinetic energy. And when these molecules are in motion, they collide with each other frequently, due to which the direction of the motion of these molecules becomes randomized. When molecules are moving but also constantly changing direction, diffusion occurs because of statistics of this movement. The diffusion process can be explained with the help of Fick's law of diffusion.

## Fick’s first law of diffusion

This law states that the molar flux due to diffusion is proportional to the concentration gradient. It can be written as:

$J=-D\frac{d\phi}{dx}$

where, $J$ is diffusion flux, $D$ is diffusivity, $\phi $ is concentration, and $x$ is position.

## Fick’s second law of diffusion

This law states that the rate of change of concentration of the solution at a point in space is proportional to the second derivative of concentration with space. It can be written as

$\frac{\partial \phi}{\partial t}=D\frac{{\partial}^{2}\phi}{\partial {x}^{2}}$

## Types of diffusion

**Thermal diffusion**

This diffusion is due to a temperature gradient. It is usually negligible unless the temperature gradient is very large.

**Forced diffusion**

This diffusion is due to an external force field acting on a molecule. Forced diffusion occurs when an electrical field is imposed on an electrolyte (for example, in charging an automobile battery)

**Pressure diffusion**

This diffusion is due to a pressure gradient. It is usually negligible unless the pressure gradient is very large.

**Knudsen diffusion**

This diffusion phenomena occurs in porous solids.

## Mass transfer in laminar and turbulent flow

The movement of molecules across the streamlines of fluid in laminar flow occurs by molecular diffusion. When a fluid flows over a surface, like a plate or a cylinder, the surface retards the adjacent fluid region, thus forming a boundary layer. This laminar sublayer, in which mass transfer occurs by molecular diffusion only, offers the main resistance to mass transfer. If flow throughout the fluid is laminar, the equation for molecular diffusion may be used to evaluate the mass transferred across the boundary layer.

Consider a gas flowing over a surface, such that equimolecular counter diffusion of components A and B occurs, where B is towards the surface and A is away from the surface. The variation in partial pressure of A with distance from the surface is shown in the figure.

## The mass exchange between phases

The process in which the mass is also transported from one phase to another is called interphase mass transfer. Many physical situations are there in our environment where two phases are in contact, and these phases are separated by an interface. Like single-phase transport, the concentration gradient of the transporting species influences the overall rate of mass transport. More accurately, transport between two phases can only occur when the phases are not in equilibrium. When a multiphase system is at equilibrium, no mass transfer will occur. Thus the equilibrium of the transporting species at the interface is of principal concern.

## Some important terms used above

### What is Reynold's Number?

It is a dimensionless number that is equal to the ratio of the inertia force and the viscous force of the flowing fluid;

$R=\frac{\rho vd}{\mu}$

where;

$\rho $ is density,

$v$ is velocity,

$d$ is the characteristic dimension, and

$\mu $ is dynamic viscosity.

### What is Nusselt Number?

It represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer. The larger the Nusselt number, the more effective the convection. A Nusselt number for a fluid layer represents heat transfer by pure conduction.

Nusselt number $\left(\mathrm{Nu}\right)=\frac{\mathrm{h\delta}}{k}\left(\delta =\mathrm{characteristic}\mathrm{length}\right)$

where,

${q}_{\mathrm{cond}}$ is conductive heat transfer,

${q}_{\mathrm{conv}}$is convective heat transfer,

$k$ is the conductive heat transfer coefficient, and

$h$ is the convective heat transfer coefficient.

### What is Schmidt Number?

It is the ratio of the molecular diffusivity of momentum to the molecular diffusivity of mass.

${S}_{c}=\frac{V}{Dm}$

where,

$\nu $ is the molecular diffusivity of momentum,

$D$ is the molecular diffusivity of mass.

### What is Prandtl Number?

It is the ratio of molecular diffusivity of momentum to the molecular diffusivity of heat.

${P}_{r}=\frac{\nu}{\alpha}$

where,

$\alpha $ is the thermal diffusivity,

$\nu $ is the momentum diffusivity.

## Context and Applications

This topic is significant in the professional exams for both graduate and postgraduate courses, especially for

- Bachelor of Technology in Mechanical Engineering
- Master of Technology in Heat Transfer and Energy Studies.

## Common Mistakes

Students often get confused about the hydrodynamic boundary layer and the thermal boundary layer. The hydrodynamic boundary layer is the layer defined by the instantaneous velocity of fluid particles in a fluid flow whereas, the thermal boundary layer is the layer defined as the temperature profile of the moving fluids at different stages of a fluid flow.

## Related Concepts

• Heat Transfer through radiation

• Thermal boundary conditions

• Boussinesq's equation

• Heat generation

• Dimensional and non-dimensional analysis

## Practice Problems

**Q1**. Diffusion is a process of

(a) rarefaction of particles.

(b) accumulation of particles on a solid surface.

(c) movement of particles from higher concentration to lower concentration.

(d) movement of particles through a semipermeable membrane.**Correct Option: (c)**

**Explanation: **In the diffusion process, there is no possibility of rarefaction of particles or collection of particles. The particles travel from higher concentration to lower concentration in the diffusion process.

**Q2**. The diffusivity of a constituent A in solution B has the units.

(a) kmol/(m²-s)

(b) m³/s

(c) m²/s

(d) m/s**Correct Option: (c)**

**Explanation:** Diffusivity is represented as the term which estimates the capability of an element to convey an alteration in temperature. In the SI system, diffusivity is measured in the units of meter square per second.

**Q3**. At steady-state, the partial pressure distribution of an ideal gas diffusing through a stagnant ideal gas B follows

(a) parabolic law

(b) exponential law

(c) linear law

(d) hyperbolic law.**Correct Option:** **(b)**

**Explanation: **The partial pressure distribution of any diffusing ideal gas A from a stagnant ideal gas B will obey the exponential law.

**Q4**. For a flow of gas through a capillary according to Poiseuille’s law, the permeability

(a) is proportional to the gas viscosity

(b) varies inversely as the gas viscosity

(c) varies as the square root of gas viscosity

(d) varies as the square of the gas diffusivity**Correct Option: (b)**

**Explanation: **According to Poiseuille’s law, the permeability of any certain gas relies on its viscosity. This means that permeability rises with a reduction in viscosity, that is, permeability is inversely related to viscosity.

**Q5**. The diffusivity has the same dimensions as

(a) density

(b) kinematic viscosity

(c) concentration

(d) absolute viscosity**Correct Option:** **(b)**

**Explanation:** In the standard international system, the diffusivity is measured in the units of meter square per second, which is equivalent to the SI unit of kinematic viscosity.

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