2. Physical origins
Fig6.1 Mass transfer by diffusion
3. Fick’s Law
Mass transfer by molecular diffusion is analogous to heat transfer by conduction or momentum transfer in laminar flow. Mass transfer by molecular diffusion may occur in a stagnant fluid or in a fluid in laminar flow. Like the Fourier’s equation of heat conduction, the concentration field of the diffusing species A is given by
where CA is the concentration of component A in a mixture of A and B in kgmol/m3, t is the time in seconds and D is the mass diffusivity in m2/s.
For one-dimensional mass diffusion,
wherey is the distance in the direction of diffusion.
Like the conduction equation
The mass transfer equation is
For one dimensional mass transfer
where NA/A is the mass flux in kgmol/m2s. The negative sign appears because the concentration gradient is negative in the direction of mass transfer.
Equation (6.1) is called Fick’s law of diffusion, which states that the mass flux of a constituent per unit area is proportional to the concentration gradient.
In Fig. 6.2 a thin partition separates the two gases A and B. When the partition is removed, the two gases diffuse through each other until equilibrium is established and the concentration of the gases is uniform throughout the box. The diffusion rate is given by Eq. (6.1). It can also be expressed in terms of mass flow:
Fig6.2 Diffusion of component A into component B and vice versa
whereis the mass flux per unit time, kg/s; CA is the mass concentration of component A per unit volume, kg/m3; and D is the proportionality constant called diffusion coefficient, m2/s.
4. Equimolar diffusion
In the gas phase, the concentrations are usually expressed in terms of partial pressures. If ideal gas law is assumed to hold good.
where is the universal gas constant.
Fick’s law could then be written as
wherepA1 is the partial pressure of A at y1 and pA2 is the partial pressure of A at y2.
Equation (6.3) is valid for equimolar counter diffusion in which gases A and B diffuse simultaneously in opposite directions through each other. The rates of diffusion are equal but in opposite direction i.e., NA = – NB.
5. Molecular diffusion through stationary gas
Let us consider a gas Adiffusing through a stationary gas B into a liquid–vapour interface where the gas A is absorbed. Since the gas A is diffusing towards the interface, there must be a partial pressure gradient for A in the direction of diffusion.
Where and which is
6. Diffusivity For Gases And Vapours
Gilliland proposed a semi-empirical equation for the diffusion coefficient in gases.
whereD is in cm2/s, T is in K, p is the total system pressure in pascals and VAand VBare the molecular volumes of constituents A and B as calculated from the atomic volumes; MAand MBare the molecular weights of constituents A and B.
7. Concentration boundary layer and mass transfer coefficient
Just as the calculation of momentum and heat transfer requires the knowledge of velocity and temperature profiles within the boundary layer, the calculation of mass transfer requires that concentration profile within the boundary layer should be known.
Fig6.3 Concentration boundary layer
Let us consider the flow of a fluid mixture on a surface as shown in figure. Let the free stream velocity and concentration be and . Let the plate surface be maintained at a concentration >. Then, species A diffuses from the surface into the fluid. A concentration boundary layer develops , thickness of which can be defined in the same way as that of the hydrodynamic or thermal boundary layer. The distance to which the boundary layer extends may be defined as the thickness at which the concentration is equal to 99% of the free-stream concentration
By analogy of heat transfer, mass transfer coefficient hm can be defined as
Unit of hm is same as velocity i.e m/s.
The molecular diffusion equation is
8. Analogy Between Momentum, Heat And Mass Transfer
The momentum and energy equations of a laminar boundary layer are
The concentration boundary layer can similarly be simplified to
Momentum transfer in laminar flow (Newton’s law) is
We know that the velocity and temperature profiles have the same shape if
Similarly, the momentum and concentration profiles will have the same shape if
Thus, the Schmidt number plays the same role in mass transfer as does the Prandtl number in heat transfer.
It is also seen that the temperature and concentration profiles will be similar if
Thus, Lewis number is important in the solution of simultaneous heat and mass transfer problems. When,
Sc = Pr = Le = 1, all the three boundary layers coincide.
The dimensionless mass transfer number corresponding to the Nusselt number is the Sherwood number defined as
wherex is the characteristic length.
Similarly, corresponding to Stanton number, which is expressed as
Corresponding to it, we have
whereu is any characteristic velocity of the system.
We know that forced convection heat transfer correlations are of the form
Likewise, in forced convection mass transfer problems we write the functional relation
Heat transfer and Mass transfer are similar very similar to one another. Similarities between them can be explained by following key points.
Where, qkis the rate of heat flux (a vector) in W/m2, dT/dx is the temperature gradient in the direction of heat flow x and k is the constant of proportionality
Mass transfer is explained with help of Fick’s law which is given as
where NA/A is the mass flux, is concentration gradient and D is constant of proportionality.
Where, Q is rate of heat transfer (W), A is area in m2, h is constant known as coefficient of convective heat transfer or film coefficient with unit W/m2K.
Mass convection equation is given as
Where is coefficient of mass convection, represent concentration difference.
Likewise, Sherwood number, Reynold’s number and Schmidt number are governing constants in convection mass transfer which can be correlated as
Diffusivity of gas and vapour
Mass flux by convection