 Material balance for distillation column
 Overhead section of distillation column
Material balance across overhead section of a distillation column
 Feed, top product and bottom product
Material balance between feed, distillate and bottom product in distillation column
 Solute balance
Solute balance in mass transfer operation
 Mass transfer coefficient
 Mole fraction and pressure based mass transfer coefficients
Relation between mole fraction based mass transfer coefficient and pressure based mass transfer coefficient
 Individual and overall mass transfer coefficients
Relation between individual and overall mass transfer coefficients


 To be updated.

 Drying
 Rate of drying (In relation to mass transfer coefficient)
Drying rate in constant drying rate period (when mass transfer coefficient is given)
 Total time of drying
Total time of drying is given by
 Total time of drying in falling rate region
Total time of drying in falling rate region is given by
 Rate of drying (In relation to partial pressures)
Drying rate in constant rate period when partial pressures are given
 Rate of drying (In relation to heat of vaporization)
 Humidity
 Percentage humidity
 Dew point humidity
At dew point humidity is 100%
 Humidity
 Molar saturation humidity
 Calculating wet bulb temperature
Finding wet bulb temperature: Heat flux = Mass flux * latent heat (λ)
 Relative saturation
Ratio of partial pressure to vapour pressure
 ChiltonColburn analogy
Chiltoncoulburn analogy between heat, mass and momentum transfer. The analogy is valid for fully developed turbulent flow in conduits with Re > 10000, 0.7 < Pr < 160, and tubes where L/d > 60 (the same constraints as the Sieder–Tate correlation)
 Mass transfer flux
 Total molar flux
Total molar flux of component A
 Mass transfer flux and humidity
Relationship between mass transfer flux and relative humidity
 Fick’s law
 Fick’s first law
First Law: Steady state equation
 Fick’s second law
Second Law: Unsteady state equation
 Distillation
 Flash distillation
Relationship between feed flow, overhead vapour flow and bottom liquid flow in flash distillation
 Feed line equation
Feed line equation for distillation column
Where q is amount of liquid added in bottom product per mole of feed. If feed is a mixture of liquid and vapour, q is the fraction that is liquid.
 Rectifying and stripping section equation
 Minumum number of plates
Fensky equation for minimum number of plates, i.e., under the conditions of total reflux
In this equation 1 is subtracted for reboiler. Nm is actual number of plates excluding reboiler.
 Underwood equation for minimum reflux
 Rayleigh equation for batch distillation
 Relative Volatility
 Minimum reflux and composition of distillate and equilibrium fractions
Relationship between minimum reflux and composition of distillate and equilibrium fractions
 Steam distillation
Benzene and water mixture in steam distillation
 Logarithmic mean concentration difference
Overall volumetric coefficient (K_{c}a) for mass transfer is given by
HTU – height of transfer unit is given by
 Unit Conversion
 Kilogram per cubic meter to ppm
 Solubilities at different temperatures
Given solubility S_{1}, cm^{3}/atm at 298^{o}K and 1atm, then solubility at 2.5atm pressure S_{2.5} = S_{1} * 2.5, and solubility at STP
 Absorption
 Operating line of absorption
To be updated
 Equal slope of equilibrium line and operating line
When change in liquid composition across a tray is independent of tray location, i.e. difference in liquid composition across tray is same for each tray in absorber, the equilibrium line and operating line are parallel.
Slope of operating line = L/V
Slope of equilibrium line = m
 Mass balance
For L/G to be minimum x_{1} will attain equilibrium value.
 Height of packing
For packed bed absorption tower, height of packing = HTU x NTU
 Height Z, of absorption tower
For dilute solution, i.e. when operating line and equilibrium line are straight
 Rate of absorption
First order rate equation of absorption
 Relationship between flooding velocity(u_{f}) and pressure (P)
Relationship between flooding velocity(u_{f}) and pressure (P) of distillation/absorption column
 Diffusion
 Diffusion between two components
General diffusion equation between two components
 Diffusion of component A into non diffusing B
For nondiffusing single component diffusion
 Dependency of diffusivity on temperature and pressure
Diffusivity is directly proportional to temperature raise to power 1.5 and inversely proportional to pressure
 Diffusivity in liquids
 Equimolar counter current diffusion
Fick’s law of equimolar counter current diffusion
 Knudsen diffusivity
 Mass transfer coefficient dependencies on diffusivity
Mass transfer coefficient dependencies on diffusivity according to various film theories
 Mean diffusivity
O_{2} is diffusing with nondiffusing gas (H_{2} and CH_{4}), mean diffusivity is
 Mass transfer theories
 Penetration theory
Transient rate of diffusion into relatively thick mass of fluid with constant surface concentration. As exposure time decreases the mass transfer coefficient increases.
 Surface Renewal theory
Surface renewal theory in Danckwert’s model of Mass transfer
S is also called fractional rate of surface renewal
 Lewis Number and Psychromatic ratio
For air water system, Le = 1 = b; for air water system adiabatic saturation temperature = WBT
 Stanton number (mass transfer)
 Sherwood number
The Sherwood number (Sh) (also called the mass transfer Nusselt number) represents the ratio of convective mass transfer to the rate of diffusive mass transport.Sherwood number is extensively used in convective mass transfer calculations. Sherwood number for flow in a pipe can be expressed as the ratio of the concentration gradient at the surface to overall concentration difference. Forced convection is relatively more effective in increasing the rate of mass transfer if the Schmidt number is larger. Therefore, higher Schmidt number will result in higher Sherwood number which means mass transfer coefficient will be higher.
Sherwood number for flow past flat plate
Sherwood number for flow in pipe can be expressed as the ratio of concentration gradient at surface to overall concentration difference.
Sherwood number for flow past sphere in case of
 Natural convection
 Forced convection
 Number of stages
Number of plates/stages for absorption, stripping and liquidliquid extraction
 Stripping
No. of plates/stages for transfer of solute from liquid phase to vapor phase (stripping)
 Absorption
no. of plates/stages for transfer of solute from vapour phase to liquid absorption
 Liquid – liquid extraction
 Leaching
42. To be updated
 Boundary layer
All the boundary layers will coincide with each other when prandtl number = schmidt number = lewis number
 Peclet number related to mass transfer
Peclet number greater than one advection dominates; Peclet number less than one diffusion dominates.
 Normality and equivalent weight
 Permeability of membrane
Relationship between permeability of membrane, solubility coefficient and diffusivity
 For solubility related problem
For solubility related problem pertaining to mass transfer use the following equation
 Liquidliquid Extraction
In liquidliquid extraction, Y = mX
 Fractional holdup of gas bubbles (ϵ)
In a gas liquid contactor, fractional holdup of gas bubbles (ϵ) is defined as
Interfacial area of bubble
 Mass Transfer Theories
 Degree of freedom in drying
Degree of freedom in drying – In drying of solids containing moisture above critical moisture content, no. of degree of freedom is F = C – P + 2 = 1 – 2 + 2 = 1.
 Local efficiency and murphee efficiency


 In small columns local efficiency is equal to Murphee efficiency. In large columns local efficiency is less than Murphee efficiency.
 In a tray column, separating a binary mixture, with non – ideal stages, point efficiency cannot exceed 100% but Murphee efficiency can exceed 100%.

 Towers and Columns


 Packed bed towers are preferred for a gasliquid mass transfer operations with foaming liquids because of lower pressure drop and in packed towers the gas is not bubbled through the liquid pool.
 Plates column are preferred when operation involves liquid containing suspended solids. Plate columns are preferred when large temperature changes are involved in distillation operation. Packed towers are preferred if liquid has high foaming tendency. Packed towers are cheaper than plate towers if highly corrosive fluid is to be handled.

 Diffusion


 Gas – Gas diffusion coefficient is in order of 10^{5} m^{2}/sec and liquid – liquid diffusion coefficient is in order of 10^{9} m^{2}/sec.
 For a given concentration difference, mass transfer flux for diffusion of only one component of mixture is greater than mass transfer flux of equimolal countercurrent diffusion.
 Diffusivity will remain unaltered for both cases, stagnant and diffusing air.
 Component A is diffusing in component B. The flux NA relative to a stationary point is equal to the flux due to molecular diffusion.

 Roult’s law
Roult’s law: The vapour pressure of a solution of a nonvolatile solution is equal to vapour pressure of the pure solvent at that temperature multiplied by its mole fraction. P = X_{solute} × P^{o}_{solute}
In an ideal solution, it takes exactly the same amount of energy for a solvent molecule to break away from the surface of the solution as it did in pure solvent. The forces of attraction between solvent and solute are exactly the same as between the original solvent molecules.
Roult’s law only works for solutes which doesn’t change their nature when they dissolve, i.e. they mustn’t ionize or associate. Remember: vapour pressure is inversely proportional to boiling point.
Positive deviation from roult’s law



 In mixtures showing positive deviation from roult’s law, the vapour pressure of the mixture is always higher than you would expect from an ideal mixture. If a mixture has a high vapour pressure it means that it will have a low boiling point. The molecules are escaping easily and one won’t have to heat the mixture much to overcome the intermolecular attractions completely.


Negetive deviation from roult’s law



 Mixture have vapour pressure which is less than would be expected by roult’s law. Minimum value for vapour pressure would be lower than that of either pure component. In a binary solution of component A and B, if component A exhibits positive deviation from roult’s law, then component B exhibits positive deviation from roult’s law.


 Dimensionless numbers Mass transfer


 At temperature 750K and 1 atm pressure the approximate value of Schmidt number for air is 0.1.
 Schmidt number is independent of pressure when ideal gas law applies as effect of pressure on density and diffusivity cancel each other and viscosity is independent of pressure.
 Boundary layer theory is for laminar flow only and Schmidt number is equal to one in this theory.

 Crystallization


 The ∆L law of crystal growth: If all crystals in magma grow in a uniform super saturation field and at the same temperature and if all the crystals grow from birth at rate governed by super saturation, then all crystals are not only in variant but also have the same growth rate that is independent of size ∆L = G ∆t. The total growth of each crystals in the magma during the same time interval ∆t is the same.
 A supersaturated solution of a sparingly soluble solute, at a concentration C is being fed to a crystallizer at volumetric flow rate of V. The solubility of solute is C_{s}. The output rate of solids from efficient crystallizer is (C – C_{s})/V.

 Distillation and Absorption


 Increasing the L/V ratio increases driving force everywhere in the column except at the top of the column.
 Reflux is added in distillate column in order to increase the concentration of the more volatile component in the distillate. If no reflux is added to the top plate, enrichment is not possible and the composition of the distillate would be same as that of original feed.
 The boiling point of a mixture of two immiscible liquids is always less than the boiling point of each individual component.
 At total reflux condition we get maximum enrichment of the top and bottom composition.
 In case of steam distillation, less number of trays are required as compared to during usage of reboiler.
 A distillation column at a pilot plant is scaled up by n times for industrial use at steady state. Then number of theoretical trays doesn’t increase n times, minimum reflux doesn’t increase by n times, feed flow rate and product flow rate are increased by n times, feed composition and product composition doesn’t increase by n times.

 Operating line and equilibrium line


 Any point on the operating line expresses the material balance equation for the transferring component between top and bottom part of packed tower. The vertical distance between operating line and equilibrium line represents the driving force for mass transfer at a particular point inside tower.

 Extraction


 In a few extraction systems, the direction of tie line slope changes and one tie line will be horizontal. Such systems are said to be solutropic system. It is involved in liquidliquid extraction.
 The commonly used solvent in supercritical extraction is carbon dioxide.

 Adsorption


 Apparent activation energy in case of adsorption E_{a} = E – ∆Hads

 Mass transfer theories

 Surface renewal theory and penetration theory is for unsteady state conditions. Boundary layer theory and film theory is for steady state conditions.