Mass transfer notes

  • Material balance for distillation column
  • Overhead section of distillation column

Material balance across overhead section of a distillation column

material balance across overhead section of distillation column

  • Feed, top product and bottom product

Material balance between feed, distillate and bottom product in distillation column

Material balance between feed, distillate and bottom product in distillation column

  • Solute balance

Solute balance in mass transfer operation

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

Mole fraction and pressure based mass transfer coefficients

  • Individual and overall mass transfer coefficients

Relation between individual and overall mass transfer coefficients

Individual and overall mass transfer coefficients

      1. 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)

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

  • Total time of drying in falling rate region

Total time of drying in falling rate region is given by

Total time of drying in falling rate region

  • Rate of drying (In relation to partial pressures)

Drying rate in constant rate period when partial pressures are given

Drying rate in constant rate period when partial pressures are given

  • Rate of drying (In relation to heat of vaporization)

Rate of drying (In relation to heat of vaporization)

  • Humidity
  • Percentage humidity

percentage humidity

  • Dew point humidity

At dew point humidity is 100%

dew point humidity

  • Humidity

humidity

  • Molar saturation humidity

molar saturation humidity

  • Calculating wet bulb temperature

Finding wet bulb temperature: Heat flux = Mass flux * latent heat (λ)

calculating wet bulb temperature

  • Relative saturation

Ratio of partial pressure to vapour pressure

relative saturation

  • Chilton-Colburn analogy

Chilton-coulburn 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)

chilton colburn analogy

  • Mass transfer flux
  • Total molar flux

Total molar flux of component A

total molar flux

  • Mass transfer flux and humidity

Relationship between mass transfer flux and relative humidity

mass transfer flux and humidity

  • Fick’s law
  • Fick’s first law

First Law: Steady state equation

ficks first law for steady state mass transfer

  • Fick’s second law

Second Law: Unsteady state equation

ficks second law unsteady state mass transfer

  • Distillation
  • Flash distillation

Relationship between feed flow, overhead vapour flow and bottom liquid flow in flash distillation

flash distillation equation

  • Feed line equation

Feed line equation for distillation column

feed line equation of distillation equation

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.

feed line equation of distillation column

  • Rectifying and stripping section equation

rectifying and stripping section equation

  • Minumum number of plates

Fensky equation for minimum number of plates, i.e., under the conditions of total reflux

fensky equation for minimum number of plates

In this equation 1 is subtracted for reboiler. Nm is actual number of plates excluding reboiler.

  • Underwood equation for minimum reflux

underwood equation for minimum reflux

  • Rayleigh equation for batch distillation

rayleigh equation for batch distillation

  • Relative Volatility

relative volatilty

  • Minimum reflux and composition of distillate and equilibrium fractions

Relationship between minimum reflux and composition of distillate and equilibrium fractions

minimum reflux and composition of distillate and equilibrium fractions

  • Steam distillation

Benzene and water mixture in steam distillation

steam distillation

  • Logarithmic mean concentration difference

lmtd of concentration difference

Overall volumetric coefficient (Kca) for mass transfer is given by

Overall volumetric coefficient

HTU – height of transfer unit is given by

height of transfer unit

  • Unit Conversion
  • Kilogram per cubic meter to ppm

kilogram per cubic meter to ppm

  • Solubilities at different temperatures

Given solubility S1, cm3/atm at 298oK and 1atm, then solubility at 2.5atm pressure S2.5 = S1 * 2.5, and solubility at STP

solubility at different temperatures

  • 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

mass balance of absorption

For L/G to be minimum x1 will attain equilibrium value.

  • Height of packing

For packed bed absorption tower, height of packing = HTU x NTU

height of packing for packed bed absorption tower

  • Height Z, of absorption tower

height Z of absorption tower

For dilute solution, i.e. when operating line and equilibrium line are straight

operating line and equilibrium line for dilute solution

  • Rate of absorption

First order rate equation of absorption

rate of absorption for first order

  • Relationship between flooding velocity(uf) and pressure (P)

Relationship between flooding velocity(uf) and pressure (P) of distillation/absorption column

relation between flooding velocity and pressure of distillation column

  • Diffusion
  • Diffusion between two components

General diffusion equation between two components

general equation for diffusion between two components

  • Diffusion of component A into non diffusing B

diffusion of component A into non diffusing B

For non-diffusing single component diffusion

non-diffusing single component

  • Dependency of diffusivity on temperature and pressure

Diffusivity is directly proportional to temperature raise to power 1.5 and inversely proportional to pressure

variation of diffusivity with temperature and pressure

  • Diffusivity in liquids

diffusivity in liquids

  • Equimolar counter current diffusion

Fick’s law of equimolar counter current diffusion

equimolar counter current diffusion

  • Knudsen diffusivity

knudsen diffusivity

  • Mass transfer coefficient dependencies on diffusivity

Mass transfer coefficient dependencies on diffusivity according to various film theories

mass transfer coefficients dependencies on diffusivity

  • Mean diffusivity

O2 is diffusing with non-diffusing gas (H2 and CH4), mean diffusivity is

mean diffusivity for diffusion with non diffusing gas

  • 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.

penetration theory of mass transfer

  • Surface Renewal theory

Surface renewal theory in Danckwert’s model of Mass transfer

danckwert's surface renewal theory

S is also called fractional rate of surface renewal

  • Lewis Number and Psychromatic ratio

lewis number and psychromatic ratio
lewis number and psychromatic ratio

For air water system, Le = 1 = b; for air water system adiabatic saturation temperature = WBT

  • Stanton number (mass transfer)

stanton number for 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

Sherwood number for flow past flat plate

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 in pipe

Sherwood number for flow past sphere in case of

  • Natural convection

sherwood number in natural convection mass transfer

  • Forced convection

sherwood number in forced convection mass transfer

  • Number of stages

Number of plates/stages for absorption, stripping and liquid-liquid extraction

  • Stripping

No. of plates/stages for transfer of solute from liquid phase to vapor phase (stripping)

number of stages in stripping

  • Absorption

no. of plates/stages for transfer of solute from vapour phase to liquid absorption

number of stages for absorption

  • Liquid – liquid extraction

number of stages for liquid liquid absorption

  • Leaching

42. To be updated

  • Boundary layer

All the boundary layers will coincide with each other when prandtl number = schmidt number = lewis number

coinciding of mass transfer

  • Peclet number related to mass transfer

mass transfer peclet number

Peclet number greater than one advection dominates; Peclet number less than one diffusion dominates.

  • Normality and equivalent weight

normality and equivalent weight

  • Permeability of membrane

Relationship between permeability of membrane, solubility coefficient and diffusivity

permeability of membrane

  • For solubility related problem

For solubility related problem pertaining to mass transfer use the following equation

mass transfer in solubility

  • Liquid-liquid Extraction

In liquid-liquid extraction, Y = mX

liquid liquid extraction

  • Fractional holdup of gas bubbles (ϵ)

In a gas liquid contactor, fractional holdup of gas bubbles (ϵ) is defined as

fractional hold up of gas bubbles

Interfacial area of bubble

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
      1. In small columns local efficiency is equal to Murphee efficiency. In large columns local efficiency is less than Murphee efficiency.
      2. 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
      1. Packed bed towers are preferred for a gas-liquid mass transfer operations with foaming liquids because of lower pressure drop and in packed towers the gas is not bubbled through the liquid pool.
      2. 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
      1. Gas – Gas diffusion coefficient is in order of 10-5 m2/sec and liquid – liquid diffusion coefficient is in order of 10-9 m2/sec.
      2. 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.
      3. Diffusivity will remain unaltered for both cases, stagnant and diffusing air.
      4. 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 non-volatile solution is equal to vapour pressure of the pure solvent at that temperature multiplied by its mole fraction. P = Xsolute × Posolute

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

        1.  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

        1.  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
      1. At temperature 750K and 1 atm pressure the approximate value of Schmidt number for air is 0.1.
      2. 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.
      3. Boundary layer theory is for laminar flow only and Schmidt number is equal to one in this theory.
  • Crystallization
      1. 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.
      2. 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 Cs. The output rate of solids from efficient crystallizer is (C – Cs)/V.
  • Distillation and Absorption
      1. Increasing the L/V ratio increases driving force everywhere in the column except at the top of the column.
      2. 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.
      3. The boiling point of a mixture of two immiscible liquids is always less than the boiling point of each individual component.
      4. At total reflux condition we get maximum enrichment of the top and bottom composition.
      5. In case of steam distillation, less number of trays are required as compared to during usage of reboiler.
      6. 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
      1. 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
      1. 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 liquid-liquid extraction.
      2. The commonly used solvent in supercritical extraction is carbon dioxide.
  • Adsorption
      1. Apparent activation energy in case of adsorption Ea = E – ∆Hads
  • Mass transfer theories
    1. Surface renewal theory and penetration theory is for unsteady state conditions. Boundary layer theory and film theory is for steady state conditions.