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Cheatsheet: Mass Transfer

1. Fundamentals of Mass Transfer

1.1 Fick's Law of Diffusion

Form Equation
First Law (Steady State) JA = -DAB(dCA/dx) where JA is molar flux, DAB is diffusivity, CA is concentration
Second Law (Unsteady State) ∂CA/∂t = DAB(∂²CA/∂x²) for one-dimensional diffusion
Equimolar Counterdiffusion NA = -DAB(dCA/dx) where NA is molar transfer rate
Diffusion Through Stagnant Film NA = -DABC/(1-yA)(dyA/dx) where yA is mole fraction

1.2 Diffusivity Correlations

System Correlation
Gas Phase (Fuller-Schettler-Giddings) DAB = 0.00143T1.75/[PM0.5(ΣvA1/3 + ΣvB1/3)²] cm²/s where T is K, P is atm, M is molecular weight
Liquid Phase (Wilke-Chang) DAB = 7.4×10-8(φMB)0.5T/(μBVA0.6) cm²/s where φ is association parameter, μ is viscosity (cP), V is molar volume
Stokes-Einstein (Large Molecules) DAB = kT/(6πμr) where k is Boltzmann constant, r is particle radius

1.3 Mass Transfer Mechanisms

  • Molecular Diffusion: Mass transfer due to concentration gradients at molecular level
  • Eddy Diffusion: Mass transfer due to turbulent eddies in fluid flow
  • Convective Mass Transfer: Mass transfer due to bulk fluid motion combined with diffusion

2. Mass Transfer Coefficients

2.1 Individual Coefficients

Coefficient Type Definition
kc (Concentration Basis) NA = kc(CA,i - CA,b) where C is concentration, i is interface, b is bulk
ky (Gas Mole Fraction) NA = ky(yA,i - yA,b) where y is mole fraction
kx (Liquid Mole Fraction) NA = kx(xA,i - xA,b) where x is mole fraction
kG (Gas Partial Pressure) NA = kG(pA,i - pA,b) where p is partial pressure
kL (Liquid Concentration) NA = kL(CA,i - CA,b) for liquid phase

2.2 Overall Coefficients

Coefficient Relationship
Overall Gas (KG) 1/KG = 1/kG + H/kL where H is Henry's constant
Overall Liquid (KL) 1/KL = 1/(HkG) + 1/kL
Overall Gas Mole Fraction (Ky) 1/Ky = 1/ky + m/kx where m is slope of equilibrium line
Overall Liquid Mole Fraction (Kx) 1/Kx = 1/(mky) + 1/kx

2.3 Conversion Between Coefficients

  • kG = kyP where P is total pressure
  • kL = kxC where C is total concentration
  • kc = kG/RT where R is gas constant, T is temperature
  • For dilute solutions: kc ≈ kxCtotal

3. Dimensionless Numbers

3.1 Key Dimensionless Groups

Number Definition & Significance
Schmidt (Sc) Sc = μ/(ρDAB) = ν/DAB; ratio of momentum diffusivity to mass diffusivity
Sherwood (Sh) Sh = kcL/DAB; dimensionless mass transfer coefficient
Peclet (Pe) Pe = uL/DAB = Re·Sc; ratio of convective to diffusive transport
Lewis (Le) Le = α/DAB = Sc/Pr; ratio of thermal to mass diffusivity
Grashof (Gr) Gr = gβΔCL³/ν²; ratio of buoyancy to viscous forces in natural convection

3.2 Analogies Between Transfer Processes

Analogy Application
Reynolds (Sc = Pr = 1) Sh/Re·Sc = Nu/Re·Pr = f/2 where f is friction factor
Chilton-Colburn jD = jH where jD = (Sh/Re·Sc)(Sc)2/3 and jH = (Nu/Re·Pr)(Pr)2/3
Prandtl-Taylor Sh/Nu = (Sc/Pr)n where n varies with flow regime

4. Convective Mass Transfer Correlations

4.1 Flow Past Flat Plate

Flow Regime Correlation
Laminar (Re <> ShL = 0.664ReL0.5Sc1/3 for average coefficient
Turbulent (Re > 5×10⁵) ShL = 0.037ReL0.8Sc1/3 for average coefficient
Local Laminar Shx = 0.332Rex0.5Sc1/3

4.2 Flow in Pipes and Tubes

Conditions Correlation
Laminar, Short Tube ShD = 1.86(Re·Sc·D/L)1/3(μ/μw)0.14 for Re·Sc·D/L > 10
Laminar, Long Tube ShD = 3.66 for fully developed flow with constant wall concentration
Turbulent (Re > 10,000) ShD = 0.023Re0.8Sc0.33 (Dittus-Boelter analog)
Turbulent, More Accurate ShD = 0.027Re0.8Sc1/3(μ/μw)0.14 (Sieder-Tate analog)

4.3 Flow Around Spheres

Regime Correlation
Stagnant Fluid ShD = 2.0 (pure diffusion limit)
Low Reynolds (Re <> ShD = 2 + 0.6Re0.5Sc1/3
General (2 < re=""><> ShD = 2 + 0.6Re0.5Sc1/3 (Ranz-Marshall)
Wide Range ShD = 2 + 0.552Re0.53Sc1/3 for 1 < re=""><>

4.4 Packed Beds

Correlation Range
jD = 1.17Rep-0.415 10 <>p < 2500="" where="">p = Dpusρ/μ
jD = 0.91Rep-0.51 Rep <>
ε·Shp = 0.765Rep0.82Sc0.33 55 <>p < 1500="" where="" ε="" is="" void="">

5. Interphase Mass Transfer

5.1 Two-Film Theory

  • Interface at equilibrium (no resistance at interface itself)
  • Resistance concentrated in thin films on each side of interface
  • Concentration profile linear in each film
  • NA = KG(pA,G - pA,i) = KL(CA,i - CA,L)

5.2 Equilibrium Relationships

Type Relationship
Henry's Law pA = HCA or yA = mxA where H is Henry's constant, m = H/P
Raoult's Law pA = xApAsat for ideal solutions
Distribution Coefficient CA,1 = KCA,2 for liquid-liquid systems

5.3 Controlling Resistance

  • Gas-phase controlled: kG <>L; low solubility gases (O₂, CO₂ in water)
  • Liquid-phase controlled: HkG <>L; high solubility gases (NH₃, SO₂ in water)
  • Controlling resistance determines which phase to manipulate for enhanced transfer

6. Gas Absorption

6.1 Operating Line Equation

Parameter Equation
Material Balance G(Y₁ - Y₂) = L(X₁ - X₂) where G is gas molar flow, L is liquid molar flow
Operating Line Y = (L/G)X + (Y₂ - (L/G)X₂) with slope L/G
Minimum L/G (L/G)min = (Y₁ - Y₂)/(X₁* - X₂) where X₁* is in equilibrium with Y₁

6.2 Tower Height Calculations

Method Equation
HTU-NTU (Gas Phase) Z = HOG·NOG where HOG = G/(KyaP), NOG = ∫dy/(y-y*)
HTU-NTU (Liquid Phase) Z = HOL·NOL where HOL = L/(KxaC), NOL = ∫dx/(x*-x)
Overall Coefficient Z = (G/KyaP)∫(dy/(y-y*)) from y₂ to y₁
Dilute Systems NOG = (y₁-y₂)/(y-y*)lm where log mean driving force used

6.3 HTU Relationships

  • HOG = HG + (mG/L)HL where HG = G/(kyaP), HL = L/(kxaC)
  • HOL = HL + (L/mG)HG
  • Lower HTU indicates better mass transfer performance
  • NTU represents separation difficulty (concentration change required)

6.4 Absorption Factor Method

Parameter Definition
Absorption Factor A = L/(mG) where m is slope of equilibrium line
Kremser Equation NOG = ln[(y₁-mx₂)/(y₂-mx₂)(1-1/A)+1/A]/(ln A) for A ≠ 1
When A = 1 NOG = (y₁-mx₂)/(y₂-mx₂)
Optimal Design A = 1.25 to 2.0 for economical design

7. Distillation

7.1 McCabe-Thiele Method

Line Equation
Rectifying Section yn+1 = (L/V)xn + (D/V)xD where L is liquid flow, V is vapor flow, D is distillate
Stripping Section ym+1 = (L'/V')xm - (B/V')xB where B is bottoms
q-line y = (q/(q-1))x - (zF/(q-1)) where zF is feed composition
Feed Condition (q) q = (HV - HF)/(HV - HL) = heat to vaporize 1 mole feed / molar latent heat

7.2 Feed Conditions

Feed State q Value
Saturated Liquid q = 1; q-line vertical
Saturated Vapor q = 0; q-line horizontal
Subcooled Liquid q > 1; q-line slope positive
Superheated Vapor q < 0;="" q-line="" slope="">
Mixed Liquid-Vapor 0 < q="">< 1;="" q-line="" slope="">

7.3 Reflux Ratio

Type Definition & Application
Reflux Ratio R = L/D where L is reflux, D is distillate
Minimum Reflux Rmin found where operating line intersects equilibrium curve at feed (infinite stages)
Total Reflux R = ∞; minimum stages, no product withdrawal
Optimal Reflux R = 1.2 to 1.5·Rmin for economic design

7.4 Fenske Equation

  • Nmin = ln[(xD/(1-xD))((1-xB)/xB)]/ln(αavg) at total reflux
  • αavg = √(αtop·αbottom) for relative volatility
  • Applies to binary systems at total reflux (minimum stages)

7.5 Relative Volatility

Definition Relationship
Relative Volatility α = (yA/xA)/(yB/xB) = KA/KB where K is equilibrium ratio
Constant α Equilibrium y = αx/(1+(α-1)x)
Ease of Separation α > 1.1 for practical distillation; α close to 1 makes separation difficult

8. Liquid-Liquid Extraction

8.1 Equilibrium Relationships

Parameter Definition
Distribution Coefficient KD = CA,extract/CA,raffinate at equilibrium
Selectivity β = (CA/CB)extract/(CA/CB)raffinate = KDA/KDB
Extraction Factor E = KD·(S/F) where S is solvent flow, F is feed flow

8.2 Single-Stage Extraction

  • Material balance: FxF = ExE + RxR where F is feed, E is extract, R is raffinate
  • Fraction extracted: f = (xF - xR)/xF = KD/(KD + F/S)
  • For dilute systems with constant KD

8.3 Multistage Crosscurrent Extraction

  • xn/x0 = 1/(1 + KDS/F)n where n is number of stages
  • Each stage uses fresh solvent (inefficient solvent use)

8.4 Countercurrent Extraction (Kremser)

Equation Application
N = ln[(xF-yS/KD)/(xR-yS/KD)(1-1/E)+1/E]/ln(E) For E ≠ 1; N is number of equilibrium stages
N = (xF-yS/KD)/(xR-yS/KD) For E = 1

8.5 Solvent Selection Criteria

  • High selectivity (β >> 1)
  • High distribution coefficient (KD >> 1)
  • Low mutual solubility with raffinate phase
  • Density difference from feed for easy phase separation
  • Low viscosity for good mass transfer
  • Chemical stability, non-toxic, low cost, easy to recover

9. Humidification and Drying

9.1 Psychrometric Properties

Property Definition/Equation
Humidity (Y) Y = mass water/mass dry air = 0.622pv/(P-pv) where pv is vapor pressure
Relative Humidity (RH) RH = pv/pv,sat = Y·P/[0.622pv,sat + Y·P]
Percentage Humidity % = 100Y/Ysat
Humid Heat (Cs) Cs = Cp,air + Y·Cp,water = 1.005 + 1.88Y kJ/kg dry air·K
Humid Volume (vH) vH = (0.082T/P)(1 + Y/0.622) m³/kg dry air where T is K
Enthalpy (H) H = CsT + Yλ where λ is latent heat of water

9.2 Adiabatic Saturation

  • Wet-bulb temperature (Twb) measured by thermometer with wetted wick in moving air
  • Adiabatic saturation: H₁ = H₂ when air-water contact is adiabatic
  • Cs(T₁ - Tas) = (Yas - Y₁)λas where Tas is adiabatic saturation temp
  • For air-water: Twb ≈ Tas

9.3 Drying Rate Periods

Period Characteristics
Constant Rate Surface remains wet; rate = hcA(Tg-Twb)/λ; surface at wet-bulb temp
Critical Moisture Xc where constant rate period ends; surface begins to dry
Falling Rate Rate decreases as moisture content decreases; internal diffusion controls
Equilibrium Moisture X* where drying stops; depends on air humidity and temperature

9.4 Drying Time Calculations

Period Equation
Constant Rate t = (Ls/ARc)(X₁ - Xc) where Ls is dry solid mass, A is area, Rc is constant rate
Falling Rate (Linear) t = (Ls/ARc)[(Xc-X*)/2]ln[(Xc-X*)/(X₂-X*)]
Total Time ttotal = tconstant + tfalling

10. Adsorption

10.1 Equilibrium Isotherms

Model Equation
Freundlich q = KC1/n where q is solid loading, C is concentration, K and n are constants
Langmuir q = qmbC/(1+bC) where qm is monolayer capacity, b is equilibrium constant
BET (Multilayer) q = qmBC/[(Cs-C)(1+(B-1)C/Cs)] where Cs is saturation concentration
Linear q = KC for dilute systems

10.2 Breakthrough Curves

  • Breakthrough time (tb): Time when outlet concentration reaches maximum acceptable level
  • Stoichiometric time (ts): Time to saturate bed based on mass balance = (ρbVbedq*)/(QC₀)
  • Mass transfer zone (MTZ): Region where adsorption actively occurring
  • Usable capacity: Amount adsorbed before breakthrough

10.3 Bed Design Parameters

Parameter Definition
Bed Length L = (u·ts·ρb·q*)/(ε·ρf·C₀) where u is velocity, ε is void fraction, ρf is fluid density
Bed Utilization η = tb/ts (fraction of bed used before breakthrough)
LUB (Length of Unused Bed) LUB = L(1 - η) = L(ts - tb)/ts

10.4 Regeneration

  • Thermal Swing: Increase temperature to desorb; slow but effective
  • Pressure Swing (PSA): Reduce pressure to desorb; faster cycling
  • Purge Gas: Use inert gas or steam to strip adsorbate
  • Displacement: Use solvent or more strongly adsorbed species

11. Membrane Separation

11.1 Membrane Transport Models

Model Equation
Solution-Diffusion Ji = (DiKi/δ)(pi,feed - pi,permeate) where K is partition coefficient, δ is thickness
Pore Flow Jv = (ε·r²·ΔP)/(8μτδ) (Hagen-Poiseuille) where τ is tortuosity
Permeability Pi = DiKi = Jiδ/Δpi

11.2 Membrane Selectivity

Parameter Definition
Ideal Selectivity αij = Pi/Pj (ratio of permeabilities)
Separation Factor α* = (yi/yj)/(xi/xj) where y is permeate, x is retentate
Rejection Coefficient R = (Cfeed - Cpermeate)/Cfeed for liquid separations

11.3 Reverse Osmosis

  • Jv = Lp(ΔP - Δπ) where Lp is hydraulic permeability, π is osmotic pressure
  • Osmotic pressure: π = iMRT where i is van't Hoff factor, M is molarity
  • Applied pressure must exceed osmotic pressure for permeation
  • Concentration polarization reduces flux: CP = Cwall/Cbulk = exp(Jv/km)

11.4 Membrane Processes

Process Driving Force & Application
Microfiltration (MF) ΔP (0.1-2 bar); particle removal 0.1-10 μm
Ultrafiltration (UF) ΔP (1-10 bar); macromolecule separation 1-100 nm
Nanofiltration (NF) ΔP (5-40 bar); divalent ion removal, organics
Reverse Osmosis (RO) ΔP (10-100 bar); desalination, ion removal
Gas Separation Δp; H₂ recovery, CO₂ removal, air separation
Pervaporation Δp (vacuum); organic-water separation, solvent dehydration

12. Leaching and Crystallization

12.1 Leaching (Solid-Liquid Extraction)

Type Description
Unsteady-State Solid contact time varies; batch operation
Steady-State Continuous countercurrent; similar to L-L extraction analysis
Equilibrium Stages Use Kremser equation with distribution coefficient KD

12.2 Crystallization Fundamentals

Concept Definition
Supersaturation S = (C - C*)/C* where C is concentration, C* is saturation
Nucleation Rate B = knSn where kn and n are constants
Crystal Growth Rate G = kgSg where kg and g are constants
Metastable Zone Region between saturation and spontaneous nucleation

12.3 Solubility Relationships

  • Solubility curves: C* = f(T) for each substance
  • Positive slope: cooling crystallization effective
  • Negative slope: evaporative crystallization preferred
  • Hydrate formation changes solubility behavior

12.4 Yield Calculations

  • Material balance: FCF = MCM + SCS where F is feed, M is mother liquor, S is crystals
  • Yield = (CF - CM)/(CF - C*) for simple systems
  • Adjust for hydrates: molecular weight changes affect yield

13. Equipment and Design

13.1 Packed Column Design

Parameter Correlation
Flooding Velocity uflood from generalized flooding correlation (Sherwood plot)
Design Velocity udesign = (0.5 to 0.7)uflood
Pressure Drop ΔP/Z = function of L/G, packing type, flow rates
HETP Height Equivalent to Theoretical Plate varies with packing (0.3-1.5 m typical)

13.2 Tray Column Design

Parameter Typical Values/Equations
Tray Spacing 0.3-0.6 m (12-24 inches)
Tray Efficiency EMV = (yn - yn+1)/(yn* - yn+1) where y* is equilibrium
Weeping Occurs when vapor velocity too low; liquid drains through holes
Flooding Occurs when vapor velocity too high; liquid entrainment excessive
Operating Range Turndown ratio 2:1 to 4:1

13.3 Common Packing Types

Packing Characteristics
Raschig Rings Low cost, moderate efficiency, a = 150-600 m²/m³
Pall Rings Improved over Raschig, lower ΔP, a = 200-350 m²/m³
Berl Saddles Better liquid distribution, a = 250-450 m²/m³
Structured Packing High efficiency, low ΔP, high cost, a = 250-500 m²/m³, HETP = 0.15-0.5 m

13.4 Tray Types

  • Sieve Tray: Simple holes, low cost, prone to weeping at low rates
  • Valve Tray: Movable valves, wider operating range, higher cost
  • Bubble Cap: Most expensive, widest turndown, rarely used for new designs

14. Key Design Equations Summary

14.1 Mass Transfer Rate Forms

Basis Rate Equation
Concentration NA = kcA(CAi - CAb)
Mole Fraction (Gas) NA = kyA(yAi - yAb)
Mole Fraction (Liquid) NA = kxA(xAi - xAb)
Partial Pressure NA = kGA(pAi - pAb)
Overall Gas NA = KyA(y - y*) or KGA(p - p*)
Overall Liquid NA = KxA(x* - x) or KLA(C* - C)

14.2 Column Height Equations

Method Equation
HTU-NTU Z = HTU × NTU
HETP-Stages Z = HETP × Nstages
Direct Integration Z = ∫(G/KyaP)dy or ∫(L/KxaC)dx

14.3 Quick Reference Conversions

  • kG = kyP; kL = kxC
  • kc = kG/RT
  • HOG = G/(KyaP); HOL = L/(KxaC)
  • 1/Ky = 1/ky + m/kx; 1/Kx = 1/(mky) + 1/kx
  • For dilute systems: mole fraction ≈ concentration/Ctotal
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Apr 20, 2026 Last updated
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