Chapter 4

Mass Transfer

Diffusion, distillation, absorption, and extraction.

Mass transfer is the study of the movement of chemical species from one location to another, typically between phases. It's fundamental to separation processes, reaction engineering, and many unit operations in chemical engineering.

Fundamentals of Mass Transfer

Diffusion

Fick's first law describes molecular diffusion:

JA=−DABdCAdxJ_A = -D_{AB} \frac{dC_A}{dx}

Where:

  • JAJ_A = molar flux of component A (mol/m²·s)
  • DABD_{AB} = diffusion coefficient of A in B (m²/s)
  • dCAdx\frac{dC_A}{dx} = concentration gradient (mol/m⁴)

Convective Mass Transfer

Mass transfer due to bulk fluid motion:

NA=kc(CA,s−CA,b)N_A = k_c (C_{A,s} - C_{A,b})

Where:

  • NAN_A = molar flux (mol/m²·s)
  • kck_c = mass transfer coefficient (m/s)
  • CA,sC_{A,s} = concentration at surface
  • CA,bC_{A,b} = concentration in bulk

Equilibrium Relationships

Phase Equilibrium

For vapor-liquid systems, the equilibrium ratio:

Ki=yixiK_i = \frac{y_i}{x_i}

Where yiy_i and xix_i are mole fractions in vapor and liquid phases.

Henry's Law

For dilute solutions:

PA=HxAP_A = H x_A

Where HH is Henry's constant.

Distribution Coefficient

For liquid-liquid extraction:

KD=CA,extractCA,raffinateK_D = \frac{C_{A,extract}}{C_{A,raffinate}}

Advanced Mass Transfer Theory

Mass Transfer Coefficients

The two-film theory provides a fundamental model for mass transfer:

Film Theory: Mass transfer occurs through stagnant films on either side of the interface:

NA=kL(CA,i−CA,b)=kG(pA,b−pA,i)N_A = k_L (C_{A,i} - C_{A,b}) = k_G (p_{A,b} - p_{A,i})

Where:

  • kLk_L = liquid-phase mass transfer coefficient
  • kGk_G = gas-phase mass transfer coefficient
  • CA,iC_{A,i}, pA,ip_{A,i} = interfacial concentration and partial pressure
  • CA,bC_{A,b}, pA,bp_{A,b} = bulk concentration and partial pressure

Overall Coefficients: For gas-liquid systems:

1KL=1kL+1HkG\frac{1}{K_L} = \frac{1}{k_L} + \frac{1}{H k_G} 1KG=1kG+HkL\frac{1}{K_G} = \frac{1}{k_G} + \frac{H}{k_L}

Where HH is Henry's constant.

Theories of Mass Transfer

Penetration Theory: Higbie's model for unsteady-state diffusion:

kL=2DABπtck_L = 2\sqrt{\frac{D_{AB}}{\pi t_c}}

Where tct_c is the contact time.

Surface Renewal Theory: Danckwerts' model accounting for surface renewal:

kL=DABsk_L = \sqrt{D_{AB} s}

Where ss is the surface renewal rate.

Multicomponent Diffusion

For systems with more than two components, use the Maxwell-Stefan equations:

−∇μi=RT∑j≠ixixjDij(vj−vi)-\nabla \mu_i = RT \sum_{j \neq i} \frac{x_i x_j}{D_{ij}} (v_j - v_i)

Where DijD_{ij} are the Maxwell-Stefan diffusivities.

Mass Transfer with Chemical Reaction

When mass transfer is accompanied by chemical reaction:

Hatta Number: Characterizes the relative rates of reaction and diffusion:

Ha=kDABkLHa = \frac{\sqrt{k D_{AB}}}{k_L}
  • Ha<0.3Ha < 0.3: Slow reaction regime
  • 0.3<Ha<30.3 < Ha < 3: Intermediate regime
  • Ha>3Ha > 3: Fast reaction regime

Enhancement Factor:

E=absorption rate with reactionabsorption rate without reactionE = \frac{\text{absorption rate with reaction}}{\text{absorption rate without reaction}}

Separation Processes

Advanced Distillation Concepts

Multicomponent Distillation: For systems with more than two components:

  • Key components selection
  • Underwood equations for minimum reflux
  • Fenske equation for minimum stages

Azeotropic and Extractive Distillation:

  • Breaking azeotropes using entrainers
  • Selection of suitable solvents
  • Economic considerations

Reactive Distillation: Combining reaction and separation in one unit:

  • Equilibrium limitations
  • Catalyst selection and placement
  • Process intensification benefits

Advanced Absorption Theory

HTU-NTU Method: Height of transfer unit and number of transfer units:

Z=HTU×NTUZ = HTU \times NTU

Where:

HTU=GMKGaPHTU = \frac{G_M}{K_G a P} NTU=∫y1y2dyy−y∗NTU = \int_{y_1}^{y_2} \frac{dy}{y - y^*}

Packed Column Design:

  • Packing selection: random vs. structured
  • Pressure drop calculations
  • Flooding and loading points

Advanced Extraction Theory

Ternary Phase Diagrams: Using triangular diagrams for system representation:

  • Tie lines and conjugate phases
  • Plait point and critical point
  • Operating lines and equilibrium stages

Supercritical Fluid Extraction: Using fluids above their critical point:

  • Enhanced solubility and mass transfer
  • Tunable solvent properties
  • Environmental advantages

Adsorption and Chromatography

Adsorption Isotherms:

  • Langmuir isotherm: θ=KP1+KP\theta = \frac{KP}{1 + KP}
  • Freundlich isotherm: q=KP1/nq = K P^{1/n}
  • BET isotherm: For multilayer adsorption

Breakthrough Curves: Analysis of fixed-bed adsorption:

  • Mass transfer zone
  • Breakthrough time
  • Regeneration strategies

Real-World Application: Natural Gas Sweetening

Removing hydrogen sulfide (H₂S) from natural gas using amine absorption:

Process Description

  • Absorber: H₂S is absorbed by amine solution
  • Stripper: Rich amine is regenerated by heating
  • Amine circulation: Continuous loop

Design Calculations

Calculate the number of theoretical stages required:

# Gas stream composition
H2S_inlet = 0.02    # mole fraction
H2S_outlet = 0.0001 # mole fraction
gas_flow = 1000     # kmol/h

# Amine properties
amine_flow = 500    # kmol/h
equilibrium_constant = 2.5

# TODO: Calculate minimum stages using Kremser equation
# Steps:
# 1. Calculate absorption factor (A = L/(K*V))
# 2. Use Kremser equation for counter-current absorption

absorption_factor = 0
N_min = 0

print(f"Absorption factor: {absorption_factor:.2f}")
print(f"Minimum theoretical stages: {N_min:.1f}")

Your Challenge: Distillation Column Design

In this exercise, you'll design a distillation column to separate a binary mixture.

Goal: Calculate key design parameters for a benzene-toluene separation.

System Description

Separate a 50-50 mixture of benzene and toluene:

  • Feed: 100 kmol/h, 50 mol% benzene
  • Distillate: 95 mol% benzene
  • Bottoms: 5 mol% benzene
  • Relative volatility: α=2.5\alpha = 2.5

McCabe-Thiele Method

Using the McCabe-Thiele graphical method principles:

# Feed conditions
F = 100          # kmol/h
zF = 0.5         # mole fraction benzene in feed
xD = 0.95        # mole fraction benzene in distillate
xB = 0.05        # mole fraction benzene in bottoms
alpha = 2.5      # relative volatility

# TODO: Calculate material balances and operating parameters
# Steps:
# 1. Calculate distillate and bottoms flow rates (D, B)
# 2. Calculate minimum reflux ratio (Rmin)
# 3. Calculate actual reflux ratio (R = 1.5 * Rmin)
# 4. Calculate number of theoretical stages

D = 0
B = 0
Rmin = 0
R_actual = 0
N_theoretical = 0

print(f"Distillate flow: {D:.1f} kmol/h")
print(f"Bottoms flow: {B:.1f} kmol/h")
print(f"Minimum reflux ratio: {Rmin:.2f}")
print(f"Actual reflux ratio: {R_actual:.2f}")
print(f"Theoretical stages: {N_theoretical:.1f}")

# Calculate equilibrium curve points for visualization
x_values = [0, 0.2, 0.4, 0.6, 0.8, 1.0]
y_values = [alpha * x / (1 + (alpha - 1) * x) for x in x_values]

print("\nEquilibrium curve (x, y):")
for x, y in zip(x_values, y_values):
    print(f"  ({x:.1f}, {y:.3f})")

What would be the effect of increasing the reflux ratio? How does relative volatility affect the number of stages required?

ELI10 Explanation

Simple analogy for better understanding

Mass transfer is about moving chemicals from one place to another, like how sugar dissolves in coffee or how a coffee filter separates grounds from liquid. It's the science of how molecules move between different phases - from liquid to gas, from one liquid to another, or from solid to liquid. Chemical engineers use mass transfer principles to design equipment like distillation columns that separate alcohol from water, absorption towers that remove pollutants from air, and extraction systems that pull valuable chemicals from mixtures.

Self-Examination

Q1.

What are the fundamental mechanisms of mass transfer and how do they differ?

Q2.

How do chemical engineers design distillation columns for separation processes?

Q3.

Why are equilibrium relationships important in mass transfer operations?