Molecular Diffusion Principles

• Molecular diffusion involves substance displacement in a molecular state, either through an immobile medium towards decreasing concentration or in laminar motion perpendicular to flow direction.
• Molecular diffusion can occur with other transfer mechanisms, its significance determined by the flow regime.
• A medium consists of at least two components that move and diffuse at different rates.
• Total fluxes (N) are relative to a stationary reference, while fluxes (J) relate to a reference moving at the local molar (u) or massic (v) average speed of the fluid.
• The first is used for engineering calculations (equipment sizing), the second to determine the importance of molecular diffusion when in the presence of other transfer mechanisms
• A component i has a diffusion velocity only if there is a concentration gradient, regardless of the coordinate system.
• Mass flux represents the amount of a component (in kilomoles or kilograms) passing per unit time through a unit area perpendicular to the vector direction.
• Molar flux for molecular diffusion relative to average local molar speed, following Fick's first law: J A,z = -D A,B * dC A / dz; DA,B is the molecular diffusion coefficient of component A in B; dC A /dz is the concentration gradient in the z direction.
• The negative sign shows diffusion occurs toward decreasing concentrations.
• For component B, the equation is J B,z = -D B,A * dC B / dz.
• Because J A,z and J B,z relate to the same average molar speed, J A,z + J B,z = 0.
• Molecular diffusion coefficients have the same measurement unit m²/s.
• Molecular diffusion coefficient values depend on temperature, pressure, and mixture composition, and differ (by order of magnitude) in gases, liquids, and solids.
• Standard practice is to use experimentally determined diffusion coefficient values found in specialized monographs. Calculation formulas derived theoretically are used when experimental data is unavailable.
• Molecular diffusion coefficients in the gaseous phase range from (0.1 - 1)×10⁻⁴ m²/s.
• Diffusion coefficients in gases do not depend on concentration at low pressures (= 1 atm).
• The Hirschfelder-Bird-Spotz relation can be used to accurately calculate the molecular diffusion coefficient in gases: D A,B = (1.858 * 10⁻⁷ * T^(3/2) * ((1/M_A) + (1/M_B))^(1/2) ) / (P * σ²_AB * Ω D)
  • T is the mixture temperature in Kelvin, MA and MB represent the molar masses, P is absolute pressure in atm, and σAB is the collision diameter in Ångströms.
  • Ω D represents the collision integral, dependent on temperature and the intermolecular potential between molecules A and B.

Empirical Relations

• For nonpolar binary mixtures the following empirical relations can be used:
  • σ A,B = 0.5(σ A + σ B )
  • ε A,B / x = (ε A / x * ε B / x)^0.5
• The molecular diffusion coefficient relies on temperature and pressure, per the relation: (D A,B ) T2,P2 = (D A,B ) T1,P1 (P1 / P2 ) * (T2 / T1 )^1.5; This is valid below 25 atm.
• The Winkelmann method is used for experimentally determining the diffusion coefficient in the gaseous phase

Winkelmann Method

• Liquid surface level shifts over time from the vaporization of liquid A, diffusing through an insoluble, inert gas (B) layer in the tube.
• The instantaneous flux can be written as N A,z = C * D A,B * (Y A,1 - Y A,2 ) / (z * Y B,m )
  • Where z is the diffusion path length at a given time.
• Molar flux N A,z can be represented as a function of the liquid level's time variation dz/dt as N A,z = (ρ A * dV) / (S * M A * dt) = (ρ A * S * dz) / (S * M A * dt) = (ρ A / M A ) * (dz / dt)
  • Where ρ A is the liquid's density in kg/m³.
• Combining the equations, ρ A / M A * dz/dt = C * D A,B * (Y A,1 - Y A,2 ) / (z * Y B,m )
• Integrated between initial and final moments: ∫[0 to t] dt = (ρ A * Y B,m ) / (M A * C * D A,B * (Y A,1 - Y A,2 )) ∫[z0 to z] zdz
  • t = (ρ A * Y B,m ) / (M A * C * D A,B *(Y A,1 - Y A,2 )) * (z² - z0²) / 2
  • D A,B = (ρ A * Y B,m ) / (M A * C * t * (Y A,1 - Y A,2 )) * (z² - z0²) / 2
• Using the relation, the diffusion coefficient value can be calculated using experimentally obtained values for z and z0.

Concentration

• In relation (14), Y B,m represents the logarithmic mean concentration of component B Y B,m = (Y B,2 - Y B,1 ) / ln(Y B,2 / Y B,1 )
  • Y B,1 = 1 - Y A,1 and Y A,1 represents the mole fraction of component A at the liquid surface, determined via Raoult's law.
  • Y B,2 = 1 because the air current does not contain component A.
  • C represents molar density, in moles mixture/m3 mixture.

Other Equations

• More precise values can be obtained with multiple readings at variable times, introduced into relation (12), resulting in:
  • z1² - z0² = (z1 - z10) * (z1 - z10 + 2z10) = (2M A * D A,B * C * t * (Y A,1 - Y A,2 )) / (ρ A * Y B,m ) Combining,
  • t / (z1 - z10) = (ρ A * Y B,m ) / (2 * M A * C * D A,B * (Y A,1 - Y A,2 )) * (z1 - z10) + (ρ A * Y B,m * z10) / (M A * C * D A,B * (Y A,1 - Y A,2 ))
• From this, a line equation is derived (y = mx + n).
  • The diffusion coefficient can be derived from the slope of the line.
  • D A,B = (ρ A * Y B,m ) / (2 * M A * C * (Y A,1 - Y A,2 ) * m)

Experimental Setup

• The scope of the work is to determine molecular diffusion coefficients for ethyl ether and acetone in air.
• Experimental values will be compared with calculated values and literature values.
• The experimental setup consists of two graduated test tubes (1 and 2) containing the two liquids.
• Air circulates through tubes 3 and 4.
• The setup can be connected to a vacuum or use a fan for air transport.

Method

• Introduce the two liquids into graduated tubes 1 and 2, noting the levels, and ensure air circulation through tubes 3 and 4.
• It involves starting a timer to track the liquid level variation in the tubes and taking readings every 15-20 minutes, for 5-6 pairs of experimental data. The operating temperature is measured.