Mass Transfer: Diffusion & Convection

Explore diffusion and Dumaine law as the main model with Keef's fix law, review diffusion concepts, and examine molecular diffusion and steady-state gas diffusion with practice problems.

Reviewing properties reveals density, concentration, and molar concepts, clarifying mass and molar fractions. Relate density, velocity, and area to flux and flow rates, including ideal gas law notes.

Review the log mean function and its use to express the difference between two points with a logarithmic mean, especially for temperature differences in mass transfer.

Review the continuity equation in mass transfer, highlighting divergence, dimensions, and one-dimensional simplifications, with generation terms and connection to the fixed law.

Define flux as the rate of species transport per unit area normal to transport, linked to concentration gradients in a fixed reference frame, distinguishing mass transport from momentum transport.

Explore how flux J is defined with a reference velocity, turning concentration times velocity into a diffusion term and showing how velocity differences reveal the actual diffusion in pipe flow.

Compute the average molar velocity in a binary system by weighting each species' velocity with its concentration, and relate this to the total flux and the diffusion reference frame.

Introduce molecular diffusion and distinguish it from mass transport and flux, highlighting diffusion as a phenomenon driven by concentration gradients. Clarify that diffusion involves momentum transport not bulk mass flow.

Explore molecular diffusion as the movement of molecules driven by concentration gradients, influenced by temperature and pressure, with examples like dye in water and bromine diffusion.

Relate flux to the driving force using Fick's law, linking diffusion to the concentration gradient and diffusivity in a one-dimensional, binary system.

Explore the full form of Fick's law, including diffusion, convection, and generation. Learn how steady state collapses time derivatives to the one-dimensional flux J_A = D ∂C_A/∂x.

We introduce the diffusivity constant in Fick's law, linking diffusion flux to the concentration gradient, and note its dependence on temperature, phase, concentration, and pressure.

This lecture proves Dab = Dba in a binary system by applying Fick's law, analyzing A and B fluxes under a concentration-driven driving force, and showing coefficient symmetry.

Explore applications of diffusivity in engineering, including steady-state and counter-diffusion in pipes, spheres, and catalysts, with emphasis on liquids and solids.

Explore diffusion through moving bulk fluid, coupling molecular diffusion with bulk flow in a one-dimensional steady binary system to predict total mass flux.

Case A explores equimolar counter-diffusion of gases A and B, where fluxes balance oppositely; derive diffusion equation from Fick's law and the ideal gas law using partial pressures and concentration.

Analyze diffusion of ammonia and nitrogen between two connected tanks at equal temperature and total pressure. Derive steady-state fluxes using ideal gas assumptions and partial pressures, showing JA = -JB.

Study unimolecular diffusion in a binary gas where A diffuses into stagnant B, using Stefan flow and Fick's law with partial pressures or mole fractions.

Analyze a steady-state diffusion problem where oxygen diffuses through carbon dioxide, compute molar flux per unit area from partial and total pressures, and estimate gas loss through a leak.

Explore additional exercises on diffusion and diffusivity under varying temperature and pressure conditions. Review steady-state dilution covered in sections 3 and 4 for deeper practice.

Explore diffusion modeling across gas, liquid, and solid phases, linking driving forces like concentration, molar fraction, and partial pressure to diffusivity, and discuss models and experiments to estimate it.

Explore diffusion in a binary, one-dimensional, steady-state system and introduce the diffusivity constant as the proportionality factor in Fick's law, including its negative flux and gas–liquid–solid diffusivity contrasts.

Explore how to obtain diffusivities (Dab) using reference tables, correlations, and experimentation; reverse engineer equations to solve for Dab in gas, liquid, and solid diffusion.

Diffusivity in gases governs diffusion for gas–gas and gas–liquid. It rises with temperature roughly as a power of 1.5–1.75 and falls with pressure, driven by collisions and path length.

Identify diffusivity coefficients in literature by consulting reputable books, reliable online databases, and technical reports, then verify units, temperature, and pressure before applying.

Explore empirical estimations to determine diffusivity coefficients in the gas phase using mathematical models based on experimental data, including the Chapman and Scott equation, with adjustments for temperature and pressure.

Apply the Fuller–Schettler–Giddings empirical equation to estimate gas–gas diffusivity in binary systems at low pressures, accounting for molecular size, temperature, and pressure adjustments.

Recognize that higher temperature raises diffusivity because kinetic energy increases molecular movement. Apply a non-linear power-law with exponent around 1.75, using D2 = D1 × (T2/T1)^1.75, to adjust diffusivity.

Apply the Takahashi model to high pressure gases. Use reduced temperature and reduced pressure derived from critical temperature and critical pressure to correlate diffusivity.

Estimate gas diffusivity using the twin-bulb experimental setup, tracking partial pressures over time to model equimolar counter-diffusion and compute diffusivity from geometry and timing.

mass transfer course exercise uses the twin-bulb method to determine CO2 diffusivity in nitrogen, using final pressures after six hours and geometry data to apply the diffusion equation.

Explore the Stefan tube method for experimentally estimating the diffusivity of water vapor diffusing into air, linking distance change, time, and partial pressures to compute diffusion constants.

Study the diffusivity in liquids, focusing on the diffusivity coefficient for liquid–liquid interactions, with temperature as a key driver and pressure largely irrelevant. Explore acetone–water and alcohol–water diffusion examples.

Explore empirical estimations for liquids in mass transfer, featuring the Wu-Tang equation and the haiduk and meanness equation, covering both dilute and non-dilute materials.

Apply the Stokes-Einstein relation to connect diffusivity with fluid viscosity and temperature, and learn how to relate temperatures and viscosities to solve for diffusivity.

Apply the Wilke-Chang equation to liquid-liquid diffusion beyond 10% composition, accounting for temperature, viscosity, solvent molecular weight, and molar volume at the normal boiling point.

Explain the diaphragm cell method for diffusion between two liquids, showing acetone transfer from high to low concentration and deriving time-dependent concentrations using volume, area, porosity, and diffusion coefficients.

Define porosity as the fraction of voids in a material, between 0 and 1 or a percentage, and show that higher porosity increases pore volume so water fills pores.

Examine tortuosity as the ratio of the total path length to the straight-line distance, showing how many turns slow diffusion and shape mass transfer in diffusion and convection.

This conclusion summarizes mass transfer concepts, diffusion coefficients, and convective mass transfer in laminar flow, with steady-state solutions and interfacial mass transfer highlighted through engineering correlations.