Unit 2 Diffusion Mass Transfer (I) PDF

Topic 2 Diffusion Mass Transfer AECH 4210 Mass Transfer & Separation Processes  Lesson Outline ▪ Mass transfer principles ✓ Fick’s law of diffusion ✓ Mass transfer coefficients ▪ Diffusion in gases ✓ Equimolar counter diffusion ✓ General case of...

Topic 2 Diffusion Mass Transfer AECH 4210 Mass Transfer & Separation Processes  Lesson Outline ▪ Mass transfer principles ✓ Fick’s law of diffusion ✓ Mass transfer coefficients ▪ Diffusion in gases ✓ Equimolar counter diffusion ✓ General case of diffusion and convection  Lesson Outcome At the end of the lesson, the students are able to: ✓ Explain the basics of mass transfer processes and Fick’s law of diffusion. ✓ Apply the principle of diffusion for flux estimation in equimolar counter diffusion processes and general case of diffusion and convection mass transfer in gases.  Mass transfer principles (Recap) Mass transfer Transport of one component from a region of higher concentration to lower concentration. Migration of a substance through another under the influence of concentration gradient. Involves the diffusion transport of some component within a single phase or between phases and remain there. Dye molecules spread throughout the water in a random fashion  Molecular Diffusion (Recap) Random movement (in liquids, called Brownian motion) Molecules of one species (A) moving through a stationary medium of another species (B) 5  Brownian Diffusion (Recap) c Molecule A Molecule B c Figure 1.1: Molecules A and B confined in a box at constant pressure and temperature Molecular diffusion of A in the direction of decreasing concentration (i.e. from left to right) Net diffusion of B in the opposite direction Diffusion continues until the concentration is uniform throughout 6  Fick’s Law of molecular Diffusion Molecular diffusion: defined as the transfer or movement of individual molecules through a fluid by means of the random, individual movements of the molecules. 🞄 This random movement of the molecules is often called a random walk process Example: If there are a greater number of A molecules near point (1) than at (2), then, since molecules diffuse randomly in both directions, more A molecules will diffuse from point (1) to (2) than from point (2) to (1). The net diffusion of A is from high to low concentration region.  Fick’s Law of molecular Diffusion Mass transfer is characterized by the general form of transport equation. Driving force Rateof transfer process = Resistance The flux is the amount or concentration of solute carried by a fluid past a plane perpendicular to direction of flow or velocity. The unit of flux is given kg mol/m2s (SI system) or g mol/cm2s (cgs system) Flux Examples of molecular diffusion flux  Fick’s Law of molecular Diffusion The basic law of diffusion called Fick’s Law by Adolf Eugen Fick in 1885 stated “ the molar flux of a species relative to an observer moving the molar average velocity is proportional to the concentration gradient of the species” If A diffuses in a binary mixture of A and B, then the diffusion flux of A is 1 J Az dc and J Az  dz dx A  J A Z = − cD A B dz let c is the total concentration of A and B in kmol (A+B)/m3, xA the mole fraction of A in a mixture of A and B. Since c is constant, then cxA =cA  cdx A = d(cx A ) = dc A  Fick’s Law of molecular Diffusion ⚫ Substituting in to Eq.(2.3) dc A J A = −D AB dz ⚫ Where D AB is the proportionality constant called the “diffusion coefficient” or the “ diffusivity” of A in a mixture of A and B. ⚫ Equation (2.5) is the mathematical representation of Fick’s law of diffusion in a binary mixture. ✓ It is often called mass transfer equation.  Fick’s Law of molecular Diffusion Similarities of the three transport equations ⚫ The three molecular transport process (momentum, heat and mass) are characterized by the same general type of equation as:  z = − d  (2.6) dz 1. Molecular diffusion equation of momentum for constant density, (Newton’s law of viscosity)  (dux  )  = − zx  dz 2. Fourier’s law of heat conduction for constant and cp d ( c p T ) q x = −A dz 3. Fick’s law of diffusion for constant total concentration, c dc A J Az = − D A B dz EXAMPLE 1: Molecular Diffusion of Helium in Nitrogen R is 8.2057x10-5 m3·atm/mol·K 𝑝𝐴1 = 0.6 𝑎𝑡𝑚 𝑝𝐴2 = 0.2 𝑎𝑡𝑚 12 EXAMPLE 1: Molecular Diffusion of Helium in Nitrogen R is 8.2057x10-5 m3·atm/mol·K 𝑝𝐴1 = 0.6 𝑎𝑡𝑚 𝑝𝐴2 = 0.2 𝑎𝑡𝑚 13 EXAMPLE 1: Molecular Diffusion of Helium in Nitrogen R is 8.2057x10-5 m3·atm/mol·K 𝑝𝐴1 = 0.6 𝑎𝑡𝑚 𝑝𝐴2 = 0.2 𝑎𝑡𝑚 14 EXAMPLE 1: Molecular Diffusion of Helium in Nitrogen R is 8.2057x10-5 m3·atm/mol·K 𝑝𝐴1 = 0.6 𝑎𝑡𝑚 𝑝𝐴2 = 0.2 𝑎𝑡𝑚 15  Diffusion in Gases Molecular diffusion in gases can be taken place in different ways as: 1. Equimolar counter-diffusion in gases 2. General case for diffusion of gases A and B plus convection 3. Special case for A diffusing through stagnant, non- diffusing B 4. Diffusion through varying cross-sectional area  1. Equimolar counter-diffusion ⚫ Consider: ◦ Two gases A and B at constant total pressure (P) in two large chambers connected by a tube and molecular diffusion is occurring at steady state. ◦ Partial pressures: pA1 > pA2 and pB2 > pB1 – Let molecules A diffuse to the right and molecules B diffuse to the left and P constant throughout the system p A1  p A2 Fig 19.1-1 pB2  p B1  1. Equimolar counter-diffusion… The moles A diffusing to right is equal to Moles of B diffusing to the left since total pressure P is constant, J Az = −J Bz (2.11) Fick’s law for B for constant total concentration c, dcB (2.12) JBz = −DBA dz Since total pressure P is constant, then c = c A + cB dc A = −dcB (1.13) Equating Eq.(2.5) and (2.12) dcA dc J Az = −D AB = −J Bz = −(−)DBA B (2.14) dz dz  1. Equimolar counter-diffusion… Substituting (2.13) into (2.14) and canceling DAB = DBA (2.15) i.e. for binary gas mixture of A and B, the diffusion coefficient (diffusivity) DAB for A diffusing into B is the same as DBA for B diffusing into A EXAMPLE 2: Equimolar Counterdiffusion Fig 19.1-1 20 EXAMPLE 2: Equimolar Counterdiffusion Fig 19.1-1 a) 21 b) 22 CLASS ACTIVITY: R is 8.314 m3·Pa/mol·K 23 www.udst.edu.qa