Part I bblee@unimp
. Introduction to Mass Transfer and Diffusion 2. Molecular Diffusion in Gasses 3. Molecular Diffusion in Liquids Part I 4. Molecular Diffusion in Biological Solutions and Gels 5. Molecular Diffusion in Solids 6. Unsteady State Diffusion Part II bblee@unimp 2
7. Convection Mass Transfer Coefficients 8. Mass Transfer Coefficients for various geometries 9. Mass Transfer to Suspensions of Small Particles Part II Part III bblee@unimp 3
. Introduction to Mass Transfer and Diffusion. Fick s law for molecular diffusion 2. Molecular Diffusion in Gasses 2. Equimolar counterdiffusion in gases 2.2 General case for diffusion of gases & B plus convection 2.3 Special case for diffusing through stagnant, nondiffusing B bblee@unimp 4
2.4 Diffusion through a varying crosssectional area 2.5 Diffusion coefficient for gases 3. Molecular Diffusion in Liquids 3. Introduction 3.2 Equation for diffusion in liquids 3.3 Diffusion coefficients for liquids 3.4 Prediction of diffusivities in liquids bblee@unimp 5
Mass transfer occurs: Water evaporates into still air. Sugar dissolves & diffuses to the surrounding solution. bblee@unimp 6
Mass transfer occurs: Liquid-liquid extraction Crystallization Distillation bblee@unimp Drying dsorption 7
Molecular diffusion (transport): the transfer or movement of individual molecules through a fluid by means of the random, individual movements of the molecules. bblee@unimp 8
Low concentration bblee@unimp High concentration Figure 6.-: Schematic diagram of molecular diffusion process. 9
Consider: the diffusion of molecules when the whole bulk fluid is not moving but is stationary. due to a concentration gradient. The Fick s law equation: J * Z cd Total concentration of &B (kg mol/m 3 ) B dx dz bblee@unimp 0 The mole fraction of in mixture of & B Distance, m The molecular diffusivity of the molecule in B, m 2 /s
If c is constant, then c =cx, cdx = d(cx )=dc For constant total concentration: J * Z D B dc dz bblee@unimp
Example 6.- bblee@unimp 2
Example 6.- bblee@unimp 3
Example 6.- bblee@unimp 4
2. Equimolar Counterdiffusion in Gases Two gases & B at constant total pressure P in two large chambers connected by a tube where molecular diffusion at steady state is occurring. Figure 6.2-: Equimolar counterdiffusion of gases and B bblee@unimp 5
Stirring in each chamber keeps the concentrations in each chamber uniform. The partial pressure p >p 2 and p B2 >p B. Molecules of diffuse to the right and B to the left. Since the total pressure P is constant throughout, the net moles of diffusing to the right must equal to the net moles of B to the left. bblee@unimp 6
This means that Writing Fick s law for B for constant c, J * B D Now since P=p +p B = constant, then * J J B dc B dz * B c c c B Differentiating both sides, bblee@unimp dc=0, dc dc dc B dc dc B 7
Equating the equations, J * Finally, D B dc dz )D dc dz This shows that for a binary gas mixture of & B, the diffusivity coefficient D B for diffusing into B is the same as D B for B diffusing into. J * B D D B B ( B B Molecular diffusivity bblee@unimp 8
Example 6.2- bblee@unimp 9
Example 6.2- bblee@unimp 20
The diffusion flux J* occurred because of the concentration gradient. The rate at which moles of passed a fixed point to the right (positive flux). This flux can be converted: J * (kgmol / s.m 2 ) m kgmol m where v D is the diffusion velocity of (m/s). v D c s 3 bblee@unimp 2
Consider when the whole fluid is moving in bulk or convective flow to the right. The molar average velocity of the whole fluid relative to a stationary point is v M m/s. Component is still diffusing to the right, but now its diffusion velocity, v d is measured relative to the moving fluid. bblee@unimp 22
To a stationary observer is moving faster than the bulk of the phase, since its diffusion velocity v d is added to that of the bulk phase v M. The velocity of relative to the stationary point is the sum of the diffusion velocity & the average or convective velocity: Where v velocity of relative to a stationary point. bblee@unimp v v d v M 23
v v d v M Multiplying by c, c v N (kgmol /s,m 2 ) c v d c v M Let N be the total convective flux: bblee@unimp N J* cv M N N B 24
So, Since J* is Fick s law, N N B N cd cd J B B * dx c c dz dx B dz N c c c B c N N N B Equimolar counterdiffusion N N B B N = -N B bblee@unimp 25
In the evaporation of a pure liquid (e.g. benzene () at the bottom of a narrow tube, where a large amount of inert or nondiffusing air (B) is passed over the top. Figure 6.2-2a: Diffusion of through stagnant, nondiffusing B: (a) benzene evaporating into air bblee@unimp 26
The benzene vapor () diffuses through the air (B) in the tube. The boundary at the liquid surface at point is impermeable to air, since air is insoluble in benzene liquid. ir (B) cannot diffuse into or away from the surface. t point 2 the partial pressure p 2 =0, since a large volume of air is passing by. bblee@unimp 27
In the absorption of NH 3 () vapor which is in air (B) by water. The water surface is impermeable to the air, since air is only very slightly soluble in water. Figure 6.2-2: Diffusion of through stagnant, nondiffusing B. (b) ammonia in air being absorbed into water. bblee@unimp 28
thus, since B cannot diffuse, N B = 0. Keeping the total pressure P constant, substituting P c p x RT bblee@unimp N cd B c c dx dz p P c c N 0 Convective flux of P 29
bblee@unimp 30 Then, B N P p dz dp RT D N dz dp RT D P p N B 2 2 p p B z z P p dp RT D dz N 2 2 B p P p P ln z z RT P D N
bblee@unimp 3 log mean value of the inert B is defined: Then, 2 2 B B p p p p P B p P p 2 2 B p P p 2 2 2 2 B B B B BM p P p P ln p p p p ln p p P 2 2 BM B p p p z z RT P D N
EXMPLE 6.2-2 bblee@unimp 32
EXMPLE 6.2-2 bblee@unimp 33
EXMPLE 6.2-3 bblee@unimp 34
EXMPLE 6.2-3 bblee@unimp 35
So far, the cross-sectional area m 2 through which the diffusion occurs has been constant with varying distance z. In some situations the area may vary. N N kg moles of / s N t steady state, will be constant but not for a varying area. bblee@unimp 36
2.4. Diffusion from a sphere The evaporation of a drop of liquid, the evaporation of a ball of naphthalene, and the diffusion of nutrients to a sphere-like microorganism in a liquid. Figure 6.2-3a: sphere of fixed radius, r (m) in an infinite gas medium. bblee@unimp 37
N N N 4πr 2 Since this is a case of diffusing through stagnant, nondiffusing B: N N 4πr 2 DB RT dp p P dr Note that dr was substituted for dz. Integrating between r and some point r 2 a large distance away: bblee@unimp 38
N N 4π r r 2 r dr r r 2 >>r, /r 2 0. 2 D D B RT P p p 2 ln P dp B 2 4π r2 RT P p p p P dr 4 N D p p N B 2 πr 2 RTr p BM bblee@unimp 39
If p is small compared to P (a dilute gas phase), p BM P, 2r =D (diameter), c =p /RT N 2D D B 2 This equation can be used for liquids, where D B is the diffusivity of in the liquid. c c bblee@unimp 40
Example 6.2-4 bblee@unimp 4
Example 6.2-4 bblee@unimp 42
If the sphere is evaporating, the radius r of the sphere decreases slowly with time. The time it takes for the sphere to evaporate completely can be derived by assuming pseudo-steady state and by equating the diffusion flux equation, where r is now a variable, to the moles of solid evaporated per dt time and per unit area as calculated from a material balance. bblee@unimp 43
Density of the sphere t F Original sphere radius p r 2 RTp BM 2M D P p p B 2 Molecular weight bblee@unimp 44
2.5. Experimental determination of diffusion coefficients To evaporate a pure liquid un a narrow tube with a gas passed over the top (see Fig 6.2-2a). The fall in liquid level is measured with time and the diffusivity calculated: D B p r 2 RTp BM 2M t P p p F 2 bblee@unimp 45
The common method is the two-bulb method (N). Figure 6.2-4: Diffusivity measurement of gases by the two-bulb method. Pure gas is added to V and pure B to V 2 at the same pressures. bblee@unimp 46
The valve is opened, diffusion proceeds for a given time, and then the valve is closed and the mixed contents of each chamber are sampled separately. ssumptions: Neglecting the capillary volume & assuming each bulb is always of a uniform concentration. bblee@unimp 47
Quasi-steady-state diffusion in the capillary, dz L The rate of diffusion of going to V 2 is equal to the rate of accumulation in V 2 : bblee@unimp J * J * D B D Concentration of in V 2 at time t dc B c2 L c D B c 2 V Concentration of in V at time t 2 c dc dt 2 48
The average value c av at equilibrium can be calculated by a material balance from the starting composition c 0 and c 20 at t=0: V V c V c 0 V c 0 2 av 2 2 similar balance at time t gives, V V c V c V c 2 av 2 2 bblee@unimp 49
Rearranging & integrating between t=0 & t=t, c c av av c c 2 0 2 exp D B V L V V 2 V 2 if c 2 is obtained by sampling at t, D B can be calculated. bblee@unimp 50
Some typical data are given in Table 6.2-, Perry & Green (984) and Reid et al., (938). Table 6.2-: Diffusivity coefficient of gases at 0.32 kpa Pressure bblee@unimp 5
The diffusivity of a binary gas mixture in the dilute gas region, that is, at low pressure near atmospheric, can be predicted using the kinetics theory of collision with another molecule, which implies that momentum is conserved. bblee@unimp 52
The final relation for predicting the diffusivity of a binary gas pair of & B molecules is: D B. 8583x0 Pζ Ω 2 B 7 T D,B 3 / 2 M M B / 2 verage collision diameter collision integral based on the Lennard-Jones potential bblee@unimp 53
The above equation is relatively complicated to use (σ B is not available or are difficult to estimate). D B. 00x0 p 7 Σν T. 75 / Sum of structural volume increments 3 M Σν B / M 3 B 2 / 2 Note: D B α /P, D B α T.75, D B α T.75 /P bblee@unimp 54
Table 6.2-2: tomic diffusion volumes for use with Fuller et al., method. bblee@unimp 55
Example 6.2-5 bblee@unimp 56
Example 6.2-5 bblee@unimp 57
3. Introduction Diffusion of solutes in liquids is important in many industrial processes: Solvent extraction bblee@unimp Gas absorption Distillation 58
Diffusion in liquids also occurs in nature: Oxygeneration of rivers and lakes by the air Diffusion of water in blood bblee@unimp 59
It should be apparent that the rate of molecular diffusion in liquids is considerably slower than in gases. The molecules of the diffusing solute will collide with molecules of liquid B more often and diffuse more slowly than in gasses. The diffusion coefficient in a gas will be on the order of magnitude of about 0 5 times greater than in a liquid. bblee@unimp 60
The flux in a gas is being only 00 times faster, since the concentrations in liquids are considerably higher than in gases. Since the molecules in a liquid are packed together much more closely than in gases, the density & resistance to diffusion in a liquid are much greater. the attractive forces between molecules play important role in diffusion. bblee@unimp 6
Diffusion of liquids: the diffusivities are often dependent on the concentration of the diffusing components. 3.2. Equimolar counterdiffusion n equation similar to gases at steady state [N =-N B ]: N bblee@unimp D B z c 2 Kg mol /s.m 2 m 2 /s z c 2 D B c Concentration of (kg mol /m 3 ) at point. av z 2 x z x 2 Mole fraction of at point. 62
c av defined by ρ cav M verage total concentration of +B (kg mol/m 3 ) The average value of D B may vary some with concentration, & the average value of c may vary with concentration Linear average of c is usually used. bblee@unimp av ρ M ρ M verage molecular weight of the solution at point (kg mass / kg mol) 2 2 2 verage density of the solution (kg/m 3 ) 63
dilute solution of propionic acid () in a water (B) solution being contacted with toulene. Only the propionic acid () diffuses through the water phase (B), to the boundary & then into the toluene phase. The toulene-water interface is a barrier to diffusion of B and N B = 0. Substituting c av bblee@unimp P RT c ρ RT x BM p P BM 64
For liquids at steady state: N z 2 D B c z av x BM xb2 xb xbm ln( xb2 xb ) Note x +x B =x 2 +x B2 =.0 For dilute solution, x BM is close to.0 c is essentially constant. N D B z x c 2 z x c 2 2 bblee@unimp 65
Example 6.3- bblee@unimp 66
Example 6.3- bblee@unimp 67
3.2.3. Experimental determination of diffusivities relatively dilute solution & a slightly more concentrated solution are placed in chambers on opposite sides of a porous membrane of sintered glass (see Fig6.3-). Figure 6.3-: Diffusion cell for determination of diffusivity in a liquid bblee@unimp 68
Quasi-steady-state diffusion in the membrane is assumed: Concentration in the lower chamber at a time, t Fraction of area of the glass open to diffusion Concentration in the upper chamber N εd B c ηδ c' bblee@unimp 69
Combining & integrating: Initial concentrations ln c c 0 c c' 2 ' 0 ε ηδv D B t Final concentrations bblee@unimp Cell constant Note: For liquids, unlike gases, the diffusivity D B does not equal D B. 70
Table 6.3-: experimental diffusivity data for binary mixtures in the liquid phase are given. The diffusivity values are quite small and in the range of about 0.5x0-9 to 5x0-9 m 2 /s for relatively nonviscous liquids. Diffusivities in gases are larger by a factor of 0 4-0 5. bblee@unimp 7
Table 6.3-: experimental diffusivity data for binary mixtures in the liquid phase are given. bblee@unimp 72
The Stokes-Einstein equation was derived for a very large spherical molecule () diffusing in a liquid solvent (B) of small molecules. Stokes law was used to describe the drag on the moving solute molecule. Diffusivity, m 2 /s bblee@unimp D B Viscosity, Pa.s or kg/m.s 9. 96x0 μv / 3 6 T K Solute molar volume at its normal boiling point, m 3 /kgmol 73
The Wilke-Chang correlation can be used for most general purposes, where the solute () is dilute in the solvent (B). D B ssociation parameter of the solvent bblee@unimp. 73 x0 6 θm B Molecular weight of solvent B Viscosity of B, Pa.s or kg/m.s / 2 μ B T V 0. 6 Solute molar volume at the boiling point (Table 6.3-2) 74
The association parameter (φ): φ Water 2.6 Benzene.0 Methanol.9 Ether.0 Ethanol.5 Heptane.0 Unassociated solvents φ.0 bblee@unimp 75
Example 6.3-2 bblee@unimp 76
Example 6.3-2 bblee@unimp 77