Diffusion 051333 Unit operation in gro-industry III Department of Biotechnology, Faculty of gro-industry Kasetsart University Lecturer: Kittipong Rattanaporn 1
Outline Definition and mechanism Theory of diffusion Molecular diffusion in gases Molecular diffusion in liquid Mass transfer 2
Definition and mechanism Diffusion is the movement of an individual component through a mixture. (the influence of a physical stimulus) Driving force is a concentration gradient of the diffusing component. concentration gradient tends to move the component in the direction to equalize concentrations and destroy the gradient. 3
Consider 2 gases B and E in a chamber, initially separated by a partition. t some instant in time, the partition is removed, and B and E diffuse in opposite direction as a result of the concentration gradients. 4
Diffusion is the characteristic of many mass transfer operations. Diffusion can be causes by an activity gradient such as by a pressure gradient, by a temperature gradient, or by the application of an external force field. Role of diffusion in mass transfer. Distillation Leaching Crystallization Humidification Membrane separation 5
Theory of Diffusion ssumption Diffusion occurs in a direction perpendicular to the interface between the phases and at a definite location. Steady state. (The concentrations at any point do not change with time) Binary mixtures. 6
Comparison of diffusion and heat transfer Driving force et flux Diffusion Concentration gradient The rate of change in concentration Heat transfer (conduction) Temperature gradient The rate of temperature change. From the similarity, the equation of heat conduction can be adapted to problems of diffusion in solids or fluids. 7
Comparison of diffusion and heat transfer Differences between heat transfer and mass transfer Heat transfer is an energy transition but diffusion is the physical flow of material. Heat transfer in a given direction is based on one temperature gradient and the average thermal conductivity. Mass transfer, there are different concentration gradients for each component and often different diffusivities. 8
Diffusion quantities Velocity, u (length/time) Flux across a plane, (mol/area time) Flux relative to a plane of zero velocity, J (mol/area time) Concentration, c and molar density ρ M (mol/volume) Concentration gradient, dc/dx 9
Molal flow rate, velocity, and flux The total molal flux (mole/time area) ρ M u 0 (1) Where ρ M is the molar density of the mixture and u 0 is the volumetric average velocity. For component and B, the molal fluxes are u c (2) B u B c B (3) 10
The molar flux of component and B J c u c u 0 c (u u 0 ) (4) J B c B u B c B u 0 c B (u B u 0 ) (5) The diffusion flux J is assumed to be proportional to the concentration gradient and the diffusivity of component (D) J D B dc dx (6) J B D B dc B dx (7) 11
Equation 6 and 7 are statements of Fick s first law of diffusion for a binary mixture. The law is based on: The flux is in moles per unit time. The diffusion velocity is relative to the volumeaverage velocity. The driving potential is in the molar concentration. 12
The Fick s law is similar to Fourier s law of heat conduction (8) and ewton s equation for shear-stress-strain relationship (9). q k T x (8) σ µ u y (9) 13
Diffusivity D B is mass diffusivity of component on component B through the mixture. The dimensions of mass diffusivity are length squared divided by time, usually given as m 2 /s or cm 2 /s 14
The magnitude of mass diffusivities for liquids or gases in solid are less than the mass diffusivities for gases in liquids. In solids, the mass diffusivities range from 10-9 to 10-1 cm 2 /s, in liquids, the mass diffusivities range from 10-6 to 10-5 cm 2 /s and for gases, the mass diffusivities range from 5x10-1 to 10-1 cm 2 /s The mass diffusivity magnitudes are a function of temperature and concentrations; in the case of gases the mass diffusivity is substantially influenced by pressure. 15
Molecular diffusion in gas 16
Using Ideal gas law p B ρ B the gas constant R B for gas B can be written in terms of the R B T (10) Universal gas constant R u Ru is the universal gas constant 8.314 (m 3 Pa)/(g-mol K) and M B is molecular weight of gas B. Thus, or ρ ρ B R B B R M u pb R T B pbm R T u B B (12) (13) (11) 17
Since ρ B is mass concentration, substitute equation 13 in equation 6 J J B B pb M DBE x RuT DBE M B p B R T x u B (14) (15) The mass diffusivity D BE refers to diffusivity of gas B in gas E. Similarly, we can express diffusion of gas E in gas B J E DEB M R T u E p x E (16) 18
Steady-State Diffusion of gases ssume the mass diffusivity does not depend on concentration, from equation 6, we obtain D dc dx By separating variables and integrating: J B (17) J J x 2 dx D B x1 c D B ( c 1 c ( x x ) 2 1 c 2 1 2 dc ) (18) (19) 19
For the condition of steady state diffusion, the concentration at the boundaries must be constant with time and diffusion is limited to molecular motion with in the solid being described. The mass diffusivities are not influenced by magnitude of concentration and no temperature gradient. 20
Example 1 Molecular diffusion of Helium in itrogen mixture of He and H2 gas is contained in a pipe at 298 K and 1 atm total pressure which is constant throughout. t one end of the pipe at point 1 the partial pressure p 1 of He is 0.60 atm and at the other end 0.2 m p 2 of is 0.20 atm. Calculate the flux of He at steady state if D B of He- 2 mixture is 0.687 x 10-4 m 2 /s 21
Equimolar Counter diffusion in Gases Molecules of diffuse to the right and B to the left. The total pressure is constant, the net moles of diffusing to the right must equal the net moles of B to the left. J z -J Bz (20) 22
Fick s law for B for constant c, dc B J B DB dz ow P p + p B constant then (21) c c + c B (22) Differentiating both sides dc - dc B (23) 23
Equating Fick s law to (21) J D B dc dz J B ( D dc B ) dz (24) Substitute (23) into (24) and canceling like term D B D B (25) For a binary mixture of and B the diffusivity coeff. D B for diffusing in B in the same as D B for B diffusing into. B 24
Example 2 Equimolar Counterdiffusion mmonia gas () is diffusing through a uniform tube 0.10 m long containing 2 gas (B) at 1.0132 x 10 5 Pa and 298 K. t one end of the pipe at point 1 the partial pressure p 1 is 1.013 x 10 4 Pa and at the other end p 2 of is 0.507 x 10 4 Pa. The diffusivity D B 0.230 x 10-4 m 2 /s. Calculate the flux J and J B at steady state. 25
General case for Gases and B Plus Convection 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 u M (m/s) Component is still diffusing to the right, its diffusion velocity u d is measured relative to the moving fluid. The velocity of relative to the stationary point is the sum of the diffusion velocity and the average or convective velocity 26
u u d u M Mutiply by c u u d + u M (26) c u c u d + c u M (27) Rewrite (27) in terms of flux J + c u (28) 27
Let is the total convective flux of the whole stream relative to the stationary point cu M + B (29) Substitute (30) to (28) u M c c + c B ( ) J + + B (30) (31) 28
When J is Fick s law. dc dz ( ) DB + + (32) Equation 32 is the final generation for diffusion plus convection. For equimolar counter diffusion, B and the convective term in (32) become zero. Then J - B -J B c c B 29
Special case for diffusing through stagnant, ondiffusing B In this case one boundary at the end of the diffusion path is impermeable to component B, so it cannot pass through. For example, the evaporation of organic solvent into air, Gas adsorption into liquid. 30
To derive the case for diffusing in stagnant, nondiffusing B, B 0 is substitute into the general equation (32) dc dz D B + Keeping the total pressure P constant, D RT dp dz B + c c p P (33) (34) 31
Rearranging and integrating, p D dp B 1 P RT dz z2 dz z1 RT D D RT B B P p2 p1 dp p 1 P ln P p 2 ( z z ) 1 P p 1 2 (34) (35) (36) 32
Eq. 36 can write in a form of an inert B log mean value. P BM p B ln( p p / p B2 B1 B1 Substitute 37 into 36 2 ) p 1 ln( P p p 2 / P 2 p 1 ) (37) RT D B P ( z z ) 2 1 P BM ( p p ) 1 2 (38) 33
Example 3 Diffusion of water through stagnant, ondiffusing ir Water in the bottom of a narrow tube is held at a constant temperature of 293 K. The total pressure of air is 1.01325x10 5 Pa (assumed dry air) and the temperature is 293 K. Water evaporates and diffuses through the air in tube and diffusion path is 0.1524 m long. Calculate the rate of evaporation at steady state The diffusivity of water vapor at 293 K and 1 atm is 0.25 x 10-4 m 2 /s The vapor pressure of water at 293 K is 0.0231 atm 34
Molecular diffusion in liquids 35
Diffusion of solutes in liquids is very important in many industrial process, especially in separation process. The rate of molecular diffusion in liquids is slower than in gases because the molecules are very close together compared to gas. In generally, the diffusion coefficient in a gas will be greater than in a liquid but the flux in a gas is not that much greater because the concentrations in liquids are considerably higher than in gases. 36
In diffusion in liquids an important difference from diffusion from gases is that the diffusivities are often quite dependent on the concentration of the diffusing component 37
38 Calculation of diffusion in liquids Equimolar counterdiffusion (39) (40) (41) B ( ) ( ) 1 2 2 1 1 2 2 1 z z x x c D z z c c D av B B 2 2 2 1 1 + M M M c av av ρ ρ ρ
Diffusion of through nondiffusing B n example is a dilute solution of propionic acid in a water solution being contacted with toluene D c z z B av ( x x ) ( ) 1 2 2 1 x BM (42) 39
Where x BM x ln( B2 (43) For dilute solution x BM is close to 1.0 and c is constant D B x B2 / x x B1 B1 ) ( x ) 1 x 2 ( z z ) 2 1 (44) 40
Example4 : Diffusion of Ethanol through Water n ethanol-water solution in the form of stagnant film 2.0 mm thick at 293 K is in contact at one surface with an organic solvent in which ethanol is soluble and water is insoluble. Hence, B 0. t point 1 the concentration of ethanol is 16.8 wt% and the solution density is 978.2 kg/m3. The diffusivity of ethanol is 0.740 x 10-9 m 2 /s. Calculate the steady state flux. 41
Molecular diffusion in solid 42
Rate of diffusion in solids is generally slower than rates in liquids and gases. For example Leaching of food drying of foods Diffusion and catalytic reaction in solid catalyst Separation of fluids by membrane 43
We can classify transport in solids into two types of diffusion Diffusion in solids following Fick s law. Diffusion in porous solids that depends on structure. 44
Diffusion in solids following Fick s law This type of diffusion does not depend on the actual structure of solid, The diffusion occurs when the the fluid or solute diffusing is actually dissolve in the solid to form a more or less homogenous solution. 45
Using the general equation for binary diffusion (Eq. 32) The bulk flow term is usually small, so it is neglected. dc dz D c c ( ) DB + + B dc dz B (45) 46
Integration of equation 45 for a solid slab at steady state D B ( c ) 1 c 2 ( ) (46) For the case of diffusion radially through a cylinder wall inner radius r 1 and outer r 2 and length L, z 2 z 1 2πrL D B dc dr (47) 47
D B 2πL ( c ) 1 c 2 ln ( r r ) (48) The diffusion coefficient is not dependent upon the pressure of gas or liquid on the outside of the solid. The solubility of gas or liquid is directly proportional to p by Henry s law. For gas, Sp kgmol (49) c 22. 4 3 m solid 2 1 48
Diffusion in porous solid that depends on structure The porous solid that have pores or interconnected voids in the solid would effect the diffusion. For the situation where the voids are filled with liquid or gas, the concentration of solute at boundary is diffusing through the solvent in the void volume takes a tortuous path which is greater than z2 z1 by a factor, τ, called tortuous. 49
For a dilute solution, D ε τ ( c ) 1 c 2 ( ) z 2 z 1 Where ε is the open void fraction B (50) τ is the correction factor of the path longer than z. 50
Combined into an effective diffusivity ε D eff D B τ For diffusion of gases in porous solids εd B τ RT ( p ) 1 p 2 ( z z ) 2 1 (51) (52) 51
Mass transfer 52
Mass transfer is the phrase commonly used in engineering for physical processes which involve molecular and convective transport of atom and molecule within physical systems. ref: http://en.wikipedia.org/wiki/mass_transfer 53
There are many difference of mass transfer but the most cases of mass transfer is treating using the same type of equations, which feature a mass transfer coefficient, k m. The mass transfer coefficient k m is defined as the rate of mass transfer per unit area per unit concentration difference (m 3 /m 2 s, m/s). 54
k m J m B ( c c ) ( c c ) B1 B2 B1 B2 (53) The coefficient represents the volume (m 3 ) of component B transported across a boundary of one square meter per second 55
For the relationship of the ideal gas law, the mass transport due to convection become mb km M R T u ( p p ) B1 B2 (54) When the specific application of mass transport is water vapor in air, the equation 54 is used for computing the convective transport of water vapor in air. mb B km M B p 0.622R T u ( W W ) 1 2 (55) 56
The Dimensional analysis for predicting the mass transfer coefficient. The variables are grouped in the dimensionless number. Sh Sc Re Le kmd D B µ ρd p c B ρud µ c k ρc D B (56) (57) (58) (59) 57
58 Consider a fluid flow over a flat plate. for the boundary layer from the leading edge of the plate, we can write the equation for concentration. (60) 2 2 y c D y c u x c u B y x +
From the equation 60, c represents concentration of component at location within the boundary layer. Pr antl number (61) The Prandtl number provides the link between velocity and temperature profile. If µ (62) ρd B then velocity and concentration profile have the same shape. Schmidt number ì ρá Pr 1 Sc µ ρd ìc k B p (63) 59
The concentration and temperature profile will have the same shape if α 1 (64) D B The ratio α Lewis number Le D (65) B The functional relationship that correlated these dimensional number for forced convection are : ( ) f, sh Re Sc (66) Sh total mass transferred total mass transferred by molecular diffusion (67) Sc molecular diffusion of momentum molecular diffusion of mass (68) 60
These correlations are based on the assumption: constant physical properties no chemical reaction in the fluid small bulk flow at the interface no viscous dissipation no interchange of radiant energy no pressure, thermal, or forced diffusion. 61
Laminar Flow Past a Flat Plate When re < 5x10 5 k x 0.322 m, x 1/ 2 1/3 sh, x Re L Sc Sc DB 0.6 (69) Km,x is the mass transfer coefficient at a fixed location. The d c used in the Sherwood and Reynolds number is the distance from the leading edge of the plate. k x 0.664 m, L 1/ 2 1/3, Sc sh L Re L Sc DB (70) when flow is laminar over the entire length of the plate. The d c is the total length of the plate.the mass transfer coefficient is the average value for the entire plate. 0.6 62
Turbulent Flow Past a Flat Plate When re > 5x10 5 k x m, x 4/5 1/3 sh, x 0.0296Re, x Sc 0.6 < Sc < 3000 (71) DB The characteristic dimension (d c ) used in the Sherwood and Reynolds number is the distance from the leading edge of the plate. km, Lx sh L 0.036 D, 0.8 Re 0.33 Sc B. The characteristic dimension (d c ) is the total length of the plate. (72) 63
Laminar Flow in a Pipe k d D Re, d Sc 1.86 Re / L dc 10,000 Where the characteristic dimension, d c, is the diameter of the pipe. 1/3 m c Sh, d < B Turbulent Flow in a Pipe (73) k d 0.8 1/3 0.023 Re m c, d > 10,000 Sh d Re, Sc DB (74) Where the characteristic dimension, d c, is the diameter of the pipe. 64
Mass Transfer for Flow over Spherical objects 1/ 2 Sh d 2.0 + (0.4 + 0.06 2/3, Re, d Re, d ) 0.4 Sc (75) For mass transfer from a freely falling liquid droplet Sh d 2.0 + 0.6 1/ 2, Re, d ) 1/3 Sc (76) 65
The end. 66