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by: Jaylan Rath


Jaylan Rath
Texas A&M
GPA 3.77

Qi Ying

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Qi Ying
Class Notes
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This 29 page Class Notes was uploaded by Jaylan Rath on Wednesday October 21, 2015. The Class Notes belongs to CVEN 619 at Texas A&M University taught by Qi Ying in Fall. Since its upload, it has received 47 views. For similar materials see /class/226104/cven-619-texas-a-m-university in Civil Engineering at Texas A&M University.

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Date Created: 10/21/15
Lecture 2 Single Phase Transport 1 Diffusion Q CVEN619 Spring 2009 1272009 Different Transport Mechanisms 0 Advection Convection Transport of substance by directional movement of a uid phase 0 Diffusion Transport of substance by random motion of the substance molecule 0 and the concentration gradient of the substance ordinary diffusion 0 and the temperature gradients of the substance thermal diffusion 0 and the pressure differences experiences by the substance pressure diffusion 0 Turbulent Diffusion Transport of material due to turbulent motion of the fluid 0 Dispersion Spreading of mass from highly concentrated areas to less concentrated areas due to non uniformity in the flow field CVEN619 Spring 2009 1272009 j Diffusion Examples 0 Mlxmg oftyvo gases 39 n of aqmmnlar dj mmu of wiutequot and black gases in a clnsed n Figure 6 Ilu T39 I39mm ch l0 mg I m 3 pmth 0 Spreading of group of gas molecules Gannowewew Spring 2009 In Random c F 1272009 ma dwsmbutlon mouons dlsmbuuun Simplified Fick s First Law 0 The Fick s First Law used in undergraduate textbook typically looks like this JDdC dx ie the diffusive flux is proportional to its concentration gradient In fact this is an approximation of a more general form of the Fick s First Law as we Will see in our following discussions K9 CVEN619 Spring 2009 1272009 j Average Velocity 0 In a multi cornponent system various species will move at different velocities 0 Average velocity can be mass based or molar based p1u1p2u2 pmum p1p2 pm unfgu2uawumZN1N5NNm qgm CVEN619 Spring 2009 n1n2nm p 1 C 1272009 j Diffusion Velocity 0 The velocity of a particular species relative to the mass average or molar average velocity is termed a diffusion velocity V 111 11 V ul u l CVEN619 Spring 2009 1272009 j Relative flux 0 The uxes relative to the coordinates moving along with the average velocity 0 Relative rnass flux j m pu upu plun pu 0 Relative molar ux Ji ci fi ciui u ciui ciu Ni ciu 2 CVEN619 Spring 2009 9 1272009 j Fick s First Law 0 It is an empirical relation first postulated by Pick 0 Diffusive molar flux of component A relative to coordinates moving with the local molar average velocity is chC A C J A cDVxA CVEN619 Spring 2009 1272009 j Fiok s First Law Fixed Coordinates 0 Combine 2 and 3 Assume a binary system JA NA cAu J2 cDABVxA it is easy to show that NA CAll CDABVXA xANANB CDABVXA B ti C e a Similarly it can be shown that for mass flux contribution gradient contribution quotA IDA IODABV CVEN619 Spring 2009 1272009 j Simpler Forms 0 In liquid diffusion problem if a there is no impressed flow due to external forces and b the concentration of the diffusing substance is low then u0 10A nA 0DABV 0 Assume constant density we have 11A 2 DABV0A or in molar flux form NA DABVCA K9 CVEN619 Spring 2009 1272009 j Simpler Forms 0 In gas diffusion problem if there is no impressed How then u0 equirnolar countercurrent diffusion CVEN619 Spring 2009 1272009 j Measurement Flow of gas B Pure liquid A Figure 261 Arnold diffusion cell CVEN619 Spring 2009 Example Diffusion Coefficient Key points 1 Narrow tube partially filled with A 2 Maintained at constant pressure P and temperature T 3 Gas B does not dissolve in A and is chemically inert to A 4 The mass flux ofA diffuses into B can be measured and used to calculate the diffusion coefficient of A 1272009 j Example Diffusion Coefficient Measurement 0 Steady State Analysis dNA dz 03 1 dNB dz 0 Flux equation 0 gt N B 0 everywhere in the tube clx clx NA xA NA NB cDABd ZAXANA 013 CDAB dxA l xA dz 2 A 0 Integrate between top and bottom of the tube 22 xA2 1 j NAdzz I CDAB dxAjNAdeiln A l xAJl XA Z2 Z1 I JCA2 CVEN619 Spring 2009 3 1272009 j 9 Modified form of the equation 0 Using 1 and 2 it can be derived concentration profiles of A and B in the tube 1 x 221 2221 x 272 a i A2 X 32 Z and B Z l xA1 XE1 x31 0 Average concentration of B in the tube l xA l xA1 22 x dz f L B x32x31 xA1xA2 B 22 21 1nszx31 111w l xA1 0 Equation 3 can be expressed as N CDAB 1n l xA l CDAB x x A 1 A1 A2 22 Zl xA2 22 ZlxB CVEN619 Spring 2009 1272009 j 9 Modified form of the equation 0 Sometimes pressure is used in the equation instead of mole fractions 0 Using ideal gas law 7 p pA amp c andx x V RT A p B p 0 The mole ux expressed using partial pressure D p AB pm pm N A RTZ2 Z1pB CVEN619 Spring 2009 1272009 j ThinFilm Theory of Mass Transfer 4 Main gas z 6 m Flow of z 0 NA Slowly moving 75 3quot as lm g Li uid A Liquid A Figure 262 Film model for mass transfer of component A into a moving gas stream CVEN619 Spring 2009 1272009 j Diffusion Coefficients in Air Table 61 Di 39nsion Coefficients of Various Chemicals in Air at I am Premium Chemical Temperature C Dutcmlg s Ammonia 0 1216 25 028 Benzene If 10 25 0033 Carbon dioxide 0 038 25 064 Chloroform 0119 Formic acid 25 0 I 5 Hydrogen l 06 25 04m Mclhane 0 0J6 Nitrogen 0 3513 Oxygen 0 CLUB 25 0206 Toluene 30 0033 Water D 0220 25 0256 Source Thibodeaua C rnlodyrmmln Environmental Mavener Qf Chemicals 0 Am wmxg p nmi gg 453 461 chrjnled by permisslonnf John Wiley amp Sons Inc 12 39 5 1 1272009 j Diffusion Coefficients in Water Table 63 Diffusion Coe icicnts of Various Chemicals in Water Chemical Temperature quot3 DARX 105 cmzfsr Acetic acid 25 088 Ammonia 20 176 Bromine 20 12 Caffeine 25 063 Carbon dioxide 20 177 Chlorine 20 122 Glucose 20 06 25 069 Hydrogen 25 585 30 542 Hydrogen sul de 20 141 Nitric acid 20 26 Nitrogen 25 19 20 40 283 Oxygen 25 25 22 40 333 Phenol 20 084 Sodium chloride 20 135 Urea 20 106 The values are the diffusion coel cients multiplied by 105 k Sourcg ilygdiegl m gamgmiC Environmental Movement of Chemicals in Air Water and Soil 1272009 pp 462 463 1979 Reprinted by permission of John Wiley amp Sons Inc Equations for Diffusivities 0 Trace gas in the air 0 FSG method Lyman et a1 1982 textbook 0 Davis 1983 3 IRTma m ma D 2 q 8AdqpaV 27 CVEN619 Spring 2009 m q dq z 45 gtlt10 8m A Avogadro39s number 0 A density of the air ma molecular weight of air m q molecular weight of trace gas R Universal gas constant T Temperature 1272009 j Empirical Equations for Diffusivities 0 Substance Dissolved in Water 0 Wilke Chang Equation WBmB 12 T DAB 74x1010 W26 wB 26 for water mB 2 molecular weight of the solvent 18 gmol for water VA 2 molar volume of the solute cm3mol u 2 Absolute viscocity of the solvent g cm 1s 1 T absolute temperature K CVEN619 Spring 2009 1272009 j Fiok s Second Law 0 Fick s second Law is for nonsteady state diffusion From mass balance on a differential element aiVNA0 6t From Fick s First Law simplified form uO cconstant NA CAu CDABVXA DABVCA We have Fick s Second Law 86 2 A DABV CA 2 0 6t CVEN619 Spring 2009 1272009 j Solution of the Diffusion Equation 0 Simple analytical solution exists for 1D case Fgtlt0 01 i E D 32 2528 35 a A 9X2 Hm i IC cx00x 0 3 BCS mam 044 0 25 CA M 8 7A 12 cXp 7 7 2 DABt ADM EaMa a 73 72 71 o 1 2 z 6 e CVEN619 Spring 2009 x 1272009 Solution of the Diffusion Equation 0 Semi Infinite Medium with constant boundary condition 526A 81 AB 6x2 ICcAxO0 Xgt0 1 m BCS CAOjt CZDCAOOJ O 09 E Ziii j C CO 1erf x 05 A A V V V V V i V V V edxIOeXpu du 1272009 j CVEN619 Spring 2009 Eaf a 03 habaetzm l I 0 X D bzkm c l a t K CA CA 1 Dabt1 quot AB III a i ll 05 04 02 02 03 CVEN619 Spring 2009 1 1272009 j Solution of the Diffusion Equation 0 Diffusion into Spherical Particles Fixed Gas Phase Concentration 66A 626 2 66 D A A 82 AB 6r2 r 6r 0 Initial and Boundary Conditions ICCArO constfor0 S r S a Boya Ofora1t20 6r r0 BC2 cAra csforalt20 ccs l I I I Z Z c 01 2 mzr n 7 D t A1 Z71 sm ex 7 2 c r H mi 01 a 01 7 W 0 I I I I 0 01 02 03 04 05 06 lnxxIL Porosity 6 total volume pore volume WJ Diffusion in Porous Media Diffusion in Porous Media 0 Mass balance over incremental volume 0 Assume low solute concentration so that a 5pA8SpB8VJa 0 0 We can derive S kelp1 6amp4 VojA 0 62 1kdpb8 Retardation coefficient Diffusion in Porous Media 0 Using the simplified Fick s First Law 8 D 86 D pA 2 AB Vsz 01 A 2 AB Vch at R at R 0 Account for tortuosity 86A DABT39 VZCA at R Tortuosity factor V DAB D e R


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