Module 1 : The equation of continuity. Lecture 5: Conservation of Mass for each species. & Fick s Law

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1 Module : The equato of cotuty Lecture 5: Coservato of Mass for each speces & Fck s Law NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg

2 2 Basc Deftos I Mass Trasfer, we usually deal wth cocetrato oe form or other. For ths we defe some symbols whch are dscussed below. Mass Cocetrato of th speces: Ths s the rato of mass of speces per ut volume. It s represeted by the symbol ρ. Thus the total mass cocetrato s gve by ρ. all Molar Cocetrato of th speces: Ths s the rato of moles of speces per ut volume. It s represeted by the symbol c. Substtutg for moles (.e. Mass/molecular weght) we get c = ρ M (3.) where M Molecular weght of the th compoet Mass fracto of the th speces: Ths s the rato of mass of speces to the total mass. It s represeted by the symbol w. sg defto of mass cocetrato, mass of speces equals volume tmes mass cocetrato of th speces. Therefore we ca wrte ρ w = (3.2) ρ NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg

3 3 Mole fracto of the th speces: Ths s the rato of moles of speces to the total moles. It s represeted by the symbol x. sg defto of molar cocetrato, we ca wrte c x = (3.3) c Veloctes For a mxture of speces I mass trasfer studes, we are terested the umber of moles of a partcular speces passg a certa pot as well as the mass of molecules of a partcular speces passg a certa pot space. ccordgly we ca defe two veloctes wth respect to axes fxed space as Mass average velocty Local Mass verage Velocty s defed as = = = ρ ρ (3.4.) where ρ s the Mass Cocetrato of the th speces ad s the velocty of the th speces relatve to statoary coordate axes. Molar average velocty Local Molar verage Velocty s defed as NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg

4 4 = = = c c (3.5) Here t mples that = c = c = c (3.6) = s the velocty of the speces relatve to statoary coordate axes. It s ot the velocty of the dvdual molecules but rather the average velocty over a small volume elemet of the flud. Dffuso veloctes : Dffuso velocty of speces wth respect to : Dffuso velocty of speces wth respect to Fluxes The flux s defed as the product of a speces cocetrato, c or desty, velocty NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg ρ ad ts. For fluxes relatve to statoary coordates we wll use (or N) whle for fluxes referred to movg coordates we wll use j (or J). For movg coordate, the speed at whch

5 5 the coordate frame s movg for cases we wll cosder wll be ether Mass average velocty or Molar average velocty.. Fluxes relatve to Statoary Coordates Mass = ρ (3.7) Molar N = c (3.8) 2. Relatve to the mass-average velocty Mass j ( ) = ρ (3.9) ( ) s the drvg force for Mass flux Molar J ( ) = (3.0) c 3. Relatve to the molar-average velocty Mass j ( ) = Molar J c ( ) ρ (3.) = (3.2) Dffuso a BINRY System ( ad B) NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg

6 6 Fck s Frst Law It relates the dffusve flux to the cocetrato feld, by postulatg that the flux moves from rego of hgher cocetrato to the rego of lower cocetrato wth a magtude.e. proportoal to the cocetrato gradet (spatal dervatve). The proportoalty costat s kow as dffuso coeffcet or dffusvty, D B (m 2 /s). j = ρd w (3.3) B where j s Mass Flux of compoet referred to axes movg at the Mass average velocty D B = dffusvty of compoet through B w = mass fracto of compoet ρ = Total mass cocetrato of mxture lso Molar flux of compoet referred to axes movg at the Molar average velocty s gve by J = c D x (3.4) B where x = mole fracto of compoet NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg

7 7 c = Molar desty of mxture The mass dffusvty D = D for a bary system B B Fg.3. Mxg of two Gases ( ad B) by Bary Dffuso Fg.3.2 Dffuso of Gas () to Lqud or Sold (B) at Gas-Lqud (Sold) terface NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg

8 8 Equvalet forms of Fck s frst law The mass flux relatve to statoary coordates s = w ( + B ) DB w ρ (3.5) The molar flux of relatve to statoary coordates s, N = ( N + N ) c D x B B x "Bulk" Flud moto Dffuso supermposed o bulk flow (3.6a) ccordg to equato (3.6) we ca wrte N = x c c D x (3.6b) B Dlute Solutos For dlute soluto of compoet a solvet B e.g. a low cocetrato of NaCl () H 2 O (B), ad equato (3.6b) becomes N Total Flux = c Covecto c D x B Dffuso (3.7) Recaptulato of equatos for the fluxes oe dmeso, e.g. y NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg

9 9 Fck s law s a emprcal law, as s the Fourer s law of heat coducto. Smlarly Newto s law of vscosty establshes a relatoshp betwee shear stress ad velocty gradet. So all these laws havg aalogy betwee them ad are dscussed below. Fck s law for ρ = costat j d dy ( ) y = DB ρ (3.8) Newto s law for ρ = costat d τ yx ν dy ( ρ ) = (3.9) x Fourer s law for ρ = costat d q y = α ρc T (3.20) dy p Thus we ca draw a cocluso from the above three relatoshps that (Flux)= - (Trasport Coeffcet) (gradet of cocetrato feld) Wth cocetrato feld = mass, mometum, or eergy, cocetrato. The aalogy does ot apply to two or three dmesos, because τ s tesor, whle j ad q are vectors. NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg