Fluxes in Multicomponent Systems

Transcription

1 Fluxes i Multicompoet Systems ChE 6603 Taylor & Krisha Friday, Jauary 14,

2 Outlie Referece velocities Types of fluxes (total, covective, diffusive) Total fluxes Diffusive Fluxes Example Coversio betwee diffusive fluxes Friday, Jauary 14,

3 T&K Table 1.2 Referece Velocities Deote the velocity of compoet i as ui. What assumptios have we made here? Molar-Averaged velocity u = x i u i xi is the mole fractio of compoet i. Mass-Averaged velocity v = ω i u i ωi is the mass fractio of compoet i. Volume-Averaged velocity u V = φ i u i φ i c i Vi is the volume fractio of compoet i. c i = x i c = ρi/mi - molar cocetratio of i. V i - partial molar volume of compoet i (EOS). Arbitrary referece velocity u a = a i u i a i =1 ai is a arbitrary weightig factor. Why ai = 1? Friday, Jauary 14,

4 Types of Fluxes Total Flux Amout of a quatity passig through a uit surface area per uit time. Covective Flux: Amout of a quatity passig through a uit surface area per uit time that is carried by some referece velocity. Diffusive Flux: Amout of a quatity passig through a uit surface area per uit time due to diffusio. The differece betwee the Total Flux ad the Covective Flux. Caot be defied idepedetly of the total & covective fluxes! Friday, Jauary 14,

5 Total Fluxes Total Fluxes may be writte i terms of a compoet s velocity ad specific desity Mass Flux of compoet i: i ω i ρu i = ρ i u i ρ - mixture mass desity Molar Flux of compoet i: N i x i cu i = c i u i Total Mass Flux: = i ρ i u i, = ρv Total Molar Flux: N t = = N i c i u i = cu Note: there is a typo i T&K eq. (1.2.5) Friday, Jauary 14,

6 Diffusive Fluxes Cocepts: The total flux is partitioed ito a covective ad diffusive flux. The covective flux is defied i terms of a average velocity. The diffusive flux is defied as the differece betwee the total ad covective fluxes. It is oly defied oce we have chose a covective flux! Mass diffusive flux of compoet i relative to a mass-averaged velocity. Total mass flux of compoet i Covective flux of compoet i due to a mass-averaged velocity j i = i ρ i v = ρ i (u i v) = i ω i t Friday, Jauary 14,

7 Diffusive Fluxes Diffusio fluxes are defied as motio relative to some referece velocity. (Diffusio flux) = (Total Flux) - (covective flux) Diffusio Flux Total Flux Covective Flux Commets j i i ρ i v Mass diffusive flux relative to a mass-averaged velocity j u i i ρ i u Mass diffusive flux relative to a molar-averaged velocity J i N i c i u Molar diffusive flux relative to a molar-averaged velocity J v i N i c i v Molar diffusive flux relative to a mass-averaged velocity j i = i ρ i v = ρ i (u i v) = i ω i t J i = N i c i u = c i (u i u) = N i x i N t Friday, Jauary 14,

8 T&K Table 1.3 Liear Depedece of Diffusive Fluxes Appropriately weighted diffusive fluxes sum to zero. Why? For mass diffusive flux relative to mass-averaged velocity: j i =0 From the previous slide: j i = ρ i (u i v) Total fluxes are all idepedet. Covective fluxes are ot all idepedet. j i = (ρ i u i ρ i v) = ρ (ω i u i ω i v), = ρv + ρ ω i u i = ρv + ρv, = 0. Note: typo i table 1.3 (missig v superscript) ω i x i J v i =0 Friday, Jauary 14,

9 Example - Stefa Tube Air Liquid Mixture z = z = 0 At steady state (1D), i = α i N i = β i We will show this later... Species Mole Fractios Acetoe(1) Methaol(2) Air(3) z (m) Molar, mass averaged velocities Species velocities u i = i Acetoe diffusive fluxes j i = ρ i (u i v) 1.6 x x 10 3 ρ i x 10 3 velocity (m/s) Molar Average Mass Average z (m) Friday, Jauary 14, 2011 Species Velocities (m/s) Acetoe Methaol Air z (m) Diffusive Fluxes for Acetoe j j u J J v z (m) 9

10 Coversio Betwee Fluxes We ca covert betwee various diffusive fluxes via liear trasformatios. Coversio betwee mass diffusive fluxes relative to mass ad molar referece velocities. (j u ) = [B uo ](j) (j) = [B ou ](j u ) To form [B uo ] -1 this must be a -1 dimesioal system of equatios! (why?) Derivatio of [B ou ]... j u i = ρ i (u i u) j i = ρ i (u i v) j u i = ρ i u i ρ i u, Let s try to get a j i u o the RHS add & subtract ρiu. 1 ρ = ρv ρu, j u i = v u. j i = ρ i (u i v), = ρ i (u i u) +ρ i (u v) j u i j i = j u i ω i j u j j=1 This is a -dimesioal set of equatios. If we derive [B ou ] from this, it will ot be full-rak. Friday, Jauary 14,

11 Derivatio of [B ou ] (cot d) j i = j u i ω i j i = j u i ω i j u j j=1 j u + 1 separate out j u j u j x i ω i j u i =0 j u = ω x 1 x i ω i j u i j=1 elimiate j u Elimiate j u from the above equatio... 1 j i = j u i ω i = 1 j=1 j=1 ω x x j ω j j u j + j u j δ ij ω i 1 ω x x j ω j, j u j gather terms o j u we have -1 of these equatios (...-1) (j) =[B ou ](j u ) B ou ij = δ ij ω i 1 ω x x j ω j The iverse [B uo ]=[B ou ] -1 ca be obtaied from equatios A.3.21-A.3.23 i T&K (ote the typo i A.3.22). [B] 1 = [A] 1 1 α [A] 1 (u)(v) T [A] 1, α = 1+(v) T [A] 1 (u) Friday, Jauary 14,