Report

Overall Shell Mass Balances I Outline 3. 4. 5. 6. Molecular Diffusion in Gases Molecular Diffusion in Liquids Molecular Diffusion in Solids Prediction of Diffusivities 7. Overall Shell Mass Balances 1. Concentration Profiles Overall Shell Mass Balance Species entering and leaving the system by Molecular Transport + by Convective Transport * May also be expressed in terms of moles Mass Generation by homogeneous Steady-State! chemical reaction Overall Shell Mass Balance * May also be expressed in terms of moles Common Boundary Conditions: 1. 2. 3. 4. Concentration is specified at the surface. The mass flux normal to a surface maybe given. At solid- fluid interfaces, convection applies: NA = kc∆cA. The rate of chemical reaction at the surface can be specified. ♪ At interfaces, concentration is not necessarily continuous. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Concentration Profiles I. Diffusion Through a Stagnant Gas Film Assumptions: 1. 2. 3. 4. Steady-state T and P are constants Gas A and B are ideal No dependence of vz on the radial coordinate At the gas-liquid interface, = Concentration Profiles I. Diffusion Through a Stagnant Gas Film Mass balance is done in this thin shell perpendicular to the direction of mass flow = − + ( + ) Concentration Profiles I. Diffusion Through a Stagnant Gas Film = − + ( + ) Since B is stagnant, = − (1 − ) Concentration Profiles I. Diffusion Through a Stagnant Gas Film = − (1 − ) Applying the mass balance, ǀ − ǀ+∆ = 0 where S = cross-sectional area of the column Concentration Profiles I. Diffusion Through a Stagnant Gas Film ǀ − ǀ+∆ = 0 Dividing by SΔz and taking the limit as Δz 0, − =0 NA = constant Concentration Profiles I. Diffusion Through a Stagnant Gas Film − =0 NA = constant But, = − (1 − ) Substituting, =0 1 − Concentration Profiles I. Diffusion Through a Stagnant Gas Film =0 1 − For ideal gases, P = cRT and so at constant P and T, c = constant DAB for gases can be assumed independent of concentration 1 =0 1 − Concentration Profiles I. Diffusion Through a Stagnant Gas Film 1 =0 1 − Integrating once, 1 = 1 1 − Integrating again, − ln 1 − = 1 + 2 Concentration Profiles I. Diffusion Through a Stagnant Gas Film − ln 1 − = 1 + 2 Let C1 = -ln K1 and C2 = -ln K2, 1 − = 1 2 B.C. at z = z1, at z = z2, xA = xA1 xA = xA2 1 − 1 − 2 = 1 − 1 1 − 1 −1 2 −1 Concentration Profiles I. Diffusion Through a Stagnant Gas Film 1 − 1 − 2 = 1 − 1 1 − 1 The molar flux then becomes = − (1 − ) −1 2 −1 1 − 2 = ln( ) 2 − 1 1 − 1 OR in terms of the driving force ΔxA *1 − 2 > 0, i.e. xA1> xA2 ǂ − i.e. z2> z1 2 1 > 0, 2 − 1 = (1 − 2 ) ( ) = (2 − 1 )( ) ln( 2 ) 1 Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Two Reaction Types: 1. Homogeneous – occurs in the entire volume of the fluid - appears in the generation term 2. Heterogeneous – occurs on a surface (catalyst) - appears in the boundary condition Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Reaction taking place 2A B 1. Reactant A diffuses to the surface of the catalyst 2. Reaction occurs on the surface 3. Product B diffuses away from the surface Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Reaction taking place 2A B Assumptions: 1. Isothermal 2. A and B are ideal gases 3. Reaction on the surface is instantaneous 4. Uni-directional transport will be considered Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction =0 = − + ( + ) Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction From stoichiometry, = −1/2 = − 1 1 − 2 Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Substitution of NA into the differential equation (− )=0 1 1 − 2 Integration twice with respect to z, 1 −2 ln 1 − = 1 + 2 = −(2 ln 1 ) − (2 ln 2 ) 2 B.C. 1: at z = 0, B.C. 2: at z = δ, xA = xA0 xA = 0 Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction The final equation is 1 1 (1− ) 1 − = (1 − 0 ) 2 2 And the molar flux of reactant through the film, 2 1 = ln( ) 1 1 − 0 2 *local rate of reaction per unit of catalytic surface Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Reading Assignment See analogous problem Example 18.3-1 of Transport Phenomena by Bird, Stewart and Lightfoot Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction 1. Gas A dissolves in liquid B and diffuses into the liquid phase 2. An irreversible 1st order homogeneous reaction takes place A + B AB Assumption: AB is negligible in the solution (pseudobinary assumption) Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction ǀ − ǀ+∆ − 1′′′ ∆ = 0 1′′′ first order rate constant for homogeneous decomposition of A S cross sectional area of the liquid Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction ǀ − ǀ+∆ − 1′′′ ∆ = 0 Dividing by SΔz and taking the limit as Δz 0, + 1′′′ = 0 Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction + 1′′′ = 0 If concentration of A is small, then the total c is almost constant and = − Combining the two equations above 2 ′′′ − 1 = 0 2 Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction 2 ′′′ − 1 = 0 2 . . 1 . . 2 = 0, = , = 0 = 0 =0 Multiplying the above equation by 2 0 gives an equation with dimensionless variables Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction 2 ′′′ − 1 = 0 2 2Γ 2 − Γ=0 2 Γ= , 0 = , = ′′′ 2 / Thiele Modulus Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction 2Γ 2Γ = 0 − 2 . . 1 = 0, . . 2 = 1, Γ=1 Γ =0 The general solution is Γ = 1 cosh + 2 sinh Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Γ = 1 cosh + 2 sinh Evaluating the constants, cosh cosh − sinh sinh cosh[ϕ 1 − ζ ] Γ= = cosh cosh Reverting to the original variables, = 0 ′′′ 2 cosh[ 1− ] ′′′ 2 cosh( ) Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Quantities that might be asked for: 1. Average concentration in the liquid phase , = 0 ( / ) 0 0 0 tanh = 2. Molar flux at the plane z = 0 ǀ=0 0 = − ǀ=0 = tanh Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Assumptions 1. Velocity field is unaffected by diffusion 2. A is slightly soluble in B 3. Viscosity of the liquid is unaffected 4. The penetration distance of A in B will be small compared to the film thickness. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Recall: The velocity of a falling film 2 cos () = 2 = 2 1−( ) 2 1−( ) Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) * CA is a function of both x and z ǀ ∆ − ǀ+∆ ∆ +ǀ ∆ − ǀ+∆ ∆ = 0 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) ǀ ∆ − ǀ+∆ ∆ +ǀ ∆ − ǀ+∆ ∆ = 0 Dividing by WΔxΔz and letting Δx 0 and Δz 0, + =0 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) + =0 The expressions for , = − + ( + ) Transport of A along the z direction is mainly by convection (bulk motion) Recall: = ∗ + = ≈ = () Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) + =0 The expressions for , = − + ( + ) Transport of A along the x direction is mainly by diffusion ≈ − Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) + =0 Substituting the expressions for , 2 = 2 Substituting the expressions vz, 2 1−( ) 2 = 2 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) 2 1−( ) 2 = 2 Boundary conditions B.C. 1 = 0, B.C. 2 = 0, B.C. 3 = , = 0 = 0 =0 BUT we can replace B.C. 3 with B.C. 3 = ∞, = 0 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) 2 1−( ) 2 =1− 0 or where 2 / / 4 exp − 2 0 = 1 − 0 erf = 2 = 2 = 2 4 2 0 exp(− ) ∞ 2 0 exp(− ) = 2 4 2 2 exp(− ) 0 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) = 1 − 0 ǀ=0 2 4 = 2 4 = − ǀ=0 = 0 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Reading Assignment See analogous problem Example 4.1-1 of Transport Phenomena by Bird, Stewart and Lightfoot Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Quantities that might be asked for: 1. Total molar flow of A across the surface at x = 0 = 0 = 0 = 0 0 ǀ=0 0 1