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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 = kcโcA. The rate of chemical reaction at the surface can be specified. โช 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, ๐ฃ๐๐ ๐๐ด ๐ฅ๐ด = ๐ Concentration Profiles I. Diffusion Through a Stagnant Gas Film Mass balance is done in this thin shell perpendicular to the direction of mass flow ๐๐ฅ๐ด ๐๐ด = −๐๐ท๐ด๐ต + ๐ฅ๐ด (๐๐ด + ๐๐ต ) ๐๐ง Concentration Profiles I. Diffusion Through a Stagnant Gas Film ๐๐ฅ๐ด ๐๐ด = −๐๐ท๐ด๐ต + ๐ฅ๐ด (๐๐ด + ๐๐ต ) ๐๐ง Since B is stagnant, ๐๐ท๐ด๐ต ๐๐ฅ๐ด ๐๐ด = − (1 − ๐ฅ๐ด ) ๐๐ง Concentration Profiles I. Diffusion Through a Stagnant Gas Film ๐๐ท๐ด๐ต ๐๐ฅ๐ด ๐๐ด = − (1 − ๐ฅ๐ด ) ๐๐ง Applying the mass balance, ๐๐๐ด ว๐ง − ๐๐๐ด ว๐ง+โ๐ง = 0 where S = cross-sectional area of the column Concentration Profiles I. Diffusion Through a Stagnant Gas Film ๐๐๐ด ว๐ง − ๐๐๐ด ว๐ง+โ๐ง = 0 Dividing by SΔz and taking the limit as Δz ๏ฎ 0, ๐๐๐ด − =0 ๐๐ง NA = constant Concentration Profiles I. Diffusion Through a Stagnant Gas Film ๐๐๐ด − =0 ๐๐ง NA = constant ๐๐ท๐ด๐ต ๐๐ฅ๐ด But, ๐๐ด = − (1 − ๐ฅ๐ด ) ๐๐ง Substituting, ๐ ๐๐ง ๐๐ท๐ด๐ต ๐๐ฅ๐ด =0 1 − ๐ฅ๐ด ๐๐ง Concentration Profiles I. Diffusion Through a Stagnant Gas Film ๐ ๐๐ง ๐๐ท๐ด๐ต ๐๐ฅ๐ด =0 1 − ๐ฅ๐ด ๐๐ง For ideal gases, P = cRT and so at constant P and T, c = constant DAB for gases can be assumed independent of concentration ๐ ๐๐ง 1 ๐๐ฅ๐ด =0 1 − ๐ฅ๐ด ๐๐ง Concentration Profiles I. Diffusion Through a Stagnant Gas Film ๐ ๐๐ง 1 ๐๐ฅ๐ด =0 1 − ๐ฅ๐ด ๐๐ง Integrating once, 1 ๐๐ฅ๐ด = ๐ถ1 1 − ๐ฅ๐ด ๐๐ง Integrating again, − ln 1 − ๐ฅ๐ด = ๐ถ1 ๐ง + ๐ถ2 Concentration Profiles I. Diffusion Through a Stagnant Gas Film − ln 1 − ๐ฅ๐ด = ๐ถ1 ๐ง + ๐ถ2 Let C1 = -ln K1 and C2 = -ln K2, 1 − ๐ฅ๐ด = ๐พ1๐ง ๐พ2 B.C. at z = z1, at z = z2, xA = xA1 xA = xA2 1 − ๐ฅ๐ด 1 − ๐ฅ๐ด2 = 1 − ๐ฅ๐ด1 1 − ๐ฅ๐ด1 ๐ง−๐ง1 ๐ง2 −๐ง1 Concentration Profiles I. Diffusion Through a Stagnant Gas Film 1 − ๐ฅ๐ด 1 − ๐ฅ๐ด2 = 1 − ๐ฅ๐ด1 1 − ๐ฅ๐ด1 The molar flux then becomes ๐๐ท๐ด๐ต ๐๐ฅ๐ด ๐๐ด = − (1 − ๐ฅ๐ด ) ๐๐ง ๐๐ด ๐ง−๐ง1 ๐ง2 −๐ง1 ๐๐ท๐ด๐ต 1 − ๐ฅ๐ด2 = ln( ) ๐ง2 − ๐ง1 1 − ๐ฅ๐ด1 OR in terms of the driving force ΔxA *๐ฅ๐ด1 − ๐ฅ๐ด2 > 0, i.e. xA1> xA2 ว๐ง − ๐ง i.e. z2> z1 2 1 > 0, ๐๐ท๐ด๐ต ๐ฅ๐ต2 − ๐ฅ๐ต1 ๐๐ด = (๐ฅ๐ด1 − ๐ฅ๐ด2 ) (๐ฅ๐ต )๐๐ = (๐ง2 − ๐ง1 )(๐ฅ๐ต )๐๐ ๐ฅ ln( ๐ต2 ) ๐ฅ๐ต1 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 ๏ฎ 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 ๏ฎ 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 ๐๐๐ด =0 ๐๐ง ๐๐ฅ๐ด ๐๐ด = −๐๐ท๐ด๐ต + ๐ฅ๐ด (๐๐ด + ๐๐ต ) ๐๐ง Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction From stoichiometry, ๐๐ต = −1/2๐๐ด ๐๐ท๐ด๐ต ๐๐ฅ๐ด ๐๐ด = − 1 1 − ๐ฅ๐ด ๐๐ง 2 Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Substitution of NA into the differential equation ๐ ๐๐ท๐ด๐ต ๐๐ฅ๐ด (− )=0 1 ๐๐ง 1 − ๐ฅ๐ด ๐๐ง 2 Integration twice with respect to z, 1 −2 ln 1 − ๐ฅ๐ด = ๐ถ1 ๐ง + ๐ถ2 = −(2 ln ๐พ1 )๐ง − (2 ln ๐พ2 ) 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 ๐ง 1 1 (1− ) 1 − ๐ฅ๐ด = (1 − ๐ฅ๐ด0 ) ๐ฟ 2 2 And the molar flux of reactant through the film, 2๐๐ท๐ด๐ต 1 ๐๐ด = ln( ) 1 ๐ฟ 1 − ๐ฅ๐ด0 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 ๏ฎ AB Assumption: AB is negligible in the solution (pseudobinary assumption) Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction ๐๐๐ด ว๐ง − ๐๐๐ด ว๐ง+โ๐ง − ๐1′′′ ๐ถ๐ด ๐โ๐ง = 0 ๐1′′′ ๏ 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 ๐๐๐ด ว๐ง − ๐๐๐ด ว๐ง+โ๐ง − ๐1′′′ ๐ถ๐ด ๐โ๐ง = 0 Dividing by SΔz and taking the limit as Δz ๏ฎ 0, ๐๐๐ด + ๐1′′′ ๐ถ๐ด = 0 ๐๐ง Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction ๐๐๐ด + ๐1′′′ ๐ถ๐ด = 0 ๐๐ง If concentration of A is small, then the total c is almost constant and ๐๐๐ด ๐๐ด = −๐ท๐ด๐ต ๐๐ง Combining the two equations above ๐2 ๐๐ด ′′′ ๐ท๐ด๐ต − ๐ 1 ๐ถ๐ด = 0 2 ๐๐ง Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction ๐2 ๐๐ด ′′′ ๐ท๐ด๐ต − ๐ 1 ๐ถ๐ด = 0 2 ๐๐ง ๐ต. ๐ถ. 1 ๐ต. ๐ถ. 2 ๐๐ก ๐ง = 0, ๐๐ก ๐ง = ๐ฟ, ๐๐ด = ๐๐ด0 ๐๐๐ด ๐๐ด = 0 ๐๐ =0 ๐๐ง Multiplying the above equation by ๐ฟ2 ๐๐ด0 ๐ท๐ด๐ต gives an equation with dimensionless variables Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction ๐2 ๐๐ด ′′′ ๐ท๐ด๐ต − ๐ 1 ๐ถ๐ด = 0 2 ๐๐ง ๐2Γ 2 − ๐ Γ=0 2 ๐๐ ๐๐ด Γ= , ๐๐ด0 ๐ง ๐= , ๐ฟ ๐= ๐ ′′′ ๐ฟ2 /๐ท๐ด๐ต Thiele Modulus Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction ๐2Γ 2Γ = 0 − ๐ ๐๐ 2 ๐ต. ๐ถ. 1 ๐๐ก ๐ = 0, ๐ต. ๐ถ. 2 ๐๐ก ๐ = 1, Γ=1 ๐Γ =0 ๐๐ The general solution is Γ = ๐ถ1 cosh ๐๐ + ๐ถ2 sinh ๐๐ Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Γ = ๐ถ1 cosh ๐๐ + ๐ถ2 sinh ๐๐ Evaluating the constants, cosh ๐ cosh ๐๐ − sinh ๐ sinh ๐๐ cosh[ฯ 1 − ζ ] Γ= = cosh ๐ cosh ๐ Reverting to the original variables, ๐๐ด = ๐๐ด0 ๐ ′′′ ๐ฟ2 ๐ง cosh[ 1− ] ๐ท๐ด๐ต ๐ฟ ๐ ′′′ ๐ฟ2 cosh( ) ๐ท๐ด๐ต Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Quantities that might be asked for: 1. Average concentration in the liquid phase ๐๐ด,๐๐ฃ๐ = ๐๐ด0 ๐ฟ (๐ /๐ )๐๐ง 0 ๐ด ๐ด0 ๐ฟ ๐๐ง 0 tanh ๐ = ๐ 2. Molar flux at the plane z = 0 ๐๐ด๐งว๐ง=0 ๐๐๐ด ๐๐ด0 ๐ท๐ด๐ต = −๐ท๐ด๐ต ว๐ง=0 = ๐ tanh ๐ ๐๐ง ๐ฟ 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 ๐๐๐ฟ 2 cos ๐ผ ๐ฃ๐ง (๐ฅ) = 2๐ ๐ฃ๐ง ๐ฅ = ๐ฃ๐๐๐ฅ ๐ฅ 2 1−( ) ๐ฟ ๐ฅ 2 1−( ) ๐ฟ Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) * CA is a function of both x and z ๐๐ด๐งว๐ง ๐โ๐ฅ − ๐๐ด๐งว๐ง+โ๐ง ๐โ๐ฅ +๐๐ด๐ฅว๐ฅ ๐โ๐ง − ๐๐ด๐ฅว๐ฅ+โ๐ฅ ๐โ๐ง = 0 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) ๐๐ด๐งว๐ง ๐โ๐ฅ − ๐๐ด๐งว๐ง+โ๐ง ๐โ๐ฅ +๐๐ด๐ฅว๐ฅ ๐โ๐ง − ๐๐ด๐ฅว๐ฅ+โ๐ฅ ๐โ๐ง = 0 Dividing by WΔxΔz and letting Δx ๏ 0 and Δz ๏ 0, ๐๐๐ด๐ง ๐๐๐ด๐ฅ + =0 ๐๐ง ๐๐ฅ Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) ๐๐๐ด๐ง ๐๐๐ด๐ฅ + =0 ๐๐ง ๐๐ฅ The expressions for ๐๐ด๐ง , ๐๐ด๐ง ๐๐๐ด = −๐ท๐ด๐ต + ๐ฅ๐ด (๐๐ด๐ง + ๐๐ต๐ง ) ๐๐ง Transport of A along the z direction is mainly by convection (bulk motion) Recall: ๐๐ด = ๐ฝ๐ด∗ + ๐๐ด ๐ฃ๐ ๐ฃ๐ = ๐๐๐๐๐ ๐๐ฃ๐๐๐๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ ๐๐ด๐ง ≈ ๐๐ด ๐ฃ๐ = ๐๐ด ๐ฃ๐ง (๐ฅ) Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) ๐๐๐ด๐ง ๐๐๐ด๐ฅ + =0 ๐๐ง ๐๐ฅ The expressions for ๐๐ด๐ฅ , ๐๐ด๐ฅ ๐๐๐ด = −๐ท๐ด๐ต + ๐ฅ๐ด (๐๐ด๐ฅ + ๐๐ต๐ฅ ) ๐๐ง Transport of A along the x direction is mainly by diffusion ๐๐ด๐ฅ ๐๐๐ด ≈ −๐ท๐ด๐ต ๐๐ง Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) ๐๐๐ด๐ง ๐๐๐ด๐ฅ + =0 ๐๐ง ๐๐ฅ Substituting the expressions for๐๐ด๐ฅ ๐๐๐ ๐๐ด๐ง , ๐ฃ๐ง ๐๐๐ด ๐ 2 ๐๐ด = ๐ท๐ด๐ต ๐๐ง ๐๐ฅ 2 Substituting the expressions vz, ๐ฃ๐๐๐ฅ ๐ฅ 2 1−( ) ๐ฟ ๐๐๐ด ๐ 2 ๐๐ด = ๐ท๐ด๐ต ๐๐ง ๐๐ฅ 2 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) ๐ฃ๐๐๐ฅ ๐ฅ 2 1−( ) ๐ฟ ๐๐๐ด ๐ 2 ๐๐ด = ๐ท๐ด๐ต ๐๐ง ๐๐ฅ 2 Boundary conditions B.C. 1 ๐๐ก ๐ง = 0, B.C. 2 ๐๐ก ๐ฅ = 0, B.C. 3 ๐๐ก ๐ฅ = ๐ฟ, ๐๐ด = 0 ๐๐ด = ๐๐ด0 ๐๐๐ด ๐๐ฅ =0 BUT we can replace B.C. 3 with B.C. 3 ๐๐ก ๐ฅ = ∞, ๐๐ด = 0 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) ๐ฃ๐๐๐ฅ ๐ฅ 2 1−( ) ๐ฟ ๐๐ด 2 =1− ๐๐ด0 ๐ or where 2 ๐ง/๐ฃ ๐ฅ/ 4๐ท๐ด๐ต ๐๐๐ฅ exp −๐ 2 ๐๐ 0 ๐๐ด = 1 − ๐๐๐ ๐๐ด0 erf ๐ฅ = ๐๐๐ด ๐ 2 ๐๐ด = ๐ท๐ด๐ต ๐๐ง ๐๐ฅ 2 ๐ฅ = ๐๐๐๐ 2 4๐ท๐ด๐ต ๐ง ๐ฃ๐๐๐ฅ ๐ฅ 2 0 exp(−๐ฅ )๐ ๐ฅ ∞ 2 0 exp(−๐ฅ )๐ ๐ฅ = ๐ฅ 2 4๐ท๐ด๐ต ๐ง ๐ฃ๐๐๐ฅ 2 ๐ฅ 2 exp(− ๐ฅ ) ๐๐ฅ ๐ฅ 0 Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) ๐๐ด = 1 − ๐๐๐ ๐๐ด0 ๐๐ด๐ฅว๐ฅ=0 ๐ฅ 2 4๐ท๐ด๐ต ๐ง ๐ฃ๐๐๐ฅ = ๐๐๐๐ ๐ฅ 2 4๐ท๐ด๐ต ๐ง ๐ฃ๐๐๐ฅ ๐๐๐ด ๐ท๐ด๐ต ๐ฃ๐๐๐ฅ = −๐ท๐ด๐ต ว๐ฅ=0 = ๐๐ด0 ๐๐ฅ ๐๐ง 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 ๐ ๐๐ด = 0 = ๐๐๐ด0 = ๐๐ด0 ๐ฟ 0 ๐๐ด๐ฅว๐ฅ=0 ๐๐ง๐๐ฆ ๐ท๐ด๐ต ๐ฃ๐๐๐ฅ ๐ ๐ท๐ด๐ต ๐ฃ๐๐๐ฅ ๐๐ฟ ๐ฟ 0 1 ๐๐ง ๐ง
