# Solution Structure for Gas-Liquid Interphase Mass Transfer ## 1. Two-Film Theory for Gas-Liquid Interphase Transport ### 1.1 Steady-State Flux Equations - **Gas Film:** \[ N_A = k_G (P_{Ab} - P_{Ai}) \] - \(N_A\): Molar flux of A - \(k_G\): Gas-phase mass transfer coefficient - \(P_{Ab}\): Bulk gas partial pressure - \(P_{Ai}\): Interfacial gas partial pressure - **Liquid Film:** \[ N_A = k_L (C_{Ai} - C_{AL}) \] - \(k_L\): Liquid-phase mass transfer coefficient - \(C_{Ai}\): Interfacial liquid concentration - \(C_{AL}\): Bulk liquid concentration #### 1.1.1 Interfacial Equilibrium Constraint (Henry's Law) At the interface, equilibrium is assumed: \[ C_{Ai} = H P_{Ai} \] - \(H\): Henry's law constant #### 1.1.2 Overall Gas-Phase Mass Transfer Coefficient - Express both fluxes in terms of a common driving force (\(P_{Ab} - P^*_{AL}\)), where \(P^*_{AL}\) is the equilibrium partial pressure corresponding to \(C_{AL}\): \[ N_A = K_G (P_{Ab} - P^*_{AL}) \] where \[ \frac{1}{K_G} = \frac{1}{k_G} + \frac{H}{k_L} \] - **Elimination of interfacial compositions:** The above relation combines individual film resistances, making use of Henry's law to relate concentrations across the interface. --- ## 2. Surface Renewal Theory Framework ### 2.1 Diffusion Equation for Unsteady-State Molecular Transport - **Governing Equation (1D):** \[ \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2} \] - \(D\): Diffusivity - \(C(x,t)\): Concentration profile - **Boundary/Initial Conditions:** - At \(t = \), \(C(x,) = C_b\) for all \(x > \) - At \(x = \), \(C(, t) = C_i\) for \(t > \) - As \(x \to \infty\), \(C(\infty, t) = C_b\) #### 2.1.1 Instantaneous Concentration Profile and Flux - **Solution for semi-infinite medium:** \[ C(x, t) = C_i + (C_b - C_i) \operatorname{erf} \left( \frac{x}{2\sqrt{Dt}} \right) \] - **Instantaneous flux at interface (\(x = \)):** \[ N_A(t) = -D \left. \frac{\partial C}{\partial x} \right|_{x=} = \frac{(C_i - C_b)}{\sqrt{\pi D t}} \] #### 2.1.2 Average Mass Transfer Coefficient (Danckwerts' Theory) - **Statistical age distribution (\(s =\) surface age):** \[ \text{Probability density: } f(s) = S e^{-S s} \] - \(S\): Fractional rate of surface renewal - **Average mass transfer coefficient:** \[ k_L = \int_^\infty \frac{D}{\sqrt{\pi D s}} S e^{-S s} ds = \sqrt{\frac{D S}{\pi}} \] --- ## **Summary Table** | Theory | Mass Transfer Coefficient | Driving Force | Notes | |--------------------|-----------------------------------------------|------------------------|-------------------------------------------------------------------------| | **Two-Film** | \( \frac{1}{K_G} = \frac{1}{k_G} + \frac{H}{k_L} \) | \(P_{Ab} - P_{AL}^*\) | Steady-state, combines gas and liquid resistances using Henry's law | | **Surface Renewal**| \( k_L = \sqrt{\frac{D S}{\pi}} \) | \(C_i - C_b\) | Unsteady-state, accounts for renewal of interface (eddies, turbulence) | --- **Diagrams (descriptions):** - First image: Schematic of two-film model with concentration and pressure profiles. - Second image: Surface renewal model showing transient concentration profiles in a semi-infinite liquid. --- ### **References** - Bird, Stewart, and Lightfoot, *Transport Phenomena* - Treybal, *Mass-Transfer Operations*