1. ON THE MASS (HEAT) TRANSFER MODELING Prof. Christo Boyadjiev ON THE MASS (HEAT) TRANSFER MODELING Prof. Christo Boyadjiev E-mail: chboyadj@bas.bgchboyadj@bas.bg

2. The mathematical structure of the model is convection-diffusion (conduction) equation: where the velocity components satisfy the hydrodynamic equations: ON THE MASS (HEAT) TRANSFER MODELING

3. The fundamental calculation problem result from the hydrodynamic equations non-linearity. The theories of first half of the 20-th century – detour of the problem: 1. - Nernst’s film theory (Langmuir, Lewis and Whitman). According this theory the mass transfer is a result of stationary diffusion trough immovable fluid film with thickness h: i.e. ON THE MASS (HEAT) TRANSFER MODELING

4. The basic disadvantages of this theory are the linear dependence of k on D, which is not confirmed experimentally, and the unknown thickness of the film h, which does not allow theoretical determination of the mass transfer coefficient. In spite of that some prerequisites and consequences of the theory are still valid. Examples for that are the assumptions that mass transfer takes place in a thin layer at the phase boundary, the existence of a thermodynamic equilibrium at the interphase, as well as the basic consequence of the theory regarding the additivity of the diffusion resistances ON THE MASS (HEAT) TRANSFER MODELING

5. 2. - Higbie’s penetration theory and in some related theories, where it is assumed that the mass transfer is non-stationary in coordinate system, which move with velocity : These theories do not give an account the velocity distribution in the boundary layer. ON THE MASS (HEAT) TRANSFER MODELING

6. The theories of second half of the 20-th century – exact theoretical methods: 1. Diffusion boundary layer theory (Landau, Levich, Boyadjiev) 2. Non-linear mass transfer theory (Krylov, Boyadjiev) ON THE MASS (HEAT) TRANSFER MODELING

7. 3. The development of the computers and numerical methods 4. Non-linear mass transfer and hydrodynamic stability (Boyadjiev, Babak) - Self-organizing dissipative structures. The theories of first half of the 21-st century Average concentration models Let’s consider diffusion model: ON THE MASS (HEAT) TRANSFER MODELING

8. The average velocity and concentration for the cross-section’s area of the column are: The velocity u(r) and concentration c(r,z) distributions may be presented by help of the average functions: where for the function of radial non-uniformities was obtained: ON THE MASS (HEAT) TRANSFER MODELING

9. The average concentration model may be obtained if put these expressions in equation, multiply with r and integrate over r in the interval [0, r 0 ]: where scale effect function is result of the velocity and concentration radial non uniformities: ON THE MASS (HEAT) TRANSFER MODELING