73511 - FLUID MECHANICS AND TRANSPORT PHENOMENA M

Conoscenze e abilità da conseguire

This course aim to provide students with advanced tools for analysing and modelling momentum, energy and mass transport in fluid or solid media. Continuum mechanics approach is used to address the discussion of fluid mechanics, heat and mass transfer problems. Successful learner in this course will be able to understand the role of local form of total mass, momentum, energy and species balance equations.

Contenuti

 Detailed contents

Review of vector and tensor algebra. Second order tensors: properties and operations. Gradients of scalars and vectors; applications to velocity and deformation fields.

Kinematics of continua. Spatial and material representation of material properties and their time derivatives. Continuity equation (Lagrangian and Eulerian expressions).

First law of mechanics of continua: integral formulation. Local equation of motion for continua: Lagrangian and Eulerian forms. Equation of motion as momentum balance. Statics of continua. Mechanical energy equation; stress power.

General formulation of a continuum thermo-mechanical problem: need and role of constitutive equations. General properties of constitutive equations.  Application of material objectivity to stress constitutive equations: A) Euler inviscid fluid; B) viscous fluids: non-linear and linear fluids; dynamic and bulk viscosity; Stokesian fluids (no bulk viscosity), incompressible Newtonian fluids.

Statics of fluids (compressible and incompressible). Calculation of force and momentum on submerged objects and surfaces. Extension to cases with uniform acceleration. Surface tension effects; contact angle; calculation of meniscus static rise in capillaries.

Motion of inviscid fluids: Euler’s equation and its properties. Bernoulli’s equation a), b) for steady and irrotational motions. Irrotational motions and velocity potential. Irrotational motions as Euler’s equation solutions. Speed of sound and estimation of relative volume changes in motions: Mach number and compressible and incompressible motions.

Classification of Newtonian fluids motions: creeping flows and inertia flows, simplifications allowed and need of a boundary layer approach. Navier-Stokes equation, dimensional analysis and order of magnitude estimate of terms: Reynolds and Froude numbers; differences between free surface and non-free surface motions. Laminar and turbulent motions. Discussion on the boundary conditions required; mass and momentum balances across discontinuity surfaces.

Two-dimensional incompressible motions of Newtonian fluids: solutions via stream function. Creeping flow around a sphere: solution for stream function, velocity components and pressure distribution; calculation of friction force and its shape and surface components.

Friction factor and its dependence on Reynolds’ number. Velocity of a falling sphere both for Stokesian and generic motions; application to design of settlers, measures of viscosity and of sphere diameter. Transient creeping flows: example of a film falling along a reservoir wall: order of magnitude estimate of the terms and feasibility of the pseudo-steady state conditions.

Lubrication theory and Mitchell bearings.

One dimensional transient motions: surface motion in a semi-infinite region; Boltzmann solution and penetration theory. Solution of transient equations of motions through Laplace transforms: examples with finite fluid regions.

Boundary layer theory: general formulation, order-of-magnitude estimate of the different terms and of the boundary layer thickness; role of the dynamic pressure distribution and boundary layer detachment. Blasius solution for the boundary layer of a fluid flowing parallel to a flat surface.

Constitutive relations for special relevant kinematics: viscometric and elongational flows, their stress features. Application examples; power law and Bingham fluids.

Rheometry: measurements of rheological properties in viscometric flows: Couette rheometer, capillary rheometer and Rabinovitch equation. Cone and plate rheometer and its use to measure viscometric viscosity, first and second normal stress difference. Transient flows for Maxwell fluids. Viscous flow in porous media and Darcy’s equation.

Several examples of fluid motions for Newtonian and non-Newtonian fluids.

Fluid motions in fixed beds, Ergun equation. Turbulent flows: phenomenology; average values and fluctuations: equations of motion for the average quantities, Reynolds stresses.

Heat transfer: Fourier’s constitutive equation, thermal conductivity for isotropic and anisotropic materials; constitutive equations for internal energy; local energy balance equation. Heat conduction in solids and quiescent fluids: problem formulation, different initial and boundary conditions. Heat conduction in a semi-infinite slab with boundary conditions on temperature or on heat flux; analogy with penetration theory. Calculation of heat transfer coefficient, heat flux and total heat exchanged. Heat conduction in two semi-infinite slabs in contact at the interface.

Two dimensional problems of steady heat conduction: use of conformal transformations. Heat conduction in fins; planar fins and efficiency. Bessel’s and modified Bessel’s equations and their solutions. Solution of heat transfer in cylindrical fins and calculation of efficiency. Solution of transient heat transfer problems in slabs and cylinders: methods of separation of variables and Laplace transform method for different boundary conditions. Solutions available in graphs.

Heat transfer in fluids under different motion regimes: a) forced convection, non-dimensional equations, Péclèt number and dependence of Nusselt number on the relevant dimensionless numbers; b) free convection, non-dimensional equations, Grashof number and dependence of Nusselt number on Grashof and Prandtl numbers.

Thermal boundary layer on flat surface: detailed solution, thickness, heat transfer coefficient, Chilton – Colbourn analogy. Discussion on analogy between heat tranfer and fluid motion. Boundary layer on flat surfaces for liquid metals.

Mass transfer: relevant variables, velocity and flux of each species, diffusive velocities and diffusive fluxes. Local mass balances in Lagrangian and Eulerian form. Constitutive equation for the diffusive mass flux (mobility and chemical potential gradients); discussion. Fick’s law, diffusivity in binary solutions; its general properties, dependence on temperature, pressure; typical orders of magnitude for different phases. Mass balance equation for Fickian mixtures; relevant boundary conditions. Discussion and analogy with heat transfer problems. Measurements of diffusivity in gases; Stefan problem of diffusion in stagnant film.

Steady state mass transfer in different geometries (planar, cylindrical and spherical) in single and multilayer walls.

Transient mass transfer: problem formulation in different geometries. Solution for transient mass transfer problems: semi-infinite slab with different boundary conditions, films of finite thickness. Calculation of mass flux, of the total sorbed mass; “short times” and “long times” methods for the measurement of diffusivities. Transient permeation through a film: use of time lag and permeability for the determination of diffusivity and solubility coefficients. Transient mass transfer in ion implantation processes.

Mass transfer in a falling film and calculation of the mass transfer coefficient. Mass transfer in a fluid in motion: dimensionless equations; dependence of the Sherwood number on the relevant dimensionless numbers: Reynolds and Prandtl in forced convection, Grashof and Prandtl in free convection. Analogy with heat transfer. Graetz problems.

Boundary layer problems in mass transfer: mass transfer from a flat surface, mass transfer boundary layer thickness; explicit solution for the concentration profile and for the local mass transfer coefficient. Levèque problem formulation and solution. Chilton – Colbourn analogy; discussion on analogy among the different transport phenomena. Calculation of the mass transfer coefficient.

Mass transfer with chemical reaction: analysis of the behavior of isothermal catalysts with different geometries (planar, cylindrical and spherical), concentration profiles and efficiency dependence on Thiele modulus. Discussion on non-isothermal catalysts behavior and efficiency.

Diffusion with surface chemical reaction: metal oxidation problems: general problem formulation and justification through order-of-magnitude analysis of the pseudo-steady state approximation; solution and oxide thickness dependence on time. Diffusion with chemical reaction in the bulk: concentration dependence on Damkholer number.

Absorption with chemical reaction: determination of the mass transfer coefficient and of the enhancement factor for the case of instantaneous reactions, Hatta’s method. Calcultion of mass transfer coefficient and enhancement factor for the case of slow and fast reactions; film theory. Elements of turbulent mass transport and on dispersion problems in laminar flows (Taylor-Aris dispersion) and in porous media.

Testi/Bibliografia

Bird – Stewart - Lightfoot, “Transport Phenomena”, Wiley, 2nd Ed. 2002

William M. Deen, “Analysis of Transport Phenomena”, Oxford University Press, 2nd Ed. 2012

Introductory textbooks:

William M. Deen, “Introduction to Chemical Engineering Fluid Mechanics ”, Cambridge University Press, 2016

Morton M. Denn, Process Fluid Mechanics.

 

Metodi didattici

Traditional lectures

Modalità di verifica e valutazione dell'apprendimento

Written exam

Strumenti a supporto della didattica

Lectures, office hours, notes, homeworks, tutoring.

Orario di ricevimento

Consulta il sito web di Maria Grazia De Angelis

Consulta il sito web di Ferruccio Doghieri