73494 - Fluid Mechanics And Transport Phenomena M

Learning outcomes

Offer the proper tools for the identification and appropriate mathematical description of the physical phenomena relevant in the technology of interest for Chemical Engineering at broad.

Offer the elements needed for innovation development, to model and simulate innovative processes, model and simulate apparatus behavior and scale-up

Course contents

Review of vector and tensor algebra. Second order tensors: properties and operations. Representation theorem of Cayley Hamilton and general expression of a regular function of a second order symmetric tensor. 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. Gradient of motion (F), gradient of displacement; polar decomposition. Comments on det(F) and its material time derivative. Conservation of mass via det(F). Reynolds’ transport theorem: general formulation and expression for mass conserving systems. Continuity equation (Lagrangian and Eulerian expressions).

First law of mechanics of continua: integral formulation. Cauchy tetrahedron analysis, Cauchy stress tensor and its properties. Local equation of motion for continua: Lagrangian and Eulerian forms. Equation of motion as momentum balance. Statics of continua. Mechanical energy equation; stress power.

First law of thermodynamics: local balance of energy in Lagrangian and Eulerian forms; Cauchy tetrahedron analysis, heat flux vector and its properties.

General formulation of a continuum thermo-mechanical problem: need and role of constitutive equations. General properties of constitutive equations. Objective scalar, vector and tensor variables: examples of objective and non-objective variables. Application to the possible constitutive equations of general (linear and nonlinear) viscous fluids with heat conduction. Application of material objectivity to stress constitutive equations: A) Euler inviscid fluid; B) viscous fluids: non-linear (Reiner Rivlin equation with comments on its comparison with experimental evidence) 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. Equation of inviscid fluid motion in terms of vorticity. Incompressible motions: vector potential (in 3 D) and stream function for 2 dimensional motions. 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; Boltzman 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. Calculation of the stress distribution and of the friction factor; estimation of the entry length in a pipe.

Viscoelastic fluids: phenomenology: recoil, stress relaxation, die-swell, Weissemberg effect, tubeless syphon. Qualitative one-dimensional models: Maxwell and Voigt models; discussion. Relaxation time and Deborah number; differential and integral models. Constitutive relations for special relevant kinematics: viscometric and elongational flows, their stress features: viscometric viscosity, first and second normal stress difference and their respective dependence on shear rate. 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. Turbulent viscosity, Prandtl mixing length and mixing velocity; universal average velocity profile in proximity of a wall.

Local energy balance (first law of Thermodynamics) and its different equivalent expressions. Second law of Thermodynamics: integral and local formulation. Use of Second Law as a constraint for the constitutive equations: general inequality and application to the case of viscous fluids with heat conduction. Derivation of the relevand consequences: independence of Helmholtz free energy, entropy, internal energy from temperature gradient and deformation rate; expression of thermal and mechanical dissipation, discussion. Representation theorem for linear fluids: the so-called Curie principle.

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.

Office hours

See the website of Ferruccio Doghieri

See the website of Giulio Cesare Sarti