88243 - Advanced Fluid Dynamics

Academic Year 2018/2019

Learning outcomes

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

Course contents

Fluid Mechanics for STEM and OFFSHORE

Introduction to the course. Exam rules. Textbooks. Eulerian and Lagrangian views. Local and material derivative. Microscopic mass balance.Microscopic momentum balance. Stress tensor in a fluid.

Deformation rate tensor components. Constituive equations for the relation between stress and deformation rate for newtonian fluids, Bingham fluids and Power law fluids. Navier Stokes equation.Laminar flows: Couette flow for the different types of fluids, Falling film flow for the different types of fluids.


Example on composite falling film (Bingham and Newtonian fluids): velocity profile, stress profile and flowrate.
Poiseuille flow in rectangular and cylindrical channels: stress profile, velocity profile, flowrate for Newtonian, Bingham and Power Law Fluids. Consideration on the solution of the Navier Stokes equation in different cases: Couette, Poiseuille and falling films.

Flow in an annulus. Velocity and stress profile for a newtonian fluid. Example: wire coating. Non dimensionalization of Navier Stokes equation. Creeping and Inertial flows. Reynolds and Strouhal number meaning. Application to the unsteady falling film problem.

Examples of visocus, bidirectional, pseudo-steady flows. Determination of the velocity profile and force exerted on a squeezing-plate viscometer.Viscometry: viscometric kinematics and viscosity. Coeutte viscometer in planar and cylindrical case.

Parallel disk viscometer: velocity profile and estimation of viscosity. Cone and plate viscometer:velocity profile and estimation of viscosity. Capillary viscometer for Newtonian fluids. Pressure profile in fluids in rigid-body rotation.

Rabinowitsch treatment of capillary viscometer data: example of application to polymeric solution following power-law behavior. Lubrication theory: study of the velocity and pressure profile in a Michell Bearing, lift force applied. Example of the falling cylinder viscometer


Solution of unsteady laminar flow problems: semiinfinite medium.
Solution of 2d problems using the stream function: Creeping flow around a sphere

Potential, inviscid and irrotational flow. Vorticity transport theorem. Euler's equation and Bernoulli's equation. Laplace's equation. Potential flow around a cylinder. D'Alembert paradox

Laminar Boundary layer around a flat plate: Blasius' derivation and numerical solution. Applciations: entrance length in a duct. Friction factor. Turbulent flow: time smoothed quantities. Time smoothed version of the continuity equation and Navier Stokes equation with inertial stress

Friction factor as interfacial coefficient in internal flow, external flow and boundary layer: analogy with heat and mass transfer case. Dimensionless diagrams for friction factor in various cases. Flow in porous media: Darcy's law and Ergun equation. Application to the filtration process and fluidization point determination

Heat Transfer for STEM and OFFSHORE

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 for STEM students

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.

Advanced Fluid Dynamics for OFFSHORE students

Dimensional analysis : fundamentals and application to offshore problems. Force exerted on a submarine pipeline; lift force on a plane wing; free surface flows, scale-up of a valve and of a ship. Mach number, speed of sound, compressibility effects. Froude number.

Pipe flow generalities: pipe flow in the most general case: momentum balance and constitutive equation for the viscous term. Laminar and turbulent flow. Friction factor. Parallelism between micro and macro approaches. Moody's diagram. Derivation of Bernoulli's equation.

Compressible pipe flow (isothermal): derivation of Bernoulli's equation for pipe flow in the case of incompressible and isothermal compressible flow. Problems with unknown mass flow rate and unkonwn pressure drop. Calculation of the pressure, density and velocity profile for natural gas flowing in a long pipeline, and determination of the total energy dissipated. Determination of compression work in isothermal compressible flow.


Compressible pipe flow (adiabatic): considerations on the energy balance and ways to determine the temperature profile. Aspects derived from the 2nd law of thermodynamics. Supersonic and subsonic flow. Minimum choking length. Examples and considerations. Rigorous and empirical solution of the prolem (Lapple diagrams)

Compressible adiabatic reversible flow in converging/diverging ducts: equations, charts. Sizing of a rupture disc in critical conditions. Pressure safety valves design.

Non newtonian pipe flow: correlations for laminar and turbulent flow for generic fluids, power law fluids, and Bingham fluids. Dimensionless numbers required. Dodge -Metzner diagram. Turbulent drag reduction and its relation to viscoelasticity and Deborah number. Empirical correlation for its calculation by Darby, Virk and Toms. Estimation of energy reduced by adding an additive for drag reduction.

Biphasic flow: flow of solids in liquids (hydraulic flow). Slip velocities. Method of Molerus for the total pressure drop in a slurry. Examples of application. Flow of solids in gases/vapors /pneumatic flow). Gas-Liquid flow. 


Readings/Bibliography

FLUID MECHANICS:

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

M.M. Denn, "Process Fluid Mechanics", 1980, out of print.

ADVANCED FLUID DYNAMICS:

R. Darby, "Chemical Engineering Fluid Mechanics"

FLUID MECHANICS, HEAT TRANSFER, MASS TRANSFER

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

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

MASS TRANSFER

E. L Cussler "Diffusion: Mass Transfer in Fluid Systems", (Cambridge Series in Chemical Engineering) 3rd Ed. 2009


Teaching methods

Traditional lectures. Use of blackboard and slides provided to the students.

Assessment methods

The exam is written and composed of problems and quizzes. Homeworks given during the course also contribute to the final grade.

Rules:

1) Tables containing formulas like the NS equations can be used during the exam. Please ask the instructors about allowed material. 
2) No books or notes, either in digital or paper form, are allowed.
3) Use of any internet-connected device is forbidden.
4) Violation to any of the rules above implies immediate cancellation of the student's exam.

Exam dates and repetition
The student can take the exam in any of the available dates  (usually 6 throughout the year). Dates are published in ALMAESAMI and the student must register in the exam list to attend the exam in the specific date. 
If the outcome of the exam is "failed", the student can repeat it in any of the possible dates. If the exam is passed, the student can accept the score, or reject it and repeat it only another time, according to the following rules:

1) once the score is published, the student has 5 days to decide whether to keep it or reject it;
2) a positive score can only be rejected once by the student; the second positive score is automatically assigned (even if its lower than the previous one).
3) the student has the possibility to withdraw before the exam end, if he/she does not want to be assessed.

 

Teaching tools

Lectures, office hours, notes, homeworks, tutoring, slides, homeworks, questionnaries.

Office hours

See the website of Maria Grazia De Angelis

See the website of Matteo Minelli

SDGs

Quality education Industry, innovation and infrastructure

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.