# 88243 - Advanced Fluid Dynamics

### SDGs

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

Modules 1 and 2, (40 hours) 1st semester

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. Constitutive equations 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.

Poiseuille flow in rectangular and cylindrical channels.
Flow in an annulus.
Non dimensionalization of Navier Stokes equation. Creeping and Inertial flows.

Viscometry: Coeutte viscometer, Parallel disk viscometer, Cone and plate viscometer, Capillary viscometer .

Nearly 1D flows: lubrication theory, study of a Michell Bearing.

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. Euler's equation and Bernoulli's equation.

Laminar Boundary layer around a flat plate. Applications: entrance length in a duct.

Friction factor as interfacial coefficient in different types of flow 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

Dimensional analysis : fundamentals and application to offshore problems.

Incompressible pipe flow generalities: momentum balance (Bernoulli's equation) and evaluation of the viscous term. Laminar and turbulent flow. Friction factor. Moody's diagram.

Compressible pipe flow (isothermal): problems with unknown mass flow rate and unkonwn pressure drop. Natural gas flowing in a long pipeline: energy required and pressure drop.
Compressible pipe flow (adiabatic): supersonic and subsonic flow. Choking flow.
Compressible adiabatic reversible flow in converging/diverging ducts: equations, charts. Rupture disks, pressure safety valves.

Non newtonian pipe flow. Turbulent drag reduction.

Biphasic flow generalities .

Module 3, 2nd semester (32 hours)

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.

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.

Slides provided to the students and available on IOL

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.

FLUID MECHANICS, HEAT TRANSFER, MASS TRANSFER

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

W.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

Lectures with presentation of the theoretical background and solution of problems.
Use of blackboard and slides projected on the screen and provided to the students.
Projection of videos.

## Assessment methods

Exam description

The exam is written and composed of

-1 Fluid Mechanics problem, in which the student is required to write the mass/momentum balance equations and boundary or initial condition for a specific case, and then solve them to find relevant properties like the velocity or stress profile, force, torque, etc.. The use of simplifying assumptions to find approximate solutions, based on geometrical or order of magnitude considerations, is also required.

-1 Heat or Mass Transfer problem, in which the student is required to write the mass/energy balance equations and boundary or initial condition for a specific case, and then solve them to find relevant properties like the concetration or temperature profiles, heat or mass fluxes, etc.. The use of simplifying assumptions to find approximate solutions, based on geometrical or order of magnitude considerations, is also required.

-A list of mixed theoretical quizzes with multiple choices, covering all the course topics.

How to prepare for the exam

-For the problems: previous exam problems, and the corresponding complete solutions, are provided on IOL as a reference to the students. Some problems are solved during the classes. The students can come to office hours for help on solution of the problems.

-For the quizzes: to prepare for the quizzes you must study the theoretical aspects using the slides provided and/or the notes taken during the classes.

-The regular and continuous attendance of the classes, although not compulsory, is an essential requisite to pass the exam.

Exam Rules:

Allowed material:
Paper tables containing the:
-Microscopic mass balance in various geometries
-Momentum balance in various geometries
-Navier Stokes equation in various geometries
-Strain rate tensor components in various geometries

A scientific calculator to carry out calculations. (It is not allowed to use mobile phone, tablet or laptop to carry out calculations)

Non allowed material
-Books or notes, either in digital or paper form.
-Any internet-connected device (mobile phone, tablet, laptop).
Violation to any of the rules above implies immediate cancellation of the student's exam.

Dates and repetition of the exam

-The student can take the exam in any of the available dates
(at least 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.
-The exam grades are published in IOL within 15 days
-If the outcome of the exam is "failed", the student can retry it in any of the possible dates.
-If the exam is "passed", the student can accept the score, or reject it and re-take 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 end of the exam, if he/she does not want to be assessed.

-Problem 1 is usually assigned a score equal to around 14-16
-Problem 2 is usually assigned a score equal to 13-15
-Quizzes are usually assigned a total score equal to around 8

-Wrong answers in the multiple choices quizzes are assigned a negative score

If the students gets a score of at least 32, the "cum laude" mention is awarded.

## Teaching tools

Lectures, office hours, notes, homeworks, tutoring, slides, homeworks, questionnaries, previous exam problems, Tables

## Office hours

See the website of Maria Grazia De Angelis

See the website of Marco Giacinti Baschetti

See the website of Camilla Luni