# 88243 - Advanced Fluid Dynamics

## 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 (16 hours)

Introduction to the course. 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 flow: 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 anulus.

Viscometry: capillary viscometer, Couette viscometer, parallel disk viscometer, cone and plate viscometer.

Modules 2 (24 hours)

Non dimensionalization of Navier-Stokes equation. Creeping and inertial flows.

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

Solution of unsteady laminar flow problems: semi-infinite 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.

Module 3 (32 hours)

Heat transport.

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 transport.

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.

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. Calculation 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 and notes from classes.

FLUID MECHANICS

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

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.

## Assessment methods

The exam is written and composed of two parts:

1. Two open questions covering any of the course topics. Answers should be approximately half-page long. This part lasts 30 minutes.
2. An exercise of fluid mechanics or heat transport or mass transport. The requirements are typically addressed by writing relevant balance equations, with boundary and/or initial condition, solving them, using simplifying assumptions or order of magnitude considerations. This part lasts 90 minutes.

The allowed material during the exam includes:

• a calculator (not on the phone or tablet or laptop)
• printed tables with formulae, provided during the course

Exam dates (6 throughout the year) are published on Almaesami and the student must register in advance to take part in the exam.

If the student fails the exam, s/he can retry on a subsequent date.

If the student passes the exam, s/he can accept the score, or reject it and retry the exam on a subsequent date. A positive score can only be rejected once, the second (most recent) positive score will be registered.

The student has the possibility to withdraw from the exam before the end, if s/he prefers not to be assessed.

## Teaching tools

Lectures, slides, homework, equation tables, office hours.

## Office hours

See the website of Camilla Luni

See the website of Marco Giacinti Baschetti