Fluid Mechanics for Chemical
EngineersOld and New
by J ames O. Wilkes
Arthur F. Thurnau Professor
Emeritus of Chemical Engineering
University of Michigan
Wednesday 16 February 2011
wilkes@umich.edu
Southampton (England) Guildhall Organ (Oct. 2010)
Published in 2006
I: Macroscopic FM
II: Microscopic FM
773 pages
82 Examples
(incl. 14 CFD)
352 Problems
Two-phase flow
Microfluidics
Computational
fluid dynamics
Two parts, with opportunities
for questions after each.
Macroscopicor relatively large-scale
phenomena: basic & simple concepts of mass,
energy, and momentum balances commensurate
with the PE and FE examinations.
Microscopicor small-scale phenomenastarts
with the relatively complicated partial differential
mass and momentum equations of fluid motion.
Solutions are often best made by computational
fluid dynamics (CFD) software.
Part I
Macroscopic Fluid Mechanics
Characteristic of a Fluid
FluidDeforms continuously when subject to a tangential
or shear force.
Velocity profile: shows how velocity varies with position;
note the no-slipcondition at each surface.
StressForce Per Unit Area, F/A
(a) Normal stress = F/A: pressure is the most important case.
(b) Shear stress = F/A: acts tangentially to an area (that due to
viscosity is important example).
(a)
(b)
Viscosity
For Newtonian fluids (a broad class), the shear stress = F/A (A =
area of plate) is proportional to the velocity gradient du/dy, in
which the constant is the viscosity (with dimensions M/LT):
=
du
dy
=
V
h
.
Volume and Mass Flow Rates (Fluxes)
(a) Uniform velocity
Volumetric flow rate: Q = uA
Mass flow rate: m = Q = uA
Momentum flow rate:
(b) Variable velocity
M
.
= mu = u
2
A
Q= udA,
A
m = udA,
A
M
.
= u
2
dA.
A
Conservation Law for X =Mass, Momentum, Energy (Only)
The diagram shows a system and transports to and from it. Ignoring the
created and destroyed terms (necessary for reactions but generally not needed in
fluid mechanics), the basic conservation or balancelaw is:
X
in
X
out
= X
System
x
in
x
out
=
dX
System
dt
.
Or, if x (lower case) denotes a rate of transfer of property X, then:
(1)
(2)
Energy Balance (Bernoullis Equation)
Assumptions:
Steady flow
No work effects (no pump or turbine)
Frictionless (OK for short runs of straight pipe)
Incompressible (constant density)
Under these circumstances, the sum of the kinetic energy, potential, and
pressure energy remains constant between points 1 and 2:
u
1
2
2
+ gz
1
+
p
1
=
u
2
2
2
+ gz
2
+
p
2
A related form is also available if there are significant work and frictional
effectsespecially useful for pumping and piping problems. Eqn. (1) can also
be divided through by g, so each term has units of length, called either the
velocity, hydrostatic, or pressure head,with H being the constant total head.
u
1
2
2g
+ z
1
+
p
1
g
= H =
u
2
2
2g
+ z
2
+
p
2
g
(1)
(2)
Dynamics of a
distillation
column
Whats the
connection
between
this photo
and the
dynamics
of a plate
distillation
column?
End of Part I
Questions?
Part II
Microscopic Fluid Mechanics
Representative Computational Fluid
Mechanics (CFD) Software
(Usually based on the finite-element, finite-
difference, or finite-volume methods)
Adina
Ansys
COMSOL
Flow-3D
Fluent
FlowLab
FloTHERM
OpenFOAM
OpenFLOWER
Etc.
For a good overview, see:
http://en.wikipedia.org/wiki/Computational_fluid_dynamics
Screw Extruder for Increasing the
Pressure of a Polymer Before a Die
r
Axis of
rotation
Metering section
Feed hopper
Barrel
Flights
Exit
to die
Screw
Primary feed
heating region
W
L
0
Compression
section
Motion of Barrel as Seen by an Observer
on ScrewCouette (Relative Motion) +
Poiseuille (Pressure-Driven) Flow
x
W
h
Flight axis
Flight
Screw
Barrel
V
x
V
y
V
Flight
z
y
Cross Section between Barrel
(Moving Left) and Screw (Fixed)
p =0
z
x
y =0
x =0
2
4
No slip
x =0.1
y =0.005
1
3 No slip No slip
=800
=500
V
x
= - 0.1
A
B
1
2
3
4
Barrel
Screw
Plots: Arrows, Streamlines, Isobars
Horace Lamb (18491934)
(to the British Association for the
Advancement of Science, 1932)
I am an old man now, and when I die and go
to heaven there are two matters on which I
hope for enlightenment. One is quantum
electrodynamics, and the other is the turbulent
motion of fluids. And about the former I am
rather optimistic.
Arrows and Streamlines for Turbulent
J ets
Turbulent Kinematic Viscosity
T
Turbulent Kinetic Energy k
Flow of a shear-thinning polymer in a die
(a) Extrusion from a pipe forming a tube
(b) Rotation, also
exploiting symmetry
Velocities
A B
A
B
x
+
+
+
+
Stern
layer
Potential
(negative)
+
+
+
+
+
+
+
+
+
+
-potential (negative)
Diffuse layer
x = 0
y
Solid
surface
E
y
= 0
+
Electric Double Layer
=
dV
dy
x
-potential
The electric potential rises to
zero over a very thin electric
double layer next to the wall
Debye length,
D
0
v
y
v
y
=
y
E
x
Velocity
Profile
Electric
Potential
The velocity changes quickly
from zero at the wall to a
constant value everywhere else
0
(constant)
Electroosmosis (Multiphysics problem:
Navier-Stokes + Conductive Media DC )
Channel Geometry
H =0.00005, L =0.0005 m, = 0.1 V
Finite-element Mesh
y
x
y =0
x =0
2
x =L
y =H
1
3
4
A
B C
D
Electric insulation
Electric insulation
1 V
0 V
Velocity Vectors (Arrow Plot)
Streamlines
Electroosmotic Switching1
Electroosmotic Switching2
End of Part II
Questions?
Postscript
Whats the
connection
between this
Memphis juke
box and fluid
mechanics?
Thanks for
your attention!
