1
CM3110
Transport I
Part I: Fluid Mechanics
Topic 1:Microscopic Balances
Topic
1:
Microscopic
Balances
© Faith A. Morrison, Michigan Tech U.
Professor Faith Morrison
Department of Chemical Engineering
Michigan Technological University
1
The hose connecting the city water supply to the washing
machine in a home burst while the homeowner was away
W t ill d t f th ½
i
i f 48h b f th
A problem from real life:
W
a
t
er sp
ill
e
d
ou
t
o
f
th
e
½
i
n p
i
pe
f
or
48
h
ours
b
e
f
ore
th
e
problem was noticed by a neighbor and the water was cut off.
How much water sprayed into the house
over the 2day period?
The water utility reports that the water pressure supplied to
© Faith A. Morrison, Michigan Tech U.
2
The
water
utility
reports
that
the
water
pressure
supplied
to
the house was approximately 60 psig.
2
Home
flood:
the
coldwater
feed to a
washing
washing
machine burst
and was
unattended for
two days
© Faith A. Morrison, Michigan Tech U.
3
Discussion:
How do we calculate the total amount of water
spilled?
© Faith A. Morrison, Michigan Tech U.
4
3
Discussion:
How do we calculate the total amount of water
spilled?
What determines flow rate through a pipe?
© Faith A. Morrison, Michigan Tech U.
5
Discussion:
How do we calculate the total amount of water
spilled?
What determines flow rate through a pipe?
What information do we need about the system to
calculate the amount of water spilled over two days?
© Faith A. Morrison, Michigan Tech U.
6
4
House flood
problem
Solution Strategy:
•Apply the laws of physics to the situation
•Calculate the velocity field in the pipe
( ill b f ti f )
© Faith A. Morrison, Michigan Tech U.
7
(
w
ill
b
e
f
unc
ti
on o
f
pressure
)
•Calculate the flow rate from the velocity
field
(as a function of pressure)
•Calculate the total amount of water spilled
crosssection A:
A
r
z
r
z
The
Situation
:
z
L
v
z
(r)
Steady flow
of water in a
pipe
R
fluid
© Faith A. Morrison, Michigan Tech U.
8
5
Next step: perform balances on flow in a tube
Because flow in a tube is a bit complicated to do as a
first problem (because of the curves), let’s consider a
h t i l bl fi t
s o m e w
h
a
t
s
i
mp
l
er pro
bl
em
fi
rs
t
.
© Faith A. Morrison, Michigan Tech U.
9
EXAMPLE I: Flow of a
Newtonian fluid down
an inclined plane
•full
y
develo
p
ed flow
晬畩f慩a
祰
镳瑥慤礠獴慴
镦汯眠楮慹敲猠⡬慭楮慲i
g
© Faith A. Morrison, Michigan Tech U.
10
6
The Laws of Physics
The
Laws
of
Physics
Mass is conserved
Momentum is conserved
Energy is conserved
© Faith A. Morrison, Michigan Tech U.
11
Physics I: Mechanics
Mass is conserved
•This was not an issue in mechanics because
we study bodies and the mass of the body does
not change
not
change
•Now we have to worry about it because we
study fluids
•Momentum is conserved
•Newton’s 2
nd
law:
F ma
© Faith A. Morrison, Michigan Tech U.
12
•The “body” is illdefined in flow
•Energy is conserved (similar issues)
on body
7
Control Volume
A chosen volume in a flow
on which we perform balances
(mass momentum energy)
•Shape, size are arbitrary; choose to be convenient
(mass
,
momentum
,
energy)
© Faith A. Morrison, Michigan Tech U.
13
•Because we are now balancing on control volumes
instead of on bodies, the laws of physics are
written differently
Mass balance, flowing system
(open system; control volume):
rate of
net mass
accumulation
flowing in
of mass
outin
steady
state
state
© Faith A. Morrison, Michigan Tech U.
14
8
Momentum balance, flowing system
(open system; control volume):
rate of
sumof forces net momentum
accumulation
acting on control vol flowing in
of momentum
outin
steady state
0
i
i
i
on
streams
the
in
outflowing
momentum
streams
the
in
inflowing
momentum
F
i
i
i
i
獴牥慭s
瑨t
楮
獴牥慭s
瑨t
楮
© Faith A. Morrison, Michigan Tech U.
15
note that momentum is
a vector quantity
EXAMPLE I: Flow of a
Newtonian fluid down
an inclined plane
•full
y
develo
p
ed flow
晬畩f慩a
祰
镳瑥慤礠獴慴
镦汯眠楮慹敲猠⡬慭楮慲i
g
© Faith A. Morrison, Michigan Tech U.
16
9
ProblemSolving Procedure  fluidmechanics problems
1. sketch system
2. choose coordinate s
y
stem
y
3. choose a control volume
4. perform a mass balance
5. perform a momentum balance
(will contain stress)
6. substitute in Newton’s law of viscosit
y,
e.
g
.
d
dv
z
yz
y,
g
7. solve the differential
equation
8. apply boundary conditions
d
y
yz
© Faith A. Morrison, Michigan Tech U.
17
x
v
0
xyz
z
x
xyz
z
y
x
v
v
v
v
v
v
0
x
z
v
xyz
z
xyz
z
y
vv
vv
0
x
z
z
v
Choose a
coordinate system
for convenience
© Faith A. Morrison, Michigan Tech U.
x
z
18
10
EXAMPLE I: Flow of a
Newtonian fluid down
an inclined plane
x
z
x
z
fluid
xv
z
air
singg
x
cosgg
z
g
cos
0
sin
g
g
g
g
g
g
z
y
x
z
© Faith A. Morrison, Michigan Tech U.
19
Choose a convenient control volume
x
x
x
© Faith A. Morrison, Michigan Tech U.
20
11
Assumptions:
(laminar flow down an incline, Newtonian)
1. no velocity in the x or ydirections (laminar
flow)
2. no shear stress at interface
3. no slip at wall
4. Newtonian fluid
5. steady state
6. well developed flow
7. no edge effects in ydirection (width)
8. constant density
© Faith A. Morrison, Michigan Tech U.
21
Flow down an Incline Plane
Boundary conditions:
0
0
)
cos(
g
0
0
0
z
xz
vHx
x
Solution:
stress matches at boundary
no slip at the wall
22
2
)
cos(
)(
x
H
g
x
v
z
© Faith A. Morrison, Michigan Tech U.
22
12
v
v
z
1.5
2.0
EXAMPLE I:
Flow of a
Newtonian
fluid down an
inclined plane
0.5
1.0
© Faith A. Morrison, Michigan Tech U.
23
0.0
W
W
z
z
dy
dx
dydxv
v
0 0
average
velocity
is the height
of the film
Engineering
Quantities of
Interest
dy
dx
0 0
volumetric
flow rate
z
W
x
vWdydxvQ
0 0
zcomponent
of force on
the wall
dzdyF
L
W
x
xzz
0 0
© Faith A. Morrison, Michigan Tech U.
(The expressions are
different in different
coordinate systems)
24
13
What is the shear stress as a
function of position for this flow?
Newton’s Law of Viscosity
x
v
z
xz
We have solved for
v
z
(x); we can now
calculate the shear
stress.
© Faith A. Morrison, Michigan Tech U.
25
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