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 2-day 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

cold-water

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

cross-section 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 ill-defined 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

Problem-Solving Procedure - fluid-mechanics 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 y-directions (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 y-direction (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

z-component

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.

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