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Momentum TransferLecture 3: Equations of Change for

Isothermal Systems

Transport Phenomena

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From last week

Shell balances for isothermal systems

Boundary conditions Where shell balances cannot be applied

Solutions for class examples on Blackboard

Go through tutorial questions to get more understanding

Shell balances as introduction to equations of continuity and

momentum

(i.e. TP is not all about shell balances, just in case you were

wondering.)

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Relevant Learning

Equations of change:

Continuity

Motion

Significance, usage, special cases (inviscid

and creeping flows)

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Equation of Continuity

[Mass accumulation] = [Mass in] [Mass out]

Substitute all equations

Divide by xyz, and asx, y, z 0, we get:

dz

v

dy

v

dx

v

t

zyx

zzzzz

yyyyy

xxxxx

vyxvyx

vzxvzx

vzyvzyt

zyx

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Equation of Continuity

Local change of with time at a fixed point of x, y, and

z.

PARTIAL time derivative

t

vxdx

vy

dy vzdz

v Divergence of v

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Equation of Continuity

What happens if we (as observers) float along with the

velocity of the flowing stream? Derivative that follows the motion

Partial time derivative (as before) + velocity x gradient of property

Total time change of a quantity as observed by an observer

following the motion of the fluid

SUBSTANTIAL time derivative

Dt

D

dz

dv

dy

dv

dx

dv

dt

dzyx

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Equation of Continuity Substantial time derivative of ?

t

vxdx

vy

dyvz

dz

vx

dx vy

dy vz

dz

DDt

.v Substantial derivative of

t vx

dx vy

dy vz

dz

vx

dxvy

dyvz

dz

Divergence of v

dz

v

dy

v

dx

v

t

zyx

v

.dt

d

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Equation of Continuity

For incompressible fluids (constant ):

v dvxdx

dvy

dydvz

dz0

d

dt0

vx

dx vx

dx

vy

dy vy

dy

vz

dz vz

dz

0

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Equation of Momentum

[Mom. accumulation] = [Mom. in] [Mom. out] + [Forces]

Momentum enters and leaves control volume throughconvective momentum transfer and viscous action through

velocity gradient

y

vx)x vx)x+x

z

x

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Equation of momentum

Rate of accumulation of x-momentum:

Considering the continuity equation:

we get:

d vx dt

d vxvx dx

d vxvy dy

dvxvz dz

dxx

dx

dyx

dy dzx

dz

dp

dx gx

t

vx

dxvy

dyvz

dz

dvx

dt vx

dvx

dx vy

dvx

dy vz

dvx

dz

dxx

dxdyx

dydzx

dz

dp

dxgx

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More on viscous flux

Recall Newtons law of viscosity:

The viscous term in the momentum equation: derivative

of the shear stress:

So, this term is a function of the second-derivative ofvelocity!

yx dvx

dy

dxx dx

dyx dy

dzx dz

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More on viscous flux

For incompressible, Newtonian fluids, we can simplify to:

d

2vx

dx 2 d

2vx

dy 2 d

2vx

dz2

2v

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Equation of momentum

For Newtonian and incompressible fluids:

Navier-Stokes equation What you learned in Fluid Mech!

Same treatment for y- and z-momentum

x

xxx

xz

xy

xx

x

gdx

dp

dz

vd

dy

vd

dx

vd

dz

dvv

dy

dvv

dx

dvv

dt

dv

2

2

2

2

2

2

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???

Further reading on this:

Welty et al. (2001), Fundamentals of Momentum, Heat and Mass

Transfer 4th ed., ch. 7-9

Bird, Stewart, Lightfoot (2007), Transport Phenomena, ch. 1-3

McCabe, Smith, Harriot (2004), Unit Operations of Chemical

Engineering, ch. 4

Note: in the first 2 books, the normal stress (xx, yy, zz) are

denoted differently (xx, yy, zz), and the pressure effects are

included in here.

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Class example

Similar to the case of parallel plates, but with the plates

vertical

Effects of gravity

Newtonian

Laminar

Incompressible

Steady state

From: Welty et al., 2001

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Class example

From: Welty et al., 2001

v0 = 0

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Other examples

Flow on inclined surface

Try to do this problem yourself!

Pipe flow

a bit more difficult, as we need to use cylindrical coordinates!

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Cylindrical coordinates

Equation of continuity:

011 zr vdz

dv

d

d

rrv

dr

d

rdt

d

cosrx

zz

sinry

x

y1tan

22 yxr

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Cylindrical coordinates

Equation of motion for fluids with constant

and :

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Pipe Flow

Fully-developed (steady) flow

Far from entrances/exits Flow is laminar and 1 directional:

vr= 0

v = 0

vz = vz(r)

p = p(z)

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Pipe Flow

Continuity eqn:

For constant density:

As vr= v = 0, we get:

011

zr vdz

dv

d

d

r

rv

dr

d

rdt

d

0dz

dvz

011

0

dzdv

ddv

rrv

drd

r

dt

d

zr

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Pipe Flow

Eqn of motion in the z-direction

Flow is symmetrical about the z-axis d2vz/d2 = 0

So, we get:

Integrating once:

z

z gdz

dp

dr

dvr

dr

d

r

rg

dz

dp

dr

dvr

dr

dz

z

1

2

2C

rg

dz

dp

dr

dvr z

z

r

Crg

dz

dp

dr

dvz

z 1

2

dvz/dr doesnt

become infinite

when r = 0

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Pipe Flow

Integrate again to get vz:

Use the boundary condition to evaluate C2, and we get:

As before, we use P = p + gL:

Another integration:

2

2

4

Cr

g

dz

dpv zz

224

1Rrg

dz

dpv zz

224

1Rr

dz

dPvz

LP

P

L

z PRrzv

0

22

04

1

220

4rR

L

PPv Lz

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Special case (Creeping flow)

Creeping flow & Stokes law

Very slow flows

Therefore, we have:

(also for other directions)

Negligible effects of inertia, Re

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Special case (Creeping flow)

Flow past a sphere

The creeping flow equation has been used to solve the velocityand pressure distribution in slow flows past a sphere

cos2

1

2

31

3

r

R

r

Rvvr

sin4

1

4

31

3

r

R

r

Rvv

0v

cos2

3 2

0

r

R

R

vgzpp

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Special case (Creeping flow)

Flow past a sphere

Total drag over the surface of the sphere = [form drag] + [viscousdrag]

Stokes equation

Valid for

RvFD 6

0.12

Re 0

Rv

Occurs as fluid needs to

change direction to pass

around the sphere

Due to shear stress at the

sphere surface

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Special case (Inviscid flow)

Euler equations for ideal fluids

Fluids with constant density and very low viscosity inviscid flow

Aerodynamics & hydrodynamics

Re >> 1 and 0

xx

zx

yx

xx g

dx

dp

dz

dvv

dy

dvv

dx

dvv

dt

dv

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Special case (Inviscid flow)

The Euler equation of motion could be re-arranged to get

the Bernoulli equation

Bird, Stewart, & Lightfoot, Section 3.3 and example 3.5-1.

Conditions:

Steady, fully-developed flow

Viscosity plays a minor role

Or, in a more familiar format:

02

112

2

2

2

2

2

1

hhgdpvvp

p

2

2

22

1

2

11

22

Pugz

Pugz

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Another example

Similar treatment andassumptions: Fully-developed (steady)

flow

Far from entrances/exits

Flow is laminar and 1

directional:vr= 0

v = 0

vz = vz(r)

p = p(z) Consider boundary

conditions to evaluatevelocity profile

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Summary

More on equations of change (isothermal)

Continuity Mass balance

Momentum equation Effects of convective flux,shear stress and viscous flux, pressure, and gravity

Common assumptions and boundary conditions

Limitations

Important concepts:

laminar and turbulent flows

Creeping and inviscid flows

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Questions?