Student Handout 08 2014
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Macroscopic momentum balance (*(!)/0( 603!. CHEE 3363 Spring 2014 Handout 8 Reading: Fox 4.4
Transcript of Student Handout 08 2014
Macroscopic momentum balance
CHEE 3363 Spring 2014 Handout 8
�Reading: Fox 4.4
Learning objectives for lecture
1. State the macroscopic conservation of linear momentum equation.�
2. Apply conservation of linear momentum to determine volume
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Macroscopic linear momentum balance 1Start with conservation of linear momentum and assume inertial reference frame:
Apply Reynolds Transport Theorem:
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F =
dP
dt
∣
∣
∣
∣
sys
Psys =
∫
M(sys)
vdm =
∫
V(sys)
vρdV
dP
dt
∣
∣
∣
∣
sys
=
∂
∂t
∫
CV
vρdV +
∫
CS
vρv·dA
F = FS + FB
Macroscopic linear momentum balance 2
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F = FS + FB =
∂
∂t
∫
CV
vρdV +
∫
CS
vρv·dA
(1) (2) (3)
(1)
(2)
(3)
(4)
(4)
Forces on a CV and its CSForces on control volume (body forces):
Forces on control surface:
Total body and surface forces:
���
control volumecontrol surface
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F = FS + FB = −
∫
CS
pndA +
∫
CS
n · τ̂dA +
∫
CV
ρgdV + FB,other
T̂ = ndF
dAFs =
∫
CS
n · T̂dA
T̂ = −pnn + τ̂ n · T̂ = −pn + n · τ̂
Fs =
∫
CS
n · T̂dA = −
∫
Cs
pndA +
∫
CS
n · τ̂dA
FB =
∫
CV
ρgdV + FB,other
Recall:
∫
CS
vρv · dA =
∑
CS
vρv · A
Momentum flux with uniform velocity
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For uniform velocities that are constant across the control surface:
The velocity v is measured with respect to control volume coordinates: Scalar product > 0 : outflow Scalar product < 0 : inflow
v1
Example: calculating momentum flux 1
Given: Tank with inlet velocity v1 (at surface A1) and outlet velocities v2 (at A2) and v3 (at A3).
Find: net rate of momentum flux
v2
v3
CS
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Assumptions:
∫
CS
vρv·dA
v1
Example: calculating momentum flux 2
Given: Tank with inlet velocity v1 (at surface A1) and outlet velocities v2 (at A2) and v3 (at A3).
Find: net rate of momentum flux
v2
v3
CS∫
CS
vρv·dA
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v = ui = umax
[
1 −
( r
R
)2]
i
Physical example 1Velocity distribution for laminar flow in a long circular tube:
Find: volume flow rate and momentum flux through section normal to the pipe axis
R
v
v = ui = umax
[
1 −
( r
R
)2]
i
Physical example 2Velocity distribution for laminar flow in a long circular tube:
Find: volume flow rate and momentum flux through section normal to the pipe axis
R
v
1
Given h, uniform velocity U
vmax = 2vmin.Find
∫
CS
vρv·dA
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Control volume:
Assumptions:
2
∫
CS
vρv·dA
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Given h, uniform velocity U
vmax = 2vmin.Find
:
CS
x y
2h
Example: momentum flux ratio 1
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Given
velocity U. At channel outlet velocity distribution is u/u = 1 - (y/h)2.Find: at channel outlet to that at inletControl volume:
:
CS
x y
2h
Example: momentum flux ratio 2
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Given
velocity U. At channel outlet velocity distribution is u/u = 1 - (y/h)2.Find: at channel outlet to that at inlet
CS
x y
2h
Example: momentum flux ratio 3
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Given
velocity U. At channel outlet velocity distribution is u/u = 1 - (y/h)2.Find: at channel outlet to that at inlet
Summary: relevant equations
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F = FS + FB =
∂
∂t
∫
CV
vρdV +
∫
CS
vρv·dA
Macroscopic momentum balance equation:
∫
CS
vρv · dA =
∑
CS
vρv · A
For uniform velocities that are constant across the control surface, in the absence of force:
