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M E 320 Professor John M. Cimbala Lecture 15
Today, we will :
- Derive and discuss the linear momentum equation for a control volume (Chapter 6)
- Discuss the momentum flux correction factor, β
- Discuss all the various forces acting on a control volume
- Do some example problems – Linear momentum equation for a control volume
E. The Linear Momentum Equation for a Control Volume (Chapter 6)
- Equations and Definitions
Recall from the RTT at the end of Chapter 4,
CV CS
d
F Vd V V n dA
dt
∑ =^ ∫ ρ +^ ∫ρ^ ρ ⋅
G G G G G
V
Rate of change of linear momentum inside the control volume
Total force ( vector ) acting on the control volume Net rate of linear momentum flow out of the control volume
Use relative velocity Vr
G
here if we have a moving or deforming control volume
- Examples
Example: Friction force in a pipe Given : Consider steady, laminar, incompressible, axisymmetric flow of a liquid in a pipe as sketched. At the inlet (1) there is a nice bell mouth, and the velocity is nearly uniform (except for a very thin boundary layer, not shown).
- At (1), u = u 1 = constant, v = 0, and w = 0. P 1 is measured.
- At (2), the flow is fully developed and parabolic:
2 2 max (^1 )
r u u R
⎛ ⎞ = (^) ⎜ − ⎟ ⎝ ⎠
. P 2 is measured.
R
x , u
L
u ( r ) u max
u 1
(1) (2)
P 2
r P 1
To do : Calculate the total friction force acting on the fluid by the pipe wall from 1 to 2.
Solution :
- Now use the approximate, most useful form of the linear momentum equation,
gravity pressure viscous other CV out^ in
d
F F F F F Vd mV mV
dt
∑ =^ ∑ +^ ∑ +^ ∑ +^ ∑ =^ ∫^ ρ +^ ∑ β −∑β
G G G G G G G G
V ^
Example: Force imparted by a water jet hitting a stationary plate Given : A horizontal water jet of area A j , average
velocity V j , and momentum flux correction factor β j
impinges normal to a stationary vertical flat plate.
To do : Calculate the horizontal force F required to keep the plate from moving.
Solution :
- First step:
- Second step: Use the approximate, most useful form of the linear momentum equation,
gravity pressure viscous other CV out^ in
d
F F F F F Vd mV mV
dt
∑ =^ ∑ +^ ∑ +^ ∑ +^ ∑ =^ ∫^ ρ +^ ∑ β −∑β
G G G G G G G G
V
V j, A j, β j
Nozzle
z x
F
