Macroscopic momentum balance 2: forces on surfaces

CHEE 3363 Spring 2014 Handout 9

Reading: Fox 4.4 (continued)

1

Learning objectives for lecture

1. Apply conservation of linear momentum to determine forces in idealized geometries.

2. Apply conservation of linear momentum to determine other physical quantities related to force.

2

Example: force exerted on pipe 1Given: 180 degree bend, gage pressure pg at inlet, discharges to patm at outlet. A1 area of inlet, A2 outlet, inlet velocity V1.

Control volume:Find: force to hold elbow Fe

A1

A2

Assumptions:

Fe

3

$"#)*"0*-&*)$#

Example: force exerted on pipe 2

A1

A2Fe

4

Continuity:

Given: 180 degree bend, gage pressure pg at inlet, discharges to patm at outlet. A1 area of inlet, A2 outlet, inlet velocity V1.Find: force to hold elbow Fe

Example: force on plate 1

5

Given: water jet of speed V impinges on plate with hole in center.Find: external force required to hold plate in place if jet leaving also has speed V.

$"#)*"0*-&*)$#

Assumptions:

Control volume:

Example: force on plate 2

6

Given: water jet of speed V impinges on plate with hole in center.Find: external force required to hold plate in place if jet leaving also has speed V.

Example: tilted plane 1

7

Given: plane jet striking inclined plane at angle . The incoming jet thickness is h, and the outgoing thicknesses are h2 and h3 as shown. There is no frictional force.Find: h2/h as a function of .

$"#)*"0*-&*)$#

Assumptions:

Control volume:

Example: tilted plane 2

8

Given: plane jet striking inclined plane at angle . The incoming jet thickness is h, and the outgoing thicknesses are h2 and h3 as shown. There is no frictional force.Find: h2/h as a function of .

Example: nozzle for spray system 1

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Given: Nozzle for spray system produces a sheet of water over 180 degrees of arc at velocity V2 and thickness t. The nozzle discharge radius is R. The diameter of the supply pipe is D and the inlet pressure is p1 (absolute). Find: axial force exerted by spray nozzle on coupling.Control volume: Conservation of momentum:

Assumptions:

Example: nozzle for spray system 2

10

Given: Nozzle for spray system produces a sheet of water over 180 degrees of arc at velocity V2 and thickness t. The nozzle discharge radius is R. The diameter of the supply pipe is D and the inlet pressure is p1 (absolute). Find: axial force exerted by spray nozzle on coupling.

Example: nozzle for spray system 3

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Given: Nozzle for spray system produces a sheet of water over 180 degrees of arc at velocity V2 and thickness t. The nozzle discharge radius is R. The diameter of the supply pipe is D and the inlet pressure is p1 (absolute). Find: axial force exerted by spray nozzle on coupling.

Example: spring + hinge 1

12

Given: Free jet of water (velocity V) with constant cross-sectional area A deflected by a hinged plate of length L supported by spring, constant k, uncompressed length x0.Find: deflection angle as a function of jet speed V.Control volume: $"#)*"0*-&*)$#

Assumptions:

Example: spring + hinge 2

13

Given: Free jet of water (velocity V) with constant cross-sectional area A deflected by a hinged plate of length L supported by spring, constant k, uncompressed length x0.Find: deflection angle as a function of jet speed V.

F = FS + FB =@

@t

ZCV

VXYZdV +

ZCS

VXYZVXYZ dA

Inertial coordinate system

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In a moving inertial frame of reference (which does not accelerate):

XYZ: stationary coordinate system xyz: coordinates attached to moving control volume

x yz

X

Y

Z V

CV

Example: non-accelerating cart 1

15

Given: Water from stationary nozzle impinges on moving vane with turning angle . Vane moves away from cart with constant speed U and receives a jet that leaves nozzle of area A with speed V.Find: Force to maintain constant vane speed.Control volume: $"#)*"0*-&*)$#(

Assumptions:

Example: non-accelerating cart 2

16

Given: Water from stationary nozzle impinges on moving vane with turning angle . Vane moves away from cart with constant speed U and receives a jet that leaves nozzle of area A with speed V.Find: Force to maintain constant vane speed.