Contents

Conservation of Linear Momentum

Lesson Outline

  • Measuring velocity and pressure using pitot-static systems (Bernoulli application)

  • Examples of CV applications for continunity

  • CV Relationships for momentum

Background

Linear momentum relates forces to changes in translational velocity. Reynolds Transport Theorem is used to generate integral expressions of momentum balances in a control volume.

Linear Momentum

../../_images/momentum-p1-11.png

Caption

momentum-p1-1 relates the extensive property of linear momentum for a system to the intensive property of momentum per unit mass which is the velocity vector. momentum-p1-2 extends to the intensive property of momentum per unit volume which is the product of density and velocity vector \(\rho \bar V\)

../../_images/momentum-p1-21.png

Caption

The result of application of theReynolds Transport Theorem to the linear momentum term is an integral equation that relates the sum of external forces to the rate of change of linear momentum in the control volume plus the net momentum leaving across the control surface as depicted in momentum-p1-3

../../_images/momentum-p1-31.png

Caption

Note

Application of momentum principles uses three primary principles:

  • Select an inertial (non-acelerating) reference frame

  • Indicate positive and negative coordinate directions

  • Draw the CV/CS and indicate

    • Forces

    • Velocities

    • Outward pointing area vectors (\(dA \bar\) vectors)

Example 1: Application of Momentum to a Jet Pump

A water jet pump has jet area of 0.01 sq.m. and jet speed of 30 m/s. The jet is within a secondary stream of water having speed 3 m/sec. The total duct area is 0.075 sq.m. The water is completely mixed in the pump and exits at uniform velocity; find the speed at the pump exit and the pressure rise in the pump.

The problem statement is explicitly repeated in momentum-ex1-1

../../_images/momentum-ex1-11.png

Caption

Apply our problem solving protocol as in momentum-ex1-2

../../_images/momentum-ex1-21.png

Caption

Note on the CV diagram to draw:

  • the +/- directions

  • the CV/CS

  • the \(\bar dA \) vectors, and

  • the \(\bar V \) vector(s)

Then apply continunity and momentum to find the unknown values as in momentum-ex1-3

../../_images/momentum-ex1-31.png

Caption

# computational thinning
u_jet = 30.0
u_approach = 3.0
a_jet = 0.01
a_approach = 0.065
a_total = 0.075
rho = 1000.0
# continunity
u_exit = (u_jet*a_jet+u_approach*a_approach)/a_total
# momentum
delta_p = -(rho/a_total)*(a_total*u_exit**2 - a_jet*u_jet**2 - a_approach*u_approach**2)
# results
print("Pump exit velocity ",round(u_exit,3),"     meters per second")
print("Added pressure     ",round(delta_p,3)," Pascals")

Pump exit velocity  6.6      meters per second
Added pressure      84240.0  Pascals

Example 2: Forces in a Pipe Fitting

Consider a pipe fitting as depicted in momentum-ex2-1

../../_images/momentum-ex2-11.png

Caption

Apply our problem solving protocol as in momentum-ex2-2

../../_images/momentum-ex2-21.png

Problem Solving Protocol: Sketch, CV Definition, Governing Principles, Knowns and Unknowns

Apply our analysis tools as in momentum-ex2-3

../../_images/momentum-ex2-31.png

Application of linear momentum in the two coordinate directions

Applications of Momentum

Momentum is used to find forces on objects that are changing the flow field.

Forces are important in things like

  • Bridge piers

  • Wind turbine support columns

  • Retaining walls

  • Dams

As an example consider the force on a sluice gate (underflow from a powerhouse) as depicted in momentum-p2-1

../../_images/momentum-p2-11.png

Schematic elevation view of a sluice gate

Upon first inspection one would be tempted to treat the gate as a submerged plate and use hydrostatic calculations, except at point A the pressure is atmospheric, same as at the free surface just upstream of the gate, so the required conditions for hydrostatic analysis do not apply in this case.

So instead of trying to find pressure on the gate, find force of gate on the water, then by eth equal-opposite action-reaction (Newtons’s 2nd law) we can find force of water on the gate .

A first step is to draw a control volume as depicted in momentum-p2-2

../../_images/momentum-p2-21.png

CV/CS for sluice gate

Using the figure as a guide draw:

  • the +/- directions

  • the CV/CS

  • the forces

  • the \(\bar dA \) vectors, and

  • the \(\bar V \) vector(s)

The resulting sketch is shown in momentum-p2-3

../../_images/momentum-p2-31.png

CV/CS for sluice gate

Some assumptions are in order; in particular the distance upstream and downstream are relatively small (a few hundred feet) and the frictional contribution is small by virtue of having only a short distance to act, so we neglect the frictional component. Using the velocity and area directions to resolve the inner products in the flux integrals produces the diagram in momentum-p2-4

../../_images/momentum-p2-41.png

CV/CS for sluice gate

The two pressure forces are some distance from the gate and are hydrostatic and computed using that equation.

\(p = \gamma \bar h\) as depicted in momentum-p2-5

../../_images/momentum-p2-51.png

Pressure force diagram

momentum-p2-6 is a skecth showing the collection of forces on the CV

../../_images/momentum-p2-61.png

Forces on CV

momentum-p2-7 completes the analysis; then we can generalize for any sluice gate (in rectangular channles) using ENGR-1330 principles as shown in the script below.

../../_images/momentum-p2-71.png

Forces on CV

Sluce Gate Calculations

d1 = 1.5 # depth upstream
d2 = 0.0563 # depth downstream
u1 = 0.2 # upstream velocity
u2 = 5.33 # downstream velocity
rho = 1000.0 # density
g = 9.8 # gravitational acceleration
w = 1.0 # channel width
Rg = (0.5*rho*g)*(d1**2 - d2**2)+rho*(d1*u1**2 - d2*u2**2)
print('Upstream Depth',d1,' m')
print('Upstream Speed',u1,'m/s')
print('Downstream Depth',d2,' m')
print('Downstream Speed',u2,'m/s')
print('Force/width',round(Rg,2),'N/m')

Upstream Depth 1.5  m
Upstream Speed 0.2 m/s
Downstream Depth 0.0563  m
Downstream Speed 5.33 m/s
Force/width 9470.05 N/m

Example 3 Shallow Wave Speed in a Channel

Consider a shallow wave in a rectangular channel as in momentum-ex3-1

../../_images/momentum-ex3-11.png

Problem Statement

First we draw a control volume as in momentum-ex3-2

../../_images/momentum-ex3-21.png

Sketch

Then apply continunity to find the apparent velocities as in momentum-ex3-3

../../_images/momentum-ex3-31.png

Continunity application

Then apply momentum as in momentum-ex3-4

../../_images/momentum-ex3-41.png

Momentum application

Complete the analysis to find the shallow wave speed as in momentum-ex3-5

../../_images/momentum-ex3-51.png

Analysis results

Readings

  1. CE-3305-2022-1 Syllabus. http://54.243.252.9/ce-3305-webroot/0-Syllabus/ce-3305-2022-1-syllabus.html

  2. Hibbeler, R.C, Fluid Mechanics, 2ed. Prentice Hall, 2018. ISBN: 9780134655413 pp. 293-355

  3. DF Elger, BC Williams, Crowe, CT and JA Roberson, Engineering Fluid Mechanics 10th edition, John Wiley & Sons, Inc., 2013. (placeholder file to get links working). http://54.243.252.9/ce-3305-webroot/3-Readings/EFM-9.pdf

  4. Cleveland, T. G. (2014) Fluid Mechanics Notes to Accompany CE 3305 at Jade-Holshule (TTU Study Abroad 2015-2019), Department of Civil, Environmental, and Construction Engineering, Whitacre College of Engineering. http://54.243.252.9/ce-3305-webroot/3-Readings/ce3305-lecture-9.pdf