iiia3_fluidflowconcepts_bernoulliapplications

Southern Methodist University

Bobby B. Lyle School of Engineering

CEE 2342/ME 2342 Fluid Mechanics

Roger O. Dickey, Ph.D., P.E.

III. BASIC EQS. OF HYDRODYNAMICS

A. Fluid Flow Concepts – Application of the Bernoulli and Continuity Equations

Reading Assignment:

Chapter 3 Elementary Fluid Dynamics … ,

Sections 3.4 and 3.8

Rate of Flow –

Consider steady flow of an incompressible fluid into a closed tank, filled with the fluid:

d1 = V1δt

d2 = V2δt

Conduit (1) Area, A1 Conduit (2) Area, A2

d1

d2

Consider the volume element in the inlet conduit

having cross-sectional area A1 perpendicular to

the velocity vector, and having length d1 giving

it a total volume, , and a mass, .

If a time interval, δt, is required for the fluid

element to move a distance equal to its length,

i.e., for the entire volume of the element to move

into the tank, then its average velocity is, tdV

11

111 dAV 11 Vm

Substituting for d1 in the expression for

velocity,

Rearranging,

The new variable is given the symbol Q, and is called the volumetric flow rate or discharge [L3/T], i.e.,

tAVV11

1

1

1

AV

111 AVt

V

tV

1

111 AVQ

Pipe Diameter, D

Pipe Area, A1 = D2/4

Velocity, V1

Volumetric Flow Rate in Pipe #1, Q1 = V1A1

For example:

Multiplying both sides of the expression for by the density of the fluid, , yields,

The new variable is given the symbol , and is called the mass flow rate or mass transport rate [M/T], i.e.,

tV

1

tV 1

1111 AVQm

111 AV

tV

m

Over the same time interval δt, consider the

volume element in the outlet conduit having

cross-sectional area A2 perpendicular to the

velocity vector, and having length d2 giving it a

total volume, . Using reasoning similar

to that used for the inlet conduit,222 AVQ

222 dAV

2222 AVQm

Since the fluid in the tank is incompressible,

every particle of fluid entering the tank through

the inlet conduit displaces an equal volume of

fluid and forces it into the outlet conduit. Thus,

This is called the continuity equation for

incompressible flow, and it derives directly from

the principle of mass conservation.

22112121

221121

or or (ii)or (i)

AVAVQQmmAVAVQQ

Many interesting and important fluid dynamics

problems can be solved with Bernoulli’s Equation,

the Continuity Equation and, especially, with the

two equations combined when written between

any two points on a streamline:

22112121

221121

222

2

211

1

or or (iii)or (ii)

22 (i)

AVAVQQmmAVAVQQ

gVpz

gVpz

*Important Point –

Using deductive reasoning for now (a formal derivation will be presented soon), the Continuity Equation can be expanded to include multiple inflow and outflow fluid streams. If they all have the same, constant fluid density, , a more general differential form of the Continuity Equation may be written:

All

outoutAll

inin AVAVdtVd

Then for steady flow, yielding:

Noting that Q = VA, this expression may be

concisely written:

Alloutout

Allinin

Alloutout

Allinin

AVAV

AVAV

0

0dtVd

All

outAll

in QQ

Refer to Handouts III.A.1. Bernoulli and

Continuity Equation Examples, and III.A.2.

Additional Continuity and Bernoulli

Equation Examples.