iiia3_fluidflowconcepts_bernoulliapplications
TRANSCRIPT
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.