CONTINUITY EQUATION
Mass and Volume Flow Rates
The amount of mass flowing through a cross section per unit time is called
the mass flow rate and is denoted by m (with dot at the top) . The dot over a symbol is used to
indicate time rate of change. A fluid flows into or out of a control volume, usually through pipes or
ducts. The differential mass flow rate of fluid flowing across a small area
element dAc in a cross section of the pipe is proportional to dAc itself, the
fluid density p, and the component of the flow velocity normal to dAc,
which we denote as Vn' and is expressed as
BERNOULLIS EQUATION
The Bernoulli equation is an approximate relation between pressure,
velocity, and elevation, and is valid in regions of steady, incompressible
flow where net frictional forces are negligible (Fig. 5–21). Despite its simplicity,
it has proven to be a very powerful tool in fluid mechanics. In this
section, we derive the Bernoulli equation by applying the conservation of
linear momentum principle, and we demonstrate both its usefulness and its
limitations.
The key approximation in the derivation of the Bernoulli equation is that
viscous effects are negligibly small compared to inertial, gravitational, and
pressure effects. Since all fluids have viscosity (there is no such thing as an
“inviscid fluid”), this approximation cannot be valid for an entire flow field
of practical interest. In other words, we cannot apply the Bernoulli equation
everywhere in a flow, no matter how small the fluid’s viscosity. However, it
turns out that the approximation is reasonable in certain regions of many
practical flows. We refer to such regions as inviscid regions of flow, and we
stress that they are not regions where the fluid itself is inviscid or frictionless,
but rather they are regions where net viscous or frictional forces are
negligibly small compared to other forces acting on fluid particles.
Care must be exercised when applying the Bernoulli equation since it is
an approximation that applies only to inviscid regions of flow. In general,
frictional effects are always important very close to solid walls (boundary
layers) and directly downstream of bodies (wakes). Thus, the Bernoulli approximation is typically useful in flow regions outside of boundary layers
and wakes, where the fluid motion is governed by the combined effects of
pressure and gravity forces.
The motion of a particle and the path it follows are described by the
velocity vector as a function of time and space coordinates and the initial
position of the particle. When the flow is steady (no change with time at a
specified location), all particles that pass through the same point follow the
same path (which is the streamline), and the velocity vectors remain tangent
to the path at every point.
here take a look with more explainations.
Mass and Volume Flow Rates
The amount of mass flowing through a cross section per unit time is called
the mass flow rate and is denoted by m (with dot at the top) . The dot over a symbol is used to
indicate time rate of change. A fluid flows into or out of a control volume, usually through pipes or
ducts. The differential mass flow rate of fluid flowing across a small area
element dAc in a cross section of the pipe is proportional to dAc itself, the
fluid density p, and the component of the flow velocity normal to dAc,
which we denote as Vn' and is expressed as
dm= p Vn dA
m= pVA
m1=m2
p1V1A1=p2V2A2
usually p or density is same, without density we get
V1A1=V2A2
thats is equal to Q (Flowrate).
BERNOULLIS EQUATION
The Bernoulli equation is an approximate relation between pressure,
velocity, and elevation, and is valid in regions of steady, incompressible
flow where net frictional forces are negligible (Fig. 5–21). Despite its simplicity,
it has proven to be a very powerful tool in fluid mechanics. In this
section, we derive the Bernoulli equation by applying the conservation of
linear momentum principle, and we demonstrate both its usefulness and its
limitations.
The key approximation in the derivation of the Bernoulli equation is that
viscous effects are negligibly small compared to inertial, gravitational, and
pressure effects. Since all fluids have viscosity (there is no such thing as an
“inviscid fluid”), this approximation cannot be valid for an entire flow field
of practical interest. In other words, we cannot apply the Bernoulli equation
everywhere in a flow, no matter how small the fluid’s viscosity. However, it
turns out that the approximation is reasonable in certain regions of many
practical flows. We refer to such regions as inviscid regions of flow, and we
stress that they are not regions where the fluid itself is inviscid or frictionless,
but rather they are regions where net viscous or frictional forces are
negligibly small compared to other forces acting on fluid particles.
Care must be exercised when applying the Bernoulli equation since it is
an approximation that applies only to inviscid regions of flow. In general,
frictional effects are always important very close to solid walls (boundary
layers) and directly downstream of bodies (wakes). Thus, the Bernoulli approximation is typically useful in flow regions outside of boundary layers
and wakes, where the fluid motion is governed by the combined effects of
pressure and gravity forces.
The motion of a particle and the path it follows are described by the
velocity vector as a function of time and space coordinates and the initial
position of the particle. When the flow is steady (no change with time at a
specified location), all particles that pass through the same point follow the
same path (which is the streamline), and the velocity vectors remain tangent
to the path at every point.
 | ||
| The Bernoulli equation states that the sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow. |
Continuity and Bernoulli's Equations
Tiada ulasan:
Catat Ulasan