mass momentum energy equations

PREPARED BY:
ARUNESWAR.G
ASHWIN
KUMAR.P

This chapter deals with four equations commonly used in fluid
mechanics: the mass, Bernoulli, Momentum and energy equations.
•The Mass equation is an expression of the conservation of mass
principle.
•The Bernoulli equation is concerned with the conservation of
kinetic, potential, and flow energies of a fluid stream and their
conversion to each other in regions of flow where net viscous forces
are negligible and where other restrictive conditions apply. The
energy equation is a statement of the conservation of energy
principle.
•In fluid mechanics, it is found convenient to separate mechanical
energy from thermal energy and to consider the conversion of
mechanical energy to thermal energy as a result of frictional effects
as mechanical energy loss. Then the energy equation becomes the
mechanical energy balance.

 The conservation of mass relation for a closed system undergoing a
change is expressed as msys = constant or dmsys/dt= 0, which is a
statement of the obvious that the mass of the system remains
constant during a process.
 For a control volume (CV) or open system, mass balance is
expressed in the rate form as
 where min and mout are the total rates of mass flow into and out of the
control volume, respectively, and dmCV/dt is the rate of change of
mass within the control volume boundaries.
 In fluid mechanics, the conservation of mass relation written for a
differential control volume is usually called the continuity equation.

 The conservation of mass principle for a control
volume can be expressed as: The net mass transfer to or from a control
volume during a time interval t is equal to the net change (increase or
decrease) in the total mass within the control volume during t. That is,

where ∆mCV= mfinal – minitialis the change in the mass of the control volume
during the process. It can also be expressed in rate form as
The Equations above are often referred to as the mass
balance and are applicable to any control volume
undergoing any kind of process.

 During a steady-flow process, the total amount of mass contained
within a control volume does not change with time (mCV = constant).
Then the conservation of mass principle requires that the total amount
of mass entering a control volume equal the total amount of mass
leaving it.
When dealing with steady-flow processes, we are not interested in the
amount of mass that flows in or out of a device over time; instead, we
are interested in the amount of mass flowing per unit time, that is, the
mass flow rate
It states that the total rate of mass entering a control volume is equal to
the total rate of mass leaving it

 Many engineering devices such as nozzles, diffusers, turbines,
compressors,and pumps involve a single stream (only one inlet and one
outlet).
 For these cases, we denote the inlet state by the subscript 1 and the
outlet state by the subscript 2, and drop the summation signs

 Many fluid systems are designed to transport a fluid from one location
to another at a specified flow rate, velocity, and elevation difference,
and the system may generate mechanical work in a turbine or it may
consume mechanical work in a pump or fan during this process.
 These systems do not involve the conversion of nuclear, chemical, or
thermal energy to mechanical energy. Also, they do not involve any heat
transfer in any significant amount, and they operate essentially at
constant temperature.
 Such systems can be analyzed conveniently by considering the
mechanical forms of energy only and the frictional effects that cause
the mechanical energy to be lost (i.e., to be converted to thermal energy
that usually cannot be used for any useful purpose).
 The mechanical energy can be defined as the form of energy that
can be converted to mechanical work completely and directly by an
ideal mechanical device.
 Kinetic and potential energies are the familiar forms of mechanical
energy.

Therefore, the mechanical energy of a flowing fluid can be expressed on
a unit-mass basis as
In the absence of any changes in flow velocity and elevation, the power
produced by an ideal hydraulic turbine is proportional to the pressure drop
of water across the turbine.

Most processes encountered in practice involve only certain forms of energy,
and in such cases it is more convenient to work with the simplified versions of
the energy balance. For systems that involve only mechanical forms of energy
and its transfer as shaft work, the conservation of energy principle can be
expressed conveniently as
where Emech, loss represents the conversion of mechanical energy to
thermal energy due to irreversibilities such as friction. For a system in
steady operation, the mechanical energy balance becomes
Emech, in = Emech, out + Emech, loss

 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 ( as shown in
the Figure below). Despite its simplicity, it has proven to be a very powerful
tool in fluid mechanics.
The Bernoulli equation is an
approximate equation that is valid
only in in viscid regions of flow
where net viscous forces are
negligibly small compared to
inertial, gravitational, or pressure
forces. Such regions occur
outside of boundary layers and
wakes.

the Bernoulli Equation is derived from the mechanical energy equation
since the we are dealing with steady flow system with out the effect
of the mechanical work and the friction on the system the first terms
become zero.
This is the famous Bernoulli equation, which is commonly used in
fluid mechanics for steady, incompressible flow along a streamline in
inviscid regions of flow.

The Bernoulli equation can also be written between any two points on the
same streamline as

 Steady flow The first limitation on the Bernoulli equation is that it is applicable to
steady flow.
 Frictionless flow Every flow involves some friction, no matter how small, and
frictional effects may or may not be negligible.
 No shaft work The Bernoulli equation was derived from a force balance on a
particle moving along a streamline.
 Incompressible flow One of the assumptions used in the derivation of the
Bernoulli equation is that ρ= constant and thus the flow is incompressible.
 No heat transfer The density of a gas is inversely proportional to temperature,
and thus the Bernoulli equation should not be used for flow sections that involve
significant temperature change such as heating or cooling sections.
 Strictly speaking, the Bernoulli equation is applicable along a
streamline, and the value of the constant C, in general, is different for different
streamlines. But when a region of the flow is irrotational, and thus there is no
vorticity in the flow field, the value of the constant C remains the same for all
streamlines, and, therefore, the Bernoulli equation becomes applicable across
streamlines as well.

EXAMPLE :Spraying Water into the Air:
Water is flowing from a hose attached to a water main at 400 kPa gage (Fig.
below). A child places his thumb to cover most of the hose outlet, causing a
thin jet of high-speed water to emerge. If the hose is held upward, what is the
maximum height that the jet could achieve?
This problem involves the conversion of flow, kinetic, and potential energies
to each other without involving any pumps, turbines, and wasteful
components with large frictional losses, and thus it is suitable for the use of
the Bernoulli equation. The water height will be maximum under the stated
assumptions. The velocity inside the hose is relatively low (V1 = 0) and we
take the hose outlet as the reference level (z1= 0). At the top of the water
trajectory V2 = 0, and atmospheric pressure pertains. Then the Bernoulli
equation simplifies to

EXAMPLE :Water Discharge from a Large Tank:
A large tank open to the atmosphere is filled with water to a height of 5 m from
the outlet tap (Fig. below). A tap near the bottom of the tank is now opened, and
water flows out from the smooth and rounded outlet. Determine the water velocity
at the outlet.
This problem involves the conversion of flow, kinetic, and potential energies to
each other without involving any pumps, turbines, and wasteful components with
large frictional losses, and thus it is suitable for the use of the Bernoulli
equation. We take point 1 to be at the free surface of water so that P1= Patm
(open to the atmosphere), V1 = 0 (the tank is large relative to the outlet), and z1=
5 m and z2 = 0 (we take the reference level at the center of the outlet).
Also, P2 = Patm (water discharges into the atmosphere).

Then the Bernoulli equation simplifies to
Solving for V2 and substituting
The relation is called the Toricelli equation.

mass momentum energy equations

mass momentum energy equations

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