Fluids in Motion,Energy of a fluid And Bernoulli’s Equation
When considering fluids in motion it is usual to assume ideal conditions. You may remember that an ideal fluid is one whose dynamic viscosity and surface tension are zero. As a result it would
not experience internal shearing forces, viscous drag would be absent and its flow would always be laminar. Furthermore, an ideal liquid would be incompressible and would never vaporize or
freeze. These assumptions enable steady fluid flow systems to be modeled mathematically and formula to be derived for calculating flow velocities, flow rates, pressures and forces. In many cases
they accurately predict the behavior of the flow systems. In cases where there is some variance, adjustments can often be made to the formula, based on experimental evidence, to give closer agreement between theory and practice.
A steady flow system is one in which the flow velocity is constant at any cross-section. The flow may be through a pipe, an open channel or a duct. Alternatively, it may be in the form of a jet issuing at a steady rate from a nozzle or an orifice. There are two basic equations which can be applied to a steady flow process.They are
(i) the continuity equation
(ii) the steady flow energy equation.
the continuity equation
Consider the steady flow of a fluid at velocity 𝝊 ms⁻¹ along a parallel pipe or duct of cross-sectional area A m², as in Figure 1. Let the volume flow rate be V m³ s⁻¹
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| FIG 1 |
The volume between sections (1) and (2) in Figure 1 has a length equal to the flow velocity. In 1 s this volume will pass through section (2), and so the volume flow rate, measured in m³ s⁻¹, is given by the formula
V =A𝝊 (m³ s⁻¹)
where :V = volume flow rate m³ s⁻¹
A = cross sectional area m²
𝝊 = fluid velocity along a parallel pipe or duct ms⁻¹
The mass flowing per second, or mass flow rate, is the volume flow rate multiplied by the fluid density
m = 𝜌V
m=𝜌A𝝊 (kg s⁻¹)
where :𝜌= the fluid density kg m⁻³
if a pipe whose cross-section changes between twostations (1) and (2) as in Figure 2 . If the fluid is a gas it is quite possible that its density will change especially if there is a change of temperature and pressure between sections (1) and (2). Irrespective of whether the fluid is a liquid or a gas, the mass per second passing section (1) will be the same as that passing section (2) if
the velocities v1 and v2 are steady. There is said to be continuity of
mass which is given by the equation
𝜌1A1𝝊1 = 𝜌₂A₂𝝊2=m (kg s⁻¹)
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| FIG 2 |
A₁v₁=A₂v₂=V (m³ s⁻¹)
The volume per second passing section (1) is now equal to the volume per second passing section (2) and there is then said to be continuity of volume.Energy of a fluid
A moving fluid can contain energy in a number of different forms. If the mass flow rate is m (kg s⁻¹), the values of these energy forms, given as the energy per second passing a particular cross-section,
are as follows:
1. Gravitational potential energy This is the work which has been done to raise the fluid to its height, z m, measured above some given datum level.
potential energy=m g z (J s⁻¹ or W)
2. Kinetic energy This is the work which has been done in accelerating the fluid from rest up to some particular velocity v ms⁻¹ .
kinetic energy= (1 /2)× mv² (J s⁻¹ or W)
3. Internal energy This is the energy contained in a fluid by virtue of the absolute temperature, T (K) to which it has been raised. It is the sum of the individual kinetic energies of the
molecules of the fluid.
internal energy=U(J s⁻¹ or W)
4. Pressure-flow energy This is also called flow work and work of introduction. It is the energy of the fluid by virtue its pressure, p (Pa) This is the pressure to which it has been raised by a pump, fan or compressor to make it flow into or through a system against the prevailing back-pressure. It is analogous to the work that must be done against friction to keep a solid body in motion.
if the steady flow of fluid entering a system as shown in Figure 3. The volume entering per second, Vm³ is the volume between sections (1) and (2) whose length is v m. The pressure-flow energy received by the fluid is the work done per second in pushing it past section (2).
pressure-flow energy = force applied×distance moved per second
pressure-flow energy = p A v
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| FIG 3 |
pressure-flow energy=pV (J s⁻¹ or W)
where :p = fluid pressure (Pa)
V = volume flow rate (m³s⁻¹)
The steady flow energy equation
When a fluid is flowing through a device such as a pump, compressor or turbine, the forms of energy listed above may all undergo a change. Furthermore, there may be a flow of energy into or out of the system in the form of heat and work. Such a system is shown in Figure 4.
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| FIG 4 |
leaving. It follows that for the system shown in Figure 4
m g z1 + (1/2)m v1² +U1+p1V1+Q = m g z2+(1/2)m v2²+U2+p2V2+W
This is known as the full steady flow energy equation (FSFEE)
Bernoulli’s equation
liquids flowing steadily through a pipe or channel, or issuing from a nozzle in the form of a jet, it is assumed that temperature changes are so small as to be negligible. This means that there is little or no change in internal energy, i.e. U1=U2, and that these terms can be eliminated from the steady flow energy equation. It is also assumed that no heat transfer takes place and that no external work is done, i.e. Q=0 and W=0, and these terms can also be eliminated. The steady flow energy equation thus
reduces to
p = pressure (gauge pressure unless otherwise specified)
w = specific weight (weight per unit volume)
v = velocity
g = acceleration due to gravity
Z = height above some specified datum
𝜌 = density
The (p/𝜌g) terms are called pressure heads.
The (v²/2g) terms are called velocity heads.
The (z) terms are called potential heads.
The terms on each side of Bernoulli’s equation give the total head at section (1) of a steady flow system which is equal to the total head at section (2). In practice there is often some energy loss,
zf due to viscous resistance and turbulence, particularly in long pipelines. This can sometimes be estimated and added to the right hand side of Bernoulli’s equation to give
When used together, the continuity equation and Bernoulli’s equation can solve many problems concerned with steady flow.
key points
1- The volume flow rate at any section through a flow system is given by the product
of the cross-sectional area and the flow velocity.
2-For any steady flow system, the mass flow rate is the same at any cross section.
3-The flow velocities at two sections through the steady flow of an incompressible fluid are in the inverse ratio of the cross-sectional areas.
4-Although energy changes might occur in a fluid system, the principle of conservation of energy tells us that the total energy of the system and its surrounding is constant.
5-The units of each term in Bernoulli’s equation are metres.
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