Principles relating to fluids at rest can be obtained with no ambiguity by purely rational methods. Some natural principles which are universally true are the basis of calculations. In flowing fluid, however, the laws that govern the motion are complex and most of the time defies mathematical expressions. Thanks to experimental data combined with mathematical theories which solved countless engineering problems in the past.

## Discharge

Also known as flow rate, discharge is the amount of fluid passing a section of a stream in unit time is called the discharge. If v is the mean velocity and A is the cross sectional area, the discharge Q is defined by Q = Av which is known as volume flow rate. Discharge is also expressed as mass flow rate and weight flow rate.

Volume flow rate, $Q = Av$

Mass flow rate, $M = \rho Q$

Weight flow rate, $W = \gamma Q$

Where:
Q = discharge in m3/sec or ft3/sec
A = cross-sectional area of flow in m2 or ft2
v = mean velocity of flow in m/sec or ft/sec
ρ = mass density of fluid in kg/m3 or slugs/ft3
γ = unit weight of fluid in N/m3 or lb/ft3

Laminar Flow
Flow is said to be laminar when the paths of the individual particles do not cross or intersect. By many careful experiments to commercial pipes of circular cross section, the flow is laminar when the Reynolds’ number Re is less than 2100.

Turbulent Flow
The flow is said to be turbulent when its path lines are irregular curves and continuously cross each other. The paths of particles of a stream flowing with turbulent motion are neither parallel nor fixed but it aggregates to forward motion of the entire stream. Reynolds’ number greater than 2100 normally defines turbulent flow but in highly controlled environment such as laboratories, laminar flow can be maintained up to values of Re as high as 50,000. However, it is very unlikely that such condition can occur in the practice.

Steady flow occurs if the discharge Q passing a given cross section of a stream is constant with time, otherwise the flow is unsteady.

Uniform Flow
The flow is said to be uniform if, with steady flow for a given length, or reach, of a stream, the average velocity at every cross-section is the same. Uniform flow usually occurs to incompressible fluids flowing in a stream of constant cross section. In streams where velocity and cross section changes, the flow is said to be non-uniform.

Continuous Flow
By the principle of conservation of mass, continuous flow occurs when at any time, the discharge Q at every section of the stream is the same.

Continuity Equations
For incompressible fluids:

$Q = A_1v_1 = A_2v_2 = A_3v_3 = \text{ constant}$

For compressible fluids:

$M = \rho_1A_1v_1 = \rho_2A_2v_2 = \rho_3A_3v_3 = \text{ constant}$

or

$W = \gamma_1A_1v_1 = \gamma_2A_2v_2 = \gamma_3A_3v_3 = \text{ constant}$

## Energy and Head of Flow

### Kinetic Energy and Potential Energy

Energy is defined as ability to do work. Both energy and work are measured in Newton-meter (or pounds-foot in English). Kinetic energy and potential energy are the two commonly recognized forms of energy. In a flowing fluid, potential energy may in turn be subdivided into energy due to position or elevation above a given datum, and energy due to pressure in the fluid. Head is the amount of energy per Newton (or per pound) of fluid.

Kinetic energy is the ability of a mass to do work by virtue of its velocity. The kinetic energy of a mass m having a velocity v is ½mv2. Since m = W/g,

$K.E. = W \dfrac{v^2}{2g}$

$\text{Velocity head} = \dfrac{K.E.}{W} = \dfrac{v^2}{2g}$

The velocity head of circular pipe of diameter D flowing full can be found as follows.
$\dfrac{v^2}{2g} = \dfrac{(Q/A)^2}{2g} = \dfrac{Q^2}{2gA^2}$

$\dfrac{v^2}{2g} = \dfrac{Q^2}{2g(\frac{1}{4}\pi D^2)^2} = \dfrac{16Q^2}{2g(\pi^2 D^4}$

$\dfrac{v^2}{2g} = \dfrac{8Q^2}{\pi^2 g D^4}$

In connection to the action of gravity, elevation energy is manifested in a fluid by virtue of its position or elevation with respect to a horizontal datum plane.

$\text{Elevation energy} = Wz$

$\text{Elevation head} = \dfrac{\text{Elevation energy}}{W} = z$

A mass of fluid acquires pressure energy when it is in contact with other masses having some form of energy. Pressure energy therefore is an energy transmitted to the fluid by another mass that possesses some energy.

$\text{Pressure energy} = W \dfrac{p}{\gamma}$

$\text{Pressure head} = \dfrac{\text{Pressure energy}}{W} = \dfrac{p}{\gamma}$

### Total Energy of Flow

The total energy or head in a fluid is the sum of kinetic and potential energies. Recall that potential energies are pressure energy and elevation energy.

Total energy = Kinetic energy + Pressure energy + Elevation energy

In symbol, the total head energy is

$E = \dfrac{v^2}{2g} + \dfrac{p}{\gamma} + z$

Where:
v = mean velocity of flow (m/sec in SI and ft/sec in English)
p = fluid pressure (N/m2 or Pa in SI and lb/ft2 or psf in English)
z = position of fluid above or below the datum plane (m in SI and ft in English)
g = gravitational acceleration (9.81 m/sec2 in SI and 32.2 ft/sec2 in English)
γ = Unit weight of fluid (N/m3 in SI and lb/ft3 in English)

### Power and Efficiency

Power is the rate of doing work per unit of time. For a fluid of unit weight γ (N/m3) flowing at the rate of Q (m3/sec) with a total energy of E (m), the power (Watt) is

$\text{Power} = Q \gamma E$

$\text{Efficiency} = \dfrac{\text{Output}}{\text{Input}} \times 100\%$

Note:
1 horsepower (hp) = 746 Watts
1 horsepower (hp) = 550 ft-lb/sec
1 Watt = 1 N-m/sec = 1 Joule/sec

## Bernoulli’s Energy Theorem

Daniel Bernoulli
(1700 - 1782)

Applying the law of conservation of energy to fluids that may be considered incompressible, Bernoulli’s theorem may be stated as follows:

Neglecting head lost, the total amount of energy per unit weight is constant at any point in the path of flow.

### Bernoulli's Energy Equations

Without head losses, the total energy at point (1) is equal to the total energy at point (2). No head lost is an ideal condition leading to theoretical values in the results.

$E_1 = E_2$

$\dfrac{{v_1}^2}{2g} + \dfrac{p_1}{\gamma} + z_1 = \dfrac{{v_2}^2}{2g} + \dfrac{p_2}{\gamma} + z_2$

The actual values can be found by considering head losses in the computation of flow energy.

$E_1 - HL_{1-2} = E_2$

$\dfrac{{v_1}^2}{2g} + \dfrac{p_1}{\gamma} + z_1 - HL_{1-2} = \dfrac{{v_2}^2}{2g} + \dfrac{p_2}{\gamma} + z_2$

Energy Equation with Pump
In most cases, pump is used to raise water from lower elevation to higher elevation. In a more technical term, the use of pump is basically to increase the energy of flow. The pump consumes electrical energy (Pinput) and delivers flow energy (Poutput).

$E_1 + {HA} - HL_{1-2} = E_2$

$\dfrac{{v_1}^2}{2g} + \dfrac{p_1}{\gamma} + z_1 + {HA} - HL_{1-2} = \dfrac{{v_2}^2}{2g} + \dfrac{p_2}{\gamma} + z_2$

$\text{Output power of pump} = Q \gamma {HA}$

Energy Equation with Turbine
Turbines extract flow energy and converted it into mechanical energy which in turn converted into electrical energy.

$E_1 - {HE} - HL_{1-2} = E_2$

$\dfrac{{v_1}^2}{2g} + \dfrac{p_1}{\gamma} + z_1 - {HE} - HL_{1-2} = \dfrac{{v_2}^2}{2g} + \dfrac{p_2}{\gamma} + z_2$

$\text{Input power of turbine} = Q \gamma {HE}$

### Hydraulic and Energy Grade Lines

Hydraulic grade line, also called hydraulic gradient and pressure gradient, is the graphical representation of the potential head (pressure head + elevation head). It is the line to which liquid rises in successive piezometer tubes. The line is always at a distance (p/γ + z) above the datum plane.

Characteristics of HGL
• HGL slopes downward in the direction of flow but it may rise or fall due to change in pressure.
• HGL is parallel to EGL for uniform pipe cross section.
• For horizontal pipes with constant cross section, the drop in pressure gradient between two points is equivalent to the head lost between these points.

Energy grade line is always above the hydraulic grade line by an amount equal to the velocity head. Thus, the distance of energy gradient above the datum plane is always (v2/2g + p/γ + z). Energy grade line therefore is the graphical representation of the total energy of flow.

Characteristics of EGL
• EGL slopes downward in the direction of flow and will only rise with the presence of pump.
• The vertical drop of EGL between two points is the head lost between those points.
• EGL is parallel to HGL for uniform pipe cross section.
• EGL is always above the HGL by v2/2g.
• Neglecting head loss, EGL is horizontal.

Illustration showing the behavior of EGL and HGL

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