9

NOTATION AND ABBREVIATIONS

a — acceleration

a —celerity of water-hammer pressure wave

A — parameter characterizing injector geometry

A — equivalent injector parameter

b/d — relative width of pump impeller

C — constant of integration

C — degrees centigrade

D, d — diameter

D₂/D_l— relative diameter of pump impeller

e — specific energy

E — energy, work

g — acceleration of gravity

G — weight of a fluid

G — weight rate of discharge

h — capillary rise or depression

h — piezometric head

h — loss of energy or head

h_con — loss of head due to contraction

h_exp — loss of head due to expansion

h_f — loss of head due to friction

h_in — inertia head of liquid moving with acceleration

h_i — local losses

H — total head

H_av — available head

H_in — inertia head of liquid in accelerated channel

H_tz — head generated by idealised centrifugal pump with definite number of vanes

H_t — head generated by idealised centrifugal pump with infinite number of vanes

H_m — mean total head

H_req — required head

H_t — theoretical or ideal head

j — resultant of body forces

j — acceleration of moving fluid

J_x — moment of inertia about an axis x

k — capillarity

k — roughness size of projections

k — any coefficient

k_eq — absolute roughness equivalent to Nikuradse's granular roughness

K — volume or bulk, modulus of elasticity

K — pipe entrance factor

K/r_Q — relative roughness

l — length

l_ent — entrance, or transition, length of pipe

l_eq — equivalent pipe length

M — mass

M — mass rate of discharge

n — rate of divergence of diffuser

n — speed of rotation

n_s — specific speed

n_x — tangential g-load

n_y — normal g-load

N — power

N_o — shaft, or brake, horsepower

N_x — tangential acceleration

N_y — normal acceleration

p — pressure

p_ab — absolute pressure

p_atm — atmospheric pressure

p_b —- buoyancy force

p_f — pressure loss due to fric­tion

p_g — gauge pressure

p_i — local loss of pressure

p_t — saturation vapour pressure

p_vac — vacuum, or negative, pressure

p_wh — water-hammer pressure

P — pressure force

Q — volume rate of discharge

Q — delivery, or capacity of pump

Q_t — theoretical, or ideal, delivery of pump

r — radius

R — surface force

R — radius of curvature

R_h — hydraulic radius, or hydraulic mean depth

Re — Reynolds number

Re_cr — critical Reynolds number

Re_t — theoretical, or ideal, Reynold’s number

S — area

t — temperature

T —friction, or shear, force torque

T — torque

u — peripheral velocity at pump impeller

v — velocity

v_cr —critical velocity

v_m —mean velocity

v_t — theoretical, or ideal, velocity of efflux

w — relative velocity at pump impeller

W — volume

y — distance

z — number of vanes in pump impeller

z — vertical distance from any datum level, elevation

α (alpha) —diffuser divergence angle

α — dimensionless coefficient accounting for nonuniform velocity distribution

β (beta) —vane angle

β_p — coefficient of compressibility

β_t — coefficient of thermal expansion

γ (gamma)—specific weight

γ_ef — effective specific weight

Γ (gamma) — circulation

δ (delta)—specific gravity

δ — pipe bend angle

δ — thickness

δ_e — laminar sublayer thickness

ε (epsilon) — coefficient of contraction of a jet

ζ (dzeta)— loss coefficientI

ζ _f — friction loss coefficient

η (eta) — efficiency

η_h — hydraulic efficiency

η_m — mechanical efficiency

η_v — volumetric efficiency

λ (lambda)—friction factor

λ_l — friction factor for laminar flow

λ_t — friction factor for turbulent flow

μ (mu) — coefficient of discharge of orifice

μ — vane number coefficient

μ — dynamic viscosity

ν (nu) — kinematic viscosity

ξ (xi) —form coefficient of orifice

π (pi) — perimeter

ρ (rho) — density

σ (sigma) — stress

σ — cavitation parameter

τ (tau) —shear stress

φ (phi) — coefficient of velocity of an orifice

ψ (psi) — coefficient taking into account stream contraction by pump impeller vanes or hub

ω (omega) — angular velocity

ABBREVIATIONS

Atm — atmosphere = 14.7 Ib/sq. in.

cm — centimetre = 0.3937 in.

cst — centistoke

hr —hour

g — gram = 0.03527 oz (avdp)

kg — kg = 1,000 g = 2.205 Ib (avdp)

lit — litre = 0.2642 gal

ln — natural (Napierian) logarithm

log — common (Briggs) logarithm

m — metre == 100 cm = 1,000 mm =3.281 ft

mm — 0,03937 in.

rpm — revolutions per minute

sec —second

kg-m—kilogram-metre = 0.009 btu

11

Introduction

1. The subject of hydraulics

The branch of mechanics, which treats of the equilibrium and motion of liquids and gases and the force interactions between them and bodies through or around which they flow is called hydromechanics or fluid mechanics. Hydraulics is an applied division of fluid mechanics covering a specific range of engineering problems and methods of their solution. It studies the laws of equilibrium and motion of fluids and their applications to the solution of practical problems.

The principal concern of hydraulics is fluid flow constrained by surrounding surfaces, i. e., flow in open and closed channels and conduits, including rivers, canals and flumes, as well as pipes, nozzles and hydraulic machine elements.

Thus, hydraulics is mainly concerned with the internal flow of fluids and it investigates what might be called "internal" problems, as distinct from "external" problems involving the flow of a continuous medium about submerged bodies, as in the case of a solid body moving in water or in the air. These "external" problems are treated in hydrodynamics and aerodynamics in connection with aircraft and ship design.

It should be noted that the term fluid as employed in hydromechanics has a broader meaning than generally implied in everyday life and includes all materials capable of an infinite change of shape under the action of the smallest external forces.

The difference between a liquid and a gas is that the former tends to gather in globules if taken in small quantities and makes a free surface in larger volumes. An important property of liquids is that, pressure or temperature changes have practically no effect on their volume, i. e., for all practical purposes they can be regarded as incompressible. Gases, on the other hand, contract readily under pressure and expand infinitely in the absence of pressure, i.e., they are highly compressible.

Despite this difference, however, under certain conditions the laws of motion of liquids and gases are practically identical. One such condition is low velocity of the gas flow as compared with the speed of sound through gas.

The science of hydraulics concerns itself mainly with the motion of liquids. The internal flow of gases is studied only insofar as the velocity of flow is much less than that of sound and, consequently, their compressibility can be disregarded. Such cases are frequently encountered in engineering, as for example, in the flow of air in ventilation systems and in gas mains.

Investigation of the flow of liquids, and even more so gases, is a much more difficult and complicated task than studying the motion of rigid bodies. In rigid-body mechanics one deals with systems of rigidly connected particles; in fluid mechanics the object of investigation is a medium consisting of a multitude of particles in constant relative motion.

To the great Galileo belongs the maxim that it is easier to study the motion of remote celestial bodies than that of a stream running at one's feet. Because of these difficulties, fluid mechanics as a science developed along two different paths.

The first was the purely theoretical path of precise mathematical analysis based on the laws of mechanics. It led to the emergence of theoretical hydromechanics, which for a long time existed as an independent science. Its methods provided an attractive and effective means of scientific research. A theoretical analysis of fluid motion, however, encounters many stumbling blocks and does not always answer the questions arising in real situations.

The urgent requirements of engineering practice, therefore, soon gave rise to a new science of fluid motion, hydraulics, in which researchers took to the second path, that of extensive experimenting and accumulation of factual data for application to engineering problems. At its origin, hydraulics was a purely empirical science. Today, though, whenever necessary it employs the methods of theoretical hydromechanics for the solution of various problems; and, conversely, in theoretical hydromechanics experiments are widely used to verify the validity of its conclusions. Thus, the difference in the methods employed by either science is gradually disappearing, and with it the boundary line between them.

The method of investigating fluid flow in hydraulics today is essentially as follows. The phenomenon under investigation is first simplified and idealised and the laws of theoretical mechanics are applied. The results are then compared with experimental data, the discrepancies are established and the theoretical formulas and solutions adjusted so as to make them suitable for practical application.

Many phenomena are so involved as to practically defy theoretical analysis and are investigated in hydraulics on the sole basis of experimental measurement, the results being expressed in empirical formulas. Thus, hydraulics can be called a semiempirical science.

At the same time it is an applied, engineering, science insofar as it emerged from the demands of life and is widely used in engineering. Hydraulics provides the methods of calculating and designing a wide range of hydraulic structures (dams, canals, weirs and pipelines for transporting fluids), machinery (pumps, turbines, fluid couplings) and other devices used in many branches of engineering, in machine-tool design, foundry practice, the manufacture of plastics, etc. Modern aircraft design, with its fluid drives, fuel and lubricating systems, hydraulic shock absorbers, etc., relies extensively on the science of hydraulics.