of liquids decreases with increasing temperature, and that of gases increases with increasing temperature.
1.2.2 Fluid Flow
If a fluid body with cross‐sectional area A is flowing at velocity C, its volumetric flow rate Q is given by
(1.25)
And the mass flow rate
(1.26)
Consider now the flow of this fluid through the control volume shown in Figure 1.6. The mass flow equations at inlet 1 and exit 2 are given by
Figure 1.6 Fluid flow through a control volume.
The continuity equation or equation of conservation of mass for this flow is obtained by equating the mass flow rates at sections 1 and 2,
(1.27)
The total energy (in units of N. m) for an element of fluid of mass m at sections 1 and 2 of the control volume shown in Figure 1.6 is given by
where
mp/ρ : flow energy required to move fluid element m against pressure p
mC2/2 : kinetic energy of element m travelling at velocity C
mgz : potential energy of the element due to its elevation z relative to a reference level
If there is no energy addition, storage, or loss between sections 1 and 2, the energy will be conserved, and E1 = E2:
(1.28)
or, in terms of pressure heads (in metres, for example)
(1.29)
If both sides of Eq. (1.28) are multiplied by ρ, it can be rewritten in terms of fluid pressure as
(1.30)
Equation (1.30) is known as Bernoulli's equation. If it is rewritten in differential form, it gives Euler's equation:
(1.31)
1.2.2.1 General Energy Equation
If there is energy addition, storage, or loss between sections 1 and 2 in Figure 1.6, the energy equation can be written as
(1.32)
where ∑ fl is the algebraic sum of all losses and gains between points 1 and 2. These could include mechanical energy gained from a booster pump, mechanical energy lost by running a fluid motor or turbine, and energy lost due to friction in the control volume.
1.2.3 Acoustic Velocity (Speed of Sound)
The acoustic velocity of a fluid is the speed of sound in the fluid under isentropic conditions and is given by
(1.33)
For a gas of molecular mass μf,
Rewriting Eq. (1.33) in terms of the universal gas constant
(1.34)
At a given temperature and ratio of specific heats, the acoustic velocity can be written as
The Mach number Ma (in honour of Ernst Mach) is defined as
(1.35)
The Mach number is used to indicate speed of flow or forward speed of aircraft and rockets and also to indicate different flow regimes:
| Mach number | Flow regime |
| Ma < 1 | Subsonic flow |
| Ma = 1 | Sonic flow |
| 1 < Ma < 5 | Supersonic flow |
| Ma > 5 | Hypersonic flow |
1.2.4 Similitude and Dimensional Analysis
Many problems in fluid mechanics can be solved analytically; however, in a large number of cases, problems can only be solved by experimentation. Similitude and dimensional analysis make it possible to use measurements obtained in a laboratory under specific conditions to describe the behaviour of other similar systems without the need for further experimentation.
1.2.4.1
