kinetic energy of a particle of mass dm (Figure 1.5) is 1/2dm(ωl)2, and the total kinetic energy (KE) for the whole rigid body having a constant angular velocity is
(1.23)
Making use of Eq. (1.19), we can write
(1.24)
Table 1.1 summarises the equations of motion of uniformly accelerating bodies in linear and rotational motion. The following notation is used in the equations:
s, v, and a: linear displacement, velocity, and acceleration
θ, ω, and α: angular displacement, velocity, and acceleration.
Subscripts i and f denote initial and final, respectively.
Table 1.1 Equations of motion for linear and rotational motions.
| Linear | Rotational |
| s = vit + at2/2 | θ = θit + αt2/2 |
| vf = vi + at | θf = θi + αt |
| s = (vi + Vf)t/2 | θ = (ωi + ωf)t/2 |
|
|
|
1.2 Fluid Mechanics
Fluid mechanics deals with the behaviour of a fluid – liquid, gas, or vapour – in quiescent state and in a state of motion. Fluids are substances that cannot preserve a shape of their own. In heat engine processes, the fluids used are predominantly in gas form and include air at various degrees of compression and products of combustion at elevated pressures and temperatures. Understanding the principles of fluid mechanics will help students to better handle the processes in the reciprocating and gas turbine engines.
1.2.1 Fluid Properties
1.2.1.1 Mass and Weight
Mass is a measure of inertia and quantity of the body of matter (fluid), m (kg).
Weight is the force with which a body of the fluid is attracted towards the earth by gravity:
Density is the amount of mass per unit volume:
Specific weight is the weight of a unit volume of a substance:
Specific gravity is
where subscripts f and w are for fluid and water, respectively.
1.2.1.2 Pressure
Pressure is the force exerted by a fluid on a unit area of its surroundings:
Pressure acts perpendicular to the walls of the container surrounding the fluid. A column of fluid of height h m having a cross sectional area of A m2 and density ρ kg/m3 will exert a pressure of
1.2.1.3 Compressibility
Compressibility is the change in volume of a substance when subjected to a change in pressure exerted on it. The usual parameter used to measure compressibility of liquids is the bulk modulus of elasticity E:
The compressibility of a gas at constant temperature is defined as
For a perfect gas:
1.2.1.4 Viscosity
Generally, the shearing stress τ developed in a moving fluid between a stationary surface and a moving fluid body is proportional to the velocity gradient Δv/Δy, and the constant of proportionality is the dynamic viscosity μ:
Fluids such as water, oil, gasoline, alcohol, kerosene, benzene, and glycerine behave in accordance with this equation and are known as Newtonian fluids. Fluids that behave otherwise (viscosity changes with stress) are known as non‐Newtonian fluids.
The previous equation can be rewritten in terms of the viscosity as
The units of μ can be developed as follows:
The ratio of dynamic viscosity to density of the fluid is the kinematic viscosity ν:
Viscosity
