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For constant volume process
W = ∫PdV
Since
dV = 0
so W = 0 for constant volume process.
For free expansion work done is zero. W = 0
The process in which high-pressure fluid is converted to low pressure by using a throttle valve is Throttling. In the throttling process enthalpy remains constant, work done is zero.

Relationship between P - V - T for a reversible adiabatic process:

IC Engines may be classified based on the state of air-fuel mixture present at the time of ignition in the engine cycle, the type of ignition employed and the nature of combustion process after ignition of the air-fuel mixture.

Physical State of Mixture

• Homogeneous Charge:
  • Premixed outside (conventional gasoline and gas engines with fuel inducted in the intake manifold)
  • Premixed in-cylinder: In- cylinder direct injection and port fuel injection
• Heterogeneous Charge (Diesel Engine)

Ignition Type

• Positive source of Ignition e.g., spark ignition
• Compression ignition

Mode of Combustion

• Flame propagation
• Spray combustion

The combustion process in a diesel engine is heterogeneous—that is, the fuel and air are not premixed prior to initiation of combustion. Because of heterogeneous material low speed and low pressure is generated as compared to SI engine where homogenous mixture of fuel is used

Efficiency of a Reversible Heat Engine:

η = 1 when T_L = 0 for any T_H
A heat engine can work on 100% efficiency when there is only heat input and no heat loss.
As W = Q_H – Q_L
All the heat supplied will be used in producing the work and there will be no losses.
All the temperature scale is taken in Kelvin.

Otto cycle: It is the standard air cycle used in the spark ignition (SI) engines or petrol engines.

Different processes of the Otto cycle are as follows:
Process 1-2: Isentropic compression
Process 2-3: Constant volume heat addition
Process 3-4: Isentropic expansion
Process 4-1: Constant volume heat rejection. 

For hollow cylinder:


The rate of heat transfer through a hollow cylinder of inner and outer radii r1 and r2, respectively, depends on ratio of radii (r2/r1).

Note:

Any identifiable/observable characteristic of a system by which the physical condition of the system may be described is called property of the system.

Some familiar properties are pressure, temperature, volume and mass.

Properties may be divided into two categories i.e. extensive (dependent of mass) and intensive (independent of mass) properties.

As W is positive to it is work output.

Note:

• Work done on the system is negative
• Work done by the system is positive

As a work done is the area under the PV diagram represents work done which is maximum for isobaric (P = c) process.

In practical design of compressors, some clearance is required between the cylinder and piston to prevent hitting of piston to crown of the cylinder. A compressor cycle with clearance is consists of two adiabatic processes and two isobaric processes.

The radiation energy emitted by a black body per unit time and per unit surface area is given by
E_b = σT⁴ (W/m²)
where Stefan – Boltzmann constant, σ = 5.67 × 10-8 W/m² K⁴, T is absolute temperature of the surface in Kelvin.
E_b = Total emissive power of a black body
For a non-black surface having an emissivity ϵ, it follows that
E_b = ϵ σT⁴
Calculation:

E₂ = 16 E₁ = 16 E

Refrigeration is based on Clausius statement of second law of thermodynamics. This law states that it is impossible to develop a device which works on a cycle and transfers heat from lower temperature to higher temperature without any external energy input. On the other hand, heat engine works on the Kelvin Planck’s statement of second law of thermodynamics.

For Carnot cycle refrigeration:

Equations of equilibrium for Non-concurrent force System: A non-concurrent force systems will be in equilibrium if the resultant of all forces and moments is zero.

• ΣF_x = 0, ΣF_y = 0 and ΣM = 0

Equations of equilibrium for concurrent force System: For the concurrent forces, the lines of action of all forces met at a point and hence the moment of those force about that point will be zero or ΣM = 0 automatically.

• ΣF_x = 0 and ΣF_y = 0

It has been observed that when a body, lying over another body, is gently pushed, it does not move because of the frictional force, which prevents the motion.

If the applied force exceeds this limit, the force of friction cannot balance it and the body begins to move, in the direction of the applied force. This maximum value of frictional force, which comes into play, when a body just begins to slide over the surface of the other body, is known as limiting friction.

Static friction is the friction experienced by a body when it is at rest or it is the friction when the body tends to move.

• Momentum is conserved in all collisions.
• In elastic collision, kinetic energy is also conserved.
• In inelastic collision, kinetic energy is not conserved. In perfectly inelastic collision, objects stick together after collision.

Conservation of momentum:
m₁v₁ + m₂v₂ = (m₁ + m₂) v_f

Kinetic energy before collision:

Kinetic energy after collision:



Change in kinetic energy:
 


 
Loss of kinetic energy:

ΣF_x = 3P – 5P = -2P
ΣF_x = P – 2P = -P
 

Consider a block of mass m lying on a frictionless inclined plane of length AB = L, height = h and angle of inclination θ.

When the block is released, it moves down the plane under a force mg sin θ.
Hence the acceleration of the block down the plane is
a = g sin θ
If the block starts from rest from point A, then its velocity when it reaches the bottom B is given by
v² – u² = 2as
v² – 0 = 2aL
 
For the block which falls vertically downward of height, h:

So, speed of both the blocks will be same.

Moment of inertia of a disc about its diameter:

Moment of inertia of a disc about its centre:

Calculation:

Two circular discs have masses in the ratio 1: 2 and radii in the ratio 2: 1. The ratio of their moment of inertia about the diameter is

Consider a mass m rotating around an axis, a distance r away.
Newton’s second law: F = ma
We know a = αr
(a = tangential acceleration; α = angular acceleration)
F = ma = mrα
Calculation:
F = ma = m r α
300 = 100 × 2 × α
α = 1.5 rad/s²

Consider an isosceles triangle:

For an equilateral triangle:

 

Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The configuration often takes the form of two parallel plates or the gap between two concentric cylinders.

In plug flow, the velocity of the fluid is assumed to be constant across any cross-section of the pipe perpendicular to the axis of the pipe. The plug flow model assumes there is no boundary layer adjacent to the inner wall of the pipe.

Stokes flow or creeping flow is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low i.e. Re≪1. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large.

Buoyancy or upthrust, is an upward force exerted by a fluid that opposes the weight of an immersed object. Archimedes' Principle states that the magnitude of the buoyant force on an object is equal to the weight of the fluid it displaces.

Barometer is used to determine the local atmospheric pressure.

Mercury is employed in the barometer because its density is sufficiently high for a relative short column to be obtained and also because it has very small vapour pressure at normal temperature. High density scales down the pressure head (h) to represent same magnitude of pressure in a tube of smaller height.

Considered a small diameter glass tube with mercury at normal temperature is inverted within a small tank of mercury, an equilibrium will be reached.

PB = ρgh + Pv

Pv is the pressure exerted on the free surface of the mercury within the glass tube.

The vapour pressure of mercury at normal temperature and atmospheric pressure is only 0.16 Pa while the quantity ρgh is equal to approximately 105 Pa i.e. an insignificant fraction can be neglected.

Due to its small vapour pressure mercury used to be the standard fluid for measuring absolute pressures.

Velocity distribution of a flow of viscous fluid through circular pipe:

The velocity is maximum when r = 0 i.e. at the centre of pipe.

Average velocity:

It is defined as the scalar function of space and time, such that its partial derivative with respect to any direction gives the velocity component at right angles to that direction.

It is denoted by ψ and defined only for two-dimensional flow.

 

Properties of Stream function:

• If stream function exists, it is a possible case of fluid flow which may be rotational or irrotational
• If the stream function satisfies the Laplace equation i.e.  it is a case of irrotational flow

Recommended Similar Important Concept:

Velocity Potential Function:

It is defined as the scalar function of space and time, such that its negative derivative with respect to any direction gives the velocity in that direction.

It is denoted by ϕ and defined for two-dimensional as well as three-dimensional flow.

 

Properties of Stream function:

• If velocity potential function exists, the flow should be irrotational
• If the velocity potential function satisfies the Laplace equation i.e. it is a case of steady incompressible irrotational flow

From the law of dimensional homogeneity, the constant must have the same dimensions as the other additive terms in the equation.
Thus P/γ and v²/2g will have same dimension as z i.e. Length.

All the terms of Bernoulli’s equation:

P/γ = P/ρg = Pressure energy per unit weight of fluid or pressure head
v²/2g = Kinetic energy per unit weight or kinetic head
z = Potential energy per unit weight or potential head

For same orifice at different height:
Q ∝ √h

For Newtonian Fluid: A = 0 and n = 1 (Example: Air, Water, Glycerin)

For Bingham Plastic: A = τ₀ and n = 1(Fluid does not move or deform till there is a critical stress. Example: Toothpaste)

For Dilatant: A = 0 and n > 1 (Fluid starts ‘thickening' with increase in its apparent viscosity. Example: starch or sand suspension or shear thickening fluid)

For Pseudo plastic: A = 0 and n < 1 (Fluid starts ‘thinning' with increase in its apparent viscosity. Example: Paint, polymer solutions, colloidal suspensions or shear thinning fluid)