Significance of Critical Pressure Ratio

*Consider a nozzle which connects vessel V_A and V_B. The vessel V_A contains steam at high pressure P_1. The vessel V_B contains steam but its pressure P₂ can be varied based on requirements.

*Initially the pressure P₂ be equal to the pressure P₁ in vessel V_A. By that time, there is no flow through nozzle.

*If the pressure P₂ is gradually reduced in vessel V_B, the discharge through the nozzle increases gradually.

*The pressure P₂, at which the discharge is maximum is called critical pressure.

*If the pressure P₂ is reduced further, the discharge through the nozzle will not increase and it remains at constant level.

* Hence it is confirmed that, at critical pressure ratio, the flow will be maximum.

*The velocity of steam at critical pressure is equal to sonic velocity. This condition will exist at throat cross section of the Convergent divergent nozzle.

Note;

*For convergent – Divergent nozzle, the flow of steam in the convergent portion of the nozzle is subsonic. i.e., the velocity of steam in the convergent section is less than the sound velocity.

*The steam flows in the divergent part of the nozzle is supersonic, i.e., the velocity of the steam in divergent portion is greater than velocity of sound.

Point of Contraflexure

# If the end portion of a beam extends beyond the support, the beam is known as overhanging beam.

Overhanging Beam

# In overhanging beams, the Bending Moment* is positive between the supports, whereas the Bending Moment is negative for overhanging portion.

# At some point on the beam, the B.M is zero after changing its sign from positive to negative or vice-versa.

# That point is known as the point of inflexion or point of contraflexure.

Note:

* Bending Moment (B.M): Algebraic sum of the moments of all the forces acting to the left or right of the section gives bending moment.

Two Reversible Adiabatic Paths Cannot Intersect Each Other

# Let us consider two reversible adiabatic paths 1-3 and 2-3 intersect each other at point 3. 

# Also a reversible constant temperature processes 1-2 be drawn in such a way that it intersects the reversible adiabatic paths at 1 and 2.

# These three reversible processes 1-2, 2-3, 3-1 constitutes a reversible cycle.

# We know that the area under the p-v plot represents the net work output in a cycle. Therefore the area under the three reversible paths represents the net work output in a cycle.

# But such a cycle is not possible, since net work is being produced ia a cycle by a heat engine by exchanging heat with a single reservoir in the processes 1-2, violates the Law of Thermodynamics (Kelvin-Plank). Therefore the consideration of the intersection of the two reversible adiabatic process is wrong.

# Through one point, only one reversible adiabatic can pass.

# As two constant property lines can never intersect each other, it is observed that a reversible adiabatic path must denote some property, which is found later that it is entropy.

Mach Number

# It is defined as the ratio of local velocity of fluid to the acoustic or sound velocity.

It is a dimensionless number designated by M.

It is expressed as, M = C/a Where, C = Velocity of fluid (air),
   a = Velocity of sound.

# Mach number can also be expressed as the square root of the ratio of inertia force
of flow to the elastic force.

M² = (Inertia force of flow)/(Elastic force of fluid)

Different Regimes of Compressible flow

Subsonic Flow:
If the flow of fluid with a mach number less than 1, (i.e., C<a) then the flow is called
a subsonic flow. It is characterized by smooth streamlines.

Sonic Flow:

It the fluid flows with Mach number is equal to unity, (i.e., C=a) then the flow is
considered as sonic flow.

Supersonic flow:

If the Mach number is greater than 1 (i.e., C>a) then the flow is called
supersonic flow. It is observed with an oblique shock.

Hypersonic flow:

It the mach number of the flowing fluid is very high, (i.e., M>5) then the flow
is a hypersonic flow. This flow is observed with severe shock in the flow field. 

Note :
M > 5 --> Hypersonic flow

M > 1 --> Supersonic flow

M = 1 --> Sonic flow

M < 1  --> Subsonic flow

COMBINED FIRST AND SECOND LAW OF THERMODYNAMICS

Combined 1^st and 2^nd Law of Thermodynamics.

By first law of thermodynamics, ΔQ = dU + ΔW

Since ΔW = pdV

ΔQ = dU + pdV ----------------- 1.

From second law of thermodynamics, (Entropy concept)

ΔQ = T. dS ----------------------- 2.

Put equation 2. in 1.

We get, TdS = dU + pdv ---------------- 3.

We know that, enthalpy h = u + pv

On differentiating, we get

dh = du + pdv + vdp,

From equation 3. , dh = Tds + vdp

Tds = dh - vdp -------------- 4.

The equations 3. and 4. are the thermodynamic equations relating the properties of system.

The following are the relations obtained from the first and second laws.

1. dQ = dE + dW: Holds good for all process, reversible or irreversible and for all systems.

2. dQ = dU + dW: Holds good for any process undergone by a closed system.

3. dQ = dU + pdV: It is good for a closed system , where pdV work is present. This relation true only for quasi-static* process.

4. dQ = TdS: This equation is true only for a reversible processes.

5. TdS = dH –Vdp: This relation hold good for any process, since there is no path function term in the equation. 

6. TdS = dU + pdV: It is good for any reversible or irreversible process, undergone by a closed system. Since the properties in the relation which are independent of the path. 

Note:

* A quasi-static process is a thermodynamic process that happens very slowly  for the system to be in equilibrium. It is reversible.

Nusselt Number (Nu)

Physical Significance of Nusselt Number

# Consider a layer of fluid of thickness L, difference in temperature  ΔT = T₂-T₁. 

Heat Transfer through a fluid layer

# Heat transfer through the fluid layer is convection when the fluid involves some motion.

# Heat transfer for this case is Q_Conv = h . A . ΔT ---------1.

# On the contrary, heat transfer through the fluid layer is conduction when the fluid is not in motion.

# Heat transfer for this case is Q_Cond = k . A.  ΔT / (L ) --------2.

# By taking their ratio of equation 1. & 2.,

Q_Conv / Q_Cond = (h .  ΔT .  L) / (k .  ΔT)

                         = (h .  L) / (k )

                         = Nu, Nusselt number.

# Nusselt number represents the improvement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid.

# The larger the Nusselt number, the more efficient the convection.

# Nusselt number, Nu is unity for fluid layer represents heat flux across the layer by pure conduction.

Note;

k - Thermal Conductivity, W/m K
h - Convection heat transfer coefficient , W/m² K

Comparison of Otto, Diesel and Dual cycles

Based on Same Compression Ratio



# The comparison of Otto, Diesel and Dual cycle for the same compression ratio in p-v plot is shown below.

P-v plot

From p-v plot,

1247➡Otto cycle 

1267 ➨Diesel cycle

12357 ➡Dual cycle


# Here the compression ratio ‘r’ is equal for all cycles.

Compression ratio, r = V₂/V₁

Heat rejected is also equal for all cycles.


# We know that the efficiency of cyclic process is given by

η = 1- Q_R/ Q_S

Q_S - Heat supplied or absorbed

Q_R - Heat rejected

For constant heat rejection Q_R , the cycle efficiency increases with increases in the value of heat supplied, Q_S.

# In T-s plot, 

T-s Plot

The area under 1247 represents Q_S for Otto cycle

The area under 1267 represents Q_S for Diesel cycle

The area under 12357 represents Q_S for Dual cycle

Since,  Q_S)_Otto > Q_S)_Dual  > Q_S)_{Diesel ,}

η_Otto > η_Dual  > η_Diesel  , for same compression ratio



Note;

Area under p-v diagram represents Work done during the processes.

Area under T-s diagram represents heat transfer during the processes.

p = C ➡ Constant pressure processes.

v = C ➡ Constant volume processes.

Pv^Ɣ = C ➡ Isentropic processes.

Slope of constant volume and constant pressure processes in T-s plot

# For constant pressure processes(p=C),

ds = m . Cp . dT/T ---------------------- 1.

T-s Plot

# The slope of the curve in T-s plot is dT/dS

Therefore from equation 1.,  dT/dS = T/(m . Cp) ---------------- 2.

For 1 kg of perfect gas,  

Equation 2. becomes

dT/dS = T/Cp    {Since m = 1 kg} -------------------- 3.


# Similarly for constant volume processes(v=C),  

dT/dS = T/Cv --------------------- 4.

# For perfect gas, we know that

Cp > Cv 

or

1/Cp < 1/Cv  --------------------- 5.

# Compare equation 5. with equation 2. & 4.

T/Cp  <  T/Cv

or

{dT/dS} of v=C > {dT/dS} of p=C ----------------6.

Here {dT/dS} is slope of the curve in T-s plot.

# Therefore the slope of the curve for constant volume process (a-b) is higher than that of constant pressure processes (a-b’). 




Note ;

Constant volume process , v=C.

Constant pressure process , p=C.

Cp, Cv - Specific heats at constant pressure and constant volume.

ds - Change in entropy.

Prandtl Number (Pr)

It is the dimensionless parameter which gives the relation between the relative thickness of the velocity boundary layer and thermal boundary layer.

It is denoted by Pr.

Pr =  (Momentum diffusivity)/(Thermal diffusivity)

     = V/α

     = (µ x ρ x Cp )/(ρ x k )

Pr = (µ x Cp )/( k )

Here,

V - Kinematic viscosity, Stroke

μ - Dynamic Viscosity, kg/m-s

α - Thermal diffusivity, k/(ρ x Cp)

Cp - Specific Heat, J/kg K

k - Thermal conductivity, W/mK

Its value is more than 100,000 for heavy oils and less than 0.01 for liquid metals.

It is in the order of 10 for water. 

For gases, it varies from 0.004 to 1, which depicts that both momentum and heat dissipate through the fluid medium at same rate.

It is clear that, heat diffuses very quickly in liquid metals (Pr < 1) relative to momentum transfer. At the same time, as in case of oils, it is very slow. (Pr > 1)

Also, the thermal boundary layer is thicker for liquid metals and thinner for oils comparative to velocity boundary layer.





Note :

The dimensionless number is named after Ludwig Prandtl , who initially set up the concept of boundary layer in  1904 and give major contributions to boundary layer theory.

Nipping of Leaf Springs

In Leaf spring, the stresses in extra full-length leaves are 50 % more than that of  in graduated leaves.

It is important to equalize the stresses in all the leaves on leaf spring.

One of the method of equalizing the stresses is to pre-stress the spring.

This can be done by bending the leaves to different radii of curvature before the assembly.

The full length leaf is given with greater radius of curvature than the adjacent graduated leaves. The shorter leaves are given with smaller radius of curvature than the previous leaves.

The initial gap formed between the extra full length leaf and graduated leaf before the assembly called a ‘nip’.

A difference in radius of curvature is made for such pre-stressing is known as ‘nipping’.

Note;

The leaves are held together by using two U bolts and a center clip.

To keep the leaves in alignment, rebound clips are also provided in order to prevent lateral shifting of leaves.

These springs are used in railway wagons, cars, trucks etc..

Unit of Refrigeration

Tonne of Refrigeration (TR) 

The amount of heat removed (i.e., refrigeration effect) from 1 tonne* of water from 0 °C to make 1 tonne of ice at 0 °C in 24 hours.

It can also be explained as the amount of heat extracted to make 1 tonne of ice from and at 0 °C in 24 hrs.

To make ice from water at 0 °C, it is important to remove latent heat**.

For water, latent heat = 335 kJ/kg.

    Therefore, one tonne of refrigeration = 1000 X 335 kJ in 24 hrs.

   =  (1000 X 335 )/24 kJ/hr

     

   = 13,958.3 kJ/hr

   =  (1000 X 335 )/(24 X 60) kJ/min

   = 232.6 kJ/min

In practice, 1 tonne of refrigeration is equivalent to 210 kJ/min or 3.5 kJ/Sec of heat.

Note:

*1 tonne is equivalent to weight of 1000 kg.

**Amount of heat transfer required to cause a change of phase in mass of a substance at constant temperature and pressure.