Mechanical

What is Shear modulus ?

Modulus of Rigidity definition

Shear modulus is the ratio of the shear stress to the shear strain.

Shear modulus is defined as the measure of the elastic shear stiffness of the material and it is also acknowledged as ‘modulus of rigidity’. So, this parameter answers the question of how rigid a body is?
Shear modulus is the material response to a deformation of body because of the shearing stress and this work as ‘the resistant of the material to shearing deformation’.

In the above figure, Side lengths of this element will not change, though the element experiences a distortion and shape of element is changing from the rectangle to a parallelogram.

Why do we calculate the modulus of rigidity of the material ?
Shear modulus equation | Modulus of Rigidity equation

Shear modulus is the ratio of the shear stress to the shear strain, which is measures the amount of distortion, is the angle (lower case Greek gamma), always ex-pressed in radians and shear stress measured in force acting on an area.
Shear modulus represented as,
G=
Where,
G= shear modulus
τ=shear stress = F/A
ϒ = shear strain=

modulus of rigidity symbol

G or S or μ

What is the SI unit of rigidity modulus ?

Shear modulus units | Unit of modulus of rigidity

Pascal or usually denoted by Giga-pascal. Shear modulus is always positive.

What is the dimensional formula of modulus of rigidity ?

Shear modulus dimensions:

Shear modulus of materials:

Shear modulus of steel | Modulus of rigidity of steel

Structural steel:79.3Gpa
Modulus of rigidity of stainless steel:77.2Gpa
Modulus of rigidity of carbon steel: 77Gpa
Nickel steel: 76Gpa

Modulus of rigidity of mild steel: 77 Gpa

What is the Rigidity modulus of copper in N/m² ?
Modulus of rigidity of copper wire:45Gpa
Shear modulus of Aluminum alloy: 27Gpa
A992 Steel: 200Gpa
Shear modulus of concrete | Modulus of rigidity of concrete: 21Gpa
Silicon shear modulus: 60Gpa
Poly ether ether ketone (PEEK):1.425Gpa
Fiberglass shear modulus: 30Gpa
Polypropylene shear modulus: 400Mpa
Polycarbonate shear modulus: 5.03Gpa
Polystyrene shear modulus:750Mpa

Shear modulus derivation | Modulus of rigidity derivation

If the co-ordinate axes (x, y, z) coincides with principle axes and intended for an isotropic element, the principal strain axes at (0x,0y,0z ) point, and considering alternative frame of reference directed at (nx1, ny1, nz1) (nx2, ny2, nz2) point and in the meantime, Ox and Oy are at 90 degree to each other.
So we can write that,
nx1nx2 + ny1ny2 + nz1nz2 = 0
Here Normal stress (σx’) and the shear-stress ( τx’y’) has been computed utilizing Cauchy’s formulation.
The resultant stress vector on the plane will have components in (x-y-z) as
τx=nx1σ1.
τy=nx2 σ2.
τz=nx3 σ3.

The normal stress on this x-y plane has been computed as the summation of the component’s projections along the normal directions and we can elaborate as
σn= σx=nx^2 σ1+nx^2 σ2+nx^2 σ3.

Similarly, the shear stress component in x and y plane nx2, ny2, nz2.
Thus
τxy=nx1nx2σ1+ny1ny2σ2+nz1nz3σ3
Considering as ε1, ε2, ε3 are the principal strains and the normal-strain is in x-direction, then we can write as
εx’x’=nx1^2ε1+ny^2ε2+nz^2ε3.
The shear strain is obtained as,

εx’=εy’

Substituting the values of σ1, σ 2 and σ 3,

τx’y’=μϒx’y’
Here, μ= shear modulus usually represented by term G.
By taking other axis as Oz¢ with direction cosines (nx3, ny3, nz3) and at right-angle with the Ox¢ and Oy¢. This Ox¢y¢z¢ will create conventional forms an orthogonal set of axes, therefore we can write as,

strain components,

Elastic constants and their relations:

Young’s modulus E:

The young’s modulus is the measure of the stiffness of the body and acts as resistance of the material when the stress is functional. The young’s modulus is considered only for linear stress-strain behavior in the direction of stress.

E=

Poisson’s ratio (μ):

The Poisson’s ratio is the measure of the deformation of the material in the directions perpendicular to the loading. Poisson’s ratio ranges between -1 to 0.5 to maintain young’s modulus, shear modulus (G), bulk modulus positive.
μ=-

Bulk Modulus:

Bulk modulus K is the ratio of the hydrostatic pressure to the volumetric strain and better represented as
K=-v

E and n are generally taken as the independent constants and G and K could be stated as follows:

G=

K=

for an isotropic material, Hooke’s law is reduced to two independent elastic constants named as Lame’s co-efficient denoted as l and m. In terms of these, the other elastic constants can be stated as follows.

If bulk modulus considered to be +ve the Poisson’s ratio never be more than 0.5 (maximum limit for incompressible material). For this case assumptions are
n = 0.5.
3G = E.
K = ∞.
⦁ In terms of principal stresses and principal strains:

⦁ In terms of rectangular stress and strain components referred to an orthogonal coordinate system XYZ:

Young’s modulus vs shear modulus | relation between young’s modulus and modulus of rigidity

Elastic Constants Relations: Shear Modulus, Bulk Modulus, Poisson’s ratio, Modulus of Elasticity.

E=3K(1-2 μ)

E=2G(1+μ)

E= 2G(1+μ)=3K(1-2 μ)

Shear Modulus of Elasticity:

Hook’s law for shear stress:
τxy=G.ϒxy
where,
τxy is represented as Shear-stress, Shear-modulus is G and Shear strain is ϒxy respectively.
Shear-Modulus is resistant to the deformation of the material in response to shear stress.

Dynamic shear modulus of soil:

Dynamic shear modulus gives information about dynamic one. Static shear-modulus gives information about static one. These are determined using shear wave velocity and density of the soil.

Shear Modulus Formula soil

Gmax=pV_s²

Where, Vs=300 m/s, ρ=2000 kg/m³, μ=0.4.

Effective shear modulus:

The ratio of the average stresses to average strains is the effective shear-modulus.

Modulus of rigidity of spring:

The modulus of rigidity of the spring is the measurement of the stiffness of the spring. It varies with the material and processing of the material.

For Closed Coil Spring:

For Open Coil Spring:

Where,
R= mean radius of the spring.
n = number of coils.
d= diameter of the wire.
N= shear modulas.
W= load.
δ=deflection.
α= Helical angle of the spring.

Modulus of Rigidity- Torsion | Modulus of Rigidity Torsion test

The rate change of strain undergoing shear stress and is a function of stress subjected to torsion loading.

The main objective of the torsion experiment is to determine the shear-modulus. The shear stress limit is also determined using the torsion test. In this test, one end of the metallic rod is subjected to torsion, and the other end is fixed.
The shear strain is calculated by using the relative angle of twist and gauge length.
γ = c * φG / LG.
Here c – cross-sectional radius.
Unit of φG measured in radian.
τ = 2T/(πc3),

shear-stress is linearly proportionate to shear-strain, if we measured at the surface.

Frequently Asked Questions:

What are the 3 Modulus of elasticity?

Young’s modulus:

This is the ratio of longitudinal stress to longitudinal strain and could be better represented as

Young’s modulus ϒ= longitudinal stress/longitudinal strain.

Bulk Modulus:

The ratio of hydrostatic pressure to volume strain is called the Bulk modulus denoted as

Bulk Modulus(K)=volume stress/volume strain.

Modulus of Rigidity:

The ratio of shear stress to the shear strain of the material may well characterized as

Shear Modulus(η)=shear stress/shear strain.

What does a Poisson ratio of 0.5 mean?

Passion’s ratio ranges between 0-0.5.at small strains, an incompressible isotropic elastic material deformation gives Poisson’s ratio of 0.5. Rubber has a higher bulk modulus than the shear-modulus and Poisson’s ratio nearly 0.5.

What is a high modulus of elasticity?

The modulus of elasticity measures the resistance of the material to the deformation of the body and if modulus increse then material required additional force for the deformation.

What does a high shear modulus mean?

A high shear-modulus means the material has more rigidity. a large amount of force is required for the deformation.

Why is shear modulus important?

The shear-modulus is the degree of the stiffness of the material and this analyze how much force is required for the deformation of the material.

Where is shear modulus used ?| What are the applications of rigidity modulus?

The Information’s of shear-modulus is used any mechanical characteristics analysis. For calculation of shear or torsion loading test etc.

Why is shear modulus always smaller than young modulus?

Young’s modulus is the function of longitudinal strain and shear modulus is a function of transverse strain. So, this gives the twisting in the body whereas young’s modulus gives the stretching of the body and Less force is required for twisting than stretching. Hence shear modulus is always smaller than the young’s modulus.

For an ideal liquid, what would be the shear modulus?

In ideal liquids shear strain is infinite, the shear modulus is the ratio of shear stress to the shear strain. So the shear modulus of ideal liquids is zero.

When the bulk modulus of a material becomes equal to the shear modulus what would be the Poisson’s ratio ?

As per the relation between bulk modulus, shear modulus and poissons ratio,
2G(1+μ)=3K(1-2 μ)
When, G=K
2(1+ μ)=3(1-2 μ)
2+2 μ=3-6 μ
8 μ=1
μ =1/8

Why the required shear stress to initiate dislocation movement is higher in BCC than FCC?

BCC structure has more shear stress values critical resolved than FCC structure.

What is the ratio of shear modulus to Young’s modulus if poissons ratio is 0.4, Calculate by considering related assumptions.

Answer.
2G(1+μ) =3K (1-2 μ)
2G (1+0.4) =3K(1-0.8)
2G(1.4) =3K(0.2)
2.8G=0.6K
G/K=0.214

Which has a higher modulus of rigidity a hallow circular rod or a solid circular rod ?

Modulus of rigidity is the ratio of shear stress to the shear strain and shear stress is the Force per unit area. Hence shear stress is inversely proportional to the area of the body. solid circular rod is stiffer and stronger than the hollow circular rod.

Modulus of Rigidity vs Modulus of Rupture:

The modulus of rupture is the fracture strength. It is the tensile strength of the beams, slabs, concrete, etc. Modulus of rigidity is the strength of material to be rigid. It is the stiffness measurement of the body.

If the radius of the wire is doubled how will the rigidity modulus vary? Explain your answer.

Modulus of rigidity does not vary by change of the dimensions and hence modulus of rigidity remains the same when the radius of the wire is doubled.

Coefficient of viscosity and modulus of rigidity:

The coefficient of viscosity is the ratio of the shear stress to the rate of shear strain which varies by the velocity change and displacement change and the modulus of rigidity is the ratio of shear stress to the shear strain where shear strain is due to transverse displacement.
The ratio of shear-modulus to the modulus of elasticity for a Poisson’s ratio of 0.25 would be
For this case we may consider that.
2G(1+μ)=3K(1-2 μ)
2G(1+0.25) =3K(1-0.5)
2G(1.25)=3K(0.5)
G/K=0.6
Answer = 0.6

What material has modulus of rigidity equal to about 0.71Gpa ?

Answer:
Nylon(0.76Gpa)
Polymers range between such low values.

For more Mechanical Engineering related article click here

Content : Dual Cycle

What is Dual cycle?

Dual Combustion Cycle | Mixed cycle | Sabathe cycle

The dual cycle is named after The Russian-German engineer Gustav Trikler. It is also known as the mixed cycle, Trinkler cycle, seiliger cycle or sabathe cycle.

Dual cycle is a combination of constant volume Otto cycle and constant pressure diesel cycle. Heat addition takes place in two parts in this cycle. Partial heat addition takes place at constant volume similar to Otto cycle while the remaining Partial heat addition takes place at constant pressure similar to diesel cycle. The significance of such method of heat addition is it gives more time to fuel for complete combustion.

Dual cycle P-V Diagram | Dual cycle T-S Diagram

The dual cycle comprises of following operations:

• Process 1-2 follows the reversible adiabatic or Isentropic compression
• In Process 2-3 constant volume partial heat Addition takes place
• In Process 3-4 constant pressure partial heat Addition takes place
• Process 4-5 follows reversible adiabatic or isentropic expansion.
• In Process 5-1 constant volume heat rejection takes place

Dual Cycle efficiency | Dual Cycle Thermal efficiency

The efficiency of dual cycle is given by

Where, r_p = Pressure ratio = P₃/P₂

r_k = compression ratio = V₁/V₂

r_c = cutoff ratio = V₄ /V₃

r_e = expansion ratio = V₅/V₄

When rc = 1, The cycle becomes Otto cycle

rp = 1, the cycle becomes diesel cycle.

Dual cycle P-V and T-S diagram

Air standard dual cycle | Dual cycle efficiency derivation

The dual cycle comprises of following operations:

• Process 1-2 follows the reversible adiabatic or Isentropic compression
• In Process 2-3 constant volume partial heat Addition takes place
• In Process 3-4 constant pressure partial heat Addition takes place
• Process 4-5 follows reversible adiabatic or isentropic expansion.
• In Process 5-1 constant volume heat rejection takes place

Total Heat supplied is given by

Where Heat supplied at constant volume

Where Heat supplied at constant pressure

Heat rejected at constant volume is given by

The efficiency of dual cycle is given by

Where, r_p = Pressure ratio = P₃/P₂

r_k = compression ratio = V₁/V₂                 

r_c = cutoff ratio = V₄ /V₃

r_e = expansion ratio = V₅/V₄

When r_c = 1, the cycle becomes Otto cycle

r_p = 1, the cycle becomes diesel cycle.

Mean effective pressure of dual cycle

The mean effective pressure of dual cycle is given by

Where, r_p = Pressure ratio = P₃/P₂

r_k = compression ratio = V₁/V₂

r_c = cutoff ratio = V₄ /V₃

r_e = expansion ratio = V₅/V₄

Otto Diesel Dual Cycle Diagram

Comparison between Otto, diesel and dual cycle

Case 1: For similar compression ratio and similar heat i/p this relationship will be

[Q_in]_otto = [Q_in]_Diesel.

[Q_R]_otto< [Q_R]_Diesel.

In this case of same compression ratio and equal heat input it will be

Case 2: In this case of, same compression ratio and same heat-rejection, this relationship will be

[Q_in]_otto> [Q_in]_Diesel.

[Q_R]_otto= [Q_R]_Diesel.

In this case of, same compression ratio and same heat-rejection.

Case 3: In this case of, same Maximum Temperature and same heat-rejection.

[Q_R]_otto= [Q_R]_Diesel

[Q_in]_Diesel>[Q_in]_otto

For same Maximum Temperature and same heat rejection

Dual fuel engine cycle | Mixed dual cycle

Dual Cycle Engine

Dual fuel engine works primarily on diesel cycle. The gaseous fuel [Natural gas] is introduced into the engine’s intake system through a supercharger at higher atmospheric pressure.

During the suction stroke, the leaner air-to-fuel ratio [air-to-natural gas mixture] is drawn into the cylinder, following the Otto cycle just as used in a spark-ignited engine. A small charge of pilot fuel is injected near the Top Dead Center and similar to CI engine it ignites near the end of the compression stroke, causing the secondary gas to burn. The combustion takes place smoothly and rapidly.

In dual-fuel engine pilot fuel and secondary fuel both burn simultaneously in a compression ignition engine. After the compression of secondary fuel at the suction stroke pilot fuel is used as a source of ignition.

The operating cost of this engine is lower than that of conventional diesel engine without compromising in delivery power, high torque and transient response.

Dual cycle application

• Dual cycle is widely used for small propulsion engines and portable heavy duty machinery like drilling machines by companies like Cummins etc. The major reason of dual cycle being used in mobile equipment’s is it provides high power to mass ratio in comparison with Otto and diesel cycle.
• They have wide range of application in aircraft and ships. Dual cycle engine are also called marine engine.

Advantage of dual cycle

1. Higher heat yield – methane has the highest thermal output per unit mass of fuel, at 50,500kJ/kg methane burned compared to 44,390kJ heat/kg petrol burned or 43,896kJ heat/kg diesel burned. Many dual-combustion engines use natural gas whose primary content is methane as starter fuels because of its higher heat output.
2. With a dual fuel combustion engine, two fuels must be purchased instead of one. This can help when the ship is low on both fuels, and the re-fueling location lacks one of the two fuels the engine takes in.
3. A potential balance between clean fuel and economical storage – natural gas needs higher storage pressure and volume but offers better combustion efficiency. Diesel is easier to store (it’s a liquid oil) but does not burn as quickly for the same temperature and pressure as the other fuels. With a dual combustion engine, one can start the diesel engine then switch to natural gas when the combustion space is hot enough.

Dual Cycle problems and solutions

A C.I engine has a compression ratio of 10. The heat liberated at constant volume is 2/3 of total Heat while the remaining is liberated at constant pressure. The initial pressure and temperature is 1 bar and 27^oC. Maximum cycle pressure is 40 bars. Find temperature at the end of compression and expansion. [PV^1.35 = C, ϒ = 1.4]

Solution: r_k = 10, P₁ = 1 bar = 100 kPa, T₁=27 C = 300K, P₃ = P₄ = 40 bar, PV^1.35 = C, ϒ = 1.4

Heat input at constant volume

An air standard dual cycle before compression air is at 100 kPa and 300K. During compression the volume changes from 0.07 m³ to 0.004m³. For constant pressure heat addition, the temperature varies from 1160 C to 1600C. Find the compression ratio; mean effective pressure and cut-off ratio for the cycle.

P₁ = 100 kPa, T₁=27 C = 300K

Compression ratio

T₃ = 1160C = 1433 K, T₄ = 1600 C = 1873 K

For isentropic compression process

Cut-off ratio

For isentropic expansion process

Total Heat Supplied

Heat rejected

Work done is given by

Mean effective pressure for dual cycle

FAQ

Q.1) where is dual cycle used?

Ans: – Dual cycle is widely used for small propulsion engines and portable heavy duty machinery like drilling machines by companies like Cummins etc. The major reason of dual cycle being used in mobile equipments is it provides high power to mass ratio in comparison with Otto and diesel cycle.

They have wide range of application in aircraft and ships. Dual cycle engine are also called marine engine.

Q.2) what is the efficiency of dual cycle?

The efficiency of dual cycle is given by

Where, r_p = Pressure ratio = P₃/P₂

r_k = compression ratio = V₁/V₂

r_c = cutoff ratio = V₄ /V₃

r_e = expansion ratio = V₅/V₄

When r_c = 1, The cycle becomes Otto cycle

r_p = 1, the cycle becomes diesel cycle.

Q.3) what are the importance of dual cycle in the diesel engine operations?

Dual fuel engine works primarily on diesel cycle. The gaseous fuel [Natural gas] is introduced into the engine’s intake system through a supercharger at higher atmospheric pressure.

During the suction stroke, the leaner air-to-fuel ratio [air-to-natural gas mixture] is drawn into the cylinder, following the Otto cycle just as used in a spark-ignited engine. A small charge of pilot fuel is injected near the Top Dead Center and similar to CI engine it ignites near the end of the compression stroke, causing the secondary gas to burn. The combustion takes place smoothly and rapidly.

In dual-fuel engine pilot fuel and secondary fuel both burn simultaneously in a compression ignition engine. After the compression of secondary fuel at the suction stroke pilot fuel is used as a source of ignition.

The operating cost of this engine is lower than that of conventional diesel engine without compromising in delivery power, high torque and transient response.

Q.4) why is the dual cycle known as a mixed cycle?

Dual fuel engine works primarily on diesel cycle. The gaseous fuel [Natural gas] is introduced into the engine’s intake system through a supercharger at higher atmospheric pressure.

During the suction stroke, the leaner air-to-fuel ratio [air-to-natural gas mixture] is drawn into the cylinder, following the Otto cycle just as used in a spark-ignited engine. A small charge of pilot fuel is injected near the Top Dead Center and similar to CI engine it ignites near the end of the compression stroke, causing the secondary gas to burn. The combustion takes place smoothly and rapidly.

In dual-fuel engine pilot fuel and secondary fuel both burn simultaneously in a compression ignition engine. After the compression of secondary fuel at the suction stroke pilot fuel is used as a source of ignition.

The operating cost of this engine is lower than that of conventional diesel engine without compromising in delivery power, high torque and transient response.

Q.5) what is cut off ratio in dual cycle?

The cut-off ratio for dual cycle is given by

r_c = cutoff ratio = V₄ /V₃

Where, V₄ = volume after partial heat addition at constant pressure

V₃ = volume after partial heat addition at constant volume

Q.6) What is Dual cycle P-V and T-S diagram ?

To see the answer Click here

Q.7) Dual cycle solved example.

A C.I engine has a compression ratio of 10. The heat liberated at constant volume is 2/3 of total Heat while the remaining is liberated at constant pressure. The initial pressure and temperature is 1 bar and 27^oC. Maximum cycle pressure is 40 bars. Find temperature at the end of compression and expansion. [PV^1.35 = C, ϒ = 1.4]

Solution: r_k = 10, P₁ = 1 bar = 100 kPa, T₁=27 C = 300K, P₃ = P₄ = 40 bar, PV^1.35 = C, ϒ = 1.4

Heat input at constant volume

An air standard dual cycle before compression air is at 100 kPa and 300K. During compression the volume changes from 0.07 m³ to 0.004m³. For constant pressure heat addition, the temperature varies from 1160 C to 1600C. Find the compression ratio; mean effective pressure and cut-off ratio for the cycle.

P₁ = 100 kPa, T₁=27 C = 300K

Compression ratio

T₃ = 1160C = 1433 K, T₄ = 1600 C = 1873 K

For isentropic compression process

Cut-off ratio

For isentropic expansion process

Total Heat Supplied

Heat rejected

Work done is given by

Mean effective pressure for dual cycle

To know about Polytropic Process (click here)and Prandtl Number (Click here)

The Otto Cycle, fundamental to gasoline engines, consists of four strokes: intake, compression, power, and exhaust. It achieves thermal efficiency up to 25-30%. The compression ratio, typically between 8:1 and 12:1, directly influences efficiency and power output.

Otto Cycle Definition

“An Otto cycle is an ideal thermodynamic cycle that explains the working of a typical spark ignition piston engine and this cycle specifically explains, what happens if mass of gas is subjected to changes due to pressure, temp, volume, heat input, and release of heat.”

The Otto cycle engine | Valve timing diagram

1. Inlet valve opens at 5-10⁰ before the Top Dead Center. This is to ensure that the inlet should open fully when piston reaches at TDC and fresh charge start entering to cylinder as early as possible after TDC.
2. Suction valve close at 20 – 30⁰ after Bottom dead center BDC to take the advantage of momentum of moving gases.
3. The spark takes place 30 – 40⁰ before TDC. This is to allow time delay between spark and completion of combustion.
4. Pressure at the end of power stroke is above atmospheric which increases the work to expel the exhaust gases. So exhaust valve opens at 20 – 30⁰ before BDC so that at BDC pressure reduces to atmospheric pressure and useful work can be saved.
5. The exhaust valve closes at 15 – 20⁰ after TDC so that inertia of exhaust gas has a tendency to to scavenge the cylinder which will increase volumetric efficiency.

Otto cycle efficiency | thermal efficiency of Otto Cycle Formula

The efficiency of Otto cycle is specified by

Where r = compression ratio.

Otto Cycle diagram

Otto cycle P-V diagram | Otto cycle T-S diagram

Otto, Diesel and Dual cycle | Comparison

Case 1: For similar compression ratio and similar heat i/p this relationship will be

[Q_in]_otto = [Q_in]_Diesel.

[Q_R]_otto< [Q_R]_Diesel.

In this case of same compression ratio and equal heat input it will be

Case 2: In this case of, same compression ratio and same heat-rejection, this relationship will be

[Q_in]_otto> [Q_in]_Diesel.

[Q_R]_otto= [Q_R]_Diesel.

In this case of, same compression ratio and same heat-rejection.

Case 3: In this case of, same Maximum Temperature and same heat-rejection.

[Q_R]_otto= [Q_R]_Diesel

[Q_in]_Diesel>[Q_in]_otto

For same Maximum Temperature and same heat rejection

Compression ratio of Otto cycle

Compression ratio of Otto cycle is defined as the ratio of volume before expansion to volume after expansion

Where V_s = Swept volume of cylinder

V_c = Clearance volume of the cylinder

In this cycle Compression ratio is generally 6 – 10. It is limited to 10 because of knocking in the engine.

Mean effective pressure formula for Otto cycle

Usually Pressure inside the cylinder on an IC engine is continuously changing; mean effective pressure is an imaginary pressure which is assumed to be constant throughout the process.

Where r_p = Pressure ratio = P₃/P₂ = P₄/P₁

Otto cycle analysis | Otto cycle calculations | Otto cycle efficiency derivation

 Consider an air standard Otto cycle with initial Pressure, Volume and temperature as P₁, V₁, T₁ respectively.

Process 1-2: Reversible adiabatic compression.

Where,

r is the compression ratio.

Process 2 -3: Heat addition at constant Volume is calculated as,

Q_in = m C_v [T­₃-T₂].

Process 3-4: Reversible adiabatic expansion is calculated as

Process 4 -1: Heat-rejection at constant Volume will be

Q_R = m C_v [T­₄-T₁]

Work done = Q_in – Q_R.

Efficiency of the Otto cycle is represented as.

Where r = compression ratio.

Working of Two stroke Engine

Two strokes’ engines work on both Otto cycle as well as diesel cycle.

Atkinson cycle vs Otto cycle

Atkinson Cycle	Otto cycle
Atkinson cycle uses slightly different valve timing diagram. The Inlet valve remains open till the start of compression stroke	Inlet valve opens at 5-100 before the Top Dead Center. This is to ensure that the inlet should open fully when piston reaches at TDC and fresh charge start entering to cylinder as early as possible after TDC.
Provides higher fuel economy as compared to Otto cycle.	Provides Lower fuel economy as compared to Atkinson cycle.
Provides Lower peak power as compared to Otto cycle.	Provides higherPeak power as compared to Atkinson cycle.
Mostly used in Hybrid vehicles where electric motor compensates the power deficiency.	Mostly used in 4-stroke and 2 – stroke SI engine where higher power is required

Brayton cycle vs Otto cycle

Brayton Cycle	Otto cycle
Constant Pressure Heat addition and heat rejection takes place in Brayton cycle.	Constant volume Heat addition and heat rejection takes place in Otto cycle.
It has capabilities to handle large volume of low-pressure gas.	Not capable of handling large volume of low-pressure gas due to restriction in reciprocating engine space.
High temperature is experienced throughout the steady state flow process.	High temperature is experienced by the engine only during Power stroke.
Suitable for gas turbine	Suitable for IC and SI engine.

Advantages and Disadvantages of Otto cycle engine

Advantages:

• This cycle has more thermal efficiency in comparison to diesel and dual cycle for identical compression ratio and equal heat input rate and same compression ratio and same heat rejection.
• This cycle engine required less maintenance and are simple and light-weight in design.
• For complete combustion pollutant emissions are low for Otto engines.

Disadvantages:

• Has lower compression ratio thus it is Poor at moving heavy loads at low speed.
• Will not be able to withstand higher stresses and strains as compared with diesel engine

Example of Otto cycle | Otto cycle problems

Q.1] A spark Ignition engine designed to have a compression ratio of 10. This is operating at low temperature and pressure at value 200⁰C and 200 kilopascals respectively. If Work O/P is 1000 kilo-Joule/kg, compute maximum possible efficiency and mean effective pressure.

Efficiency of this cycle is given by

Where r = compression ratio = 10

For compression process

For expansion process, we can assume that

Net work done can be computed by the formula

According to ideal gas theory, we know

P₁v₁ = RT₁

v₁=(RT₁)/(P₁)=(0.287*473)/200=0.6788 m³/kg

Q.2] what will be the effect on the efficiency of an Otto cycle having a compression ratio 6, if C_v increases by 20%. For the purpose of calculation Assume, that  C_v is 0.718 kJ/kg.K.

Taking log on both sides

Differentiating both sides

I.e. If C_v increases by 2% then η decrease by 1.36%.

Frequently Asked Questions

What is the difference between Otto and Diesel cycle?

In Otto cycle heat addition takes place at constant volume while in diesel cycle, heat addition at constant pressure takes place and  Otto-cycle has lower compression ratio below 12 while, diesel cycle has higher compression ratio up to 22.  Otto-cycle uses spark plug for ignition while diesel cycle needs no assistance for ignition. Otto-cycle has lower efficiency as compared to diesel cycle.

Which fuel is used in Otto cycle ? | What is 4-stroke fuel?

Generally Petrol or gasoline mixed with 3-5% ethanol is used in Otto engine. In air standard Otto-cycle, air is assumed to be as a fuel.

Which is more efficient Otto or Diesel cycle?

The normal range of compression ratio for diesel cycle is 16-20 while in Otto-cycle compression ratio is 6 – 10 and due to higher compression ratio used in diesel cycle, the efficiency of diesel cycle is greater than Otto-cycle.

How does Otto cycle works?

1. Inlet valve opens at 5-10⁰ before the Top Dead Center. This is to ensure that the inlet should open fully when piston reaches at TDC and fresh charge start entering to cylinder as early as possible after TDC.
2. Suction valve close at 20 – 30⁰ after Bottom dead center BDC to take the advantage of momentum of moving gases.
3. The spark takes place 30 – 40⁰ before TDC. This is to allow time delay between spark and completion of combustion.
4. Pressure at the end of power stroke is above atmospheric which increases the work to expel the exhaust gases. So exhaust valve opens at 20 – 30⁰ before BDC so that at BDC pressure reduces to atmospheric pressure and useful work can be saved.
5. The exhaust valve closes at 15 – 20⁰ after TDC so that inertia of exhaust gases tends to scavenge the cylinder which will increase volumetric efficiency.

Process 1-2: Reversible adiabatic compression

Where r = compression ratio

Process 2 -3: Heat additions at constant Volume

Q_in = m C_v [T­₃-T₂]

Process 3-4: Reversible adiabatic expansion

Process 4 -1: Heat-rejection at constant Volume will be

Q_R = m C_v [T­₄-T₁]

Work done = Q_in – Q_R.

Efficiency of the Otto-cycle is represented as.

Where r = compression ratio.

Difference between Otto cycle Diesel cycle and Dual cycle

Otto cycle vs Dual cycle

Otto cycle vs Carnot cycle

Carnot CycleOtto cycleIt consists of two reversible isothermal process and two reversible adiabatic processes.	Ideal air standard Otto-cycle consists of two Isochoric process and two reversible adiabatic processes.
It is a hypothetical cycle and is not practically possible to construct.	It is a real cycle and is the basis of working of modern Spark ignition engine.
It serves as a yardstick to measure the performance of other engine cycles.	It does not serve as a yardstick to measure the performance of other engine cycles.
It has 100% efficiency.	It has overall thermal efficiency in the range of 50 – 70 %.
It can be reversed to obtain Carnot refrigeration / heat pump with maximum coefficient of performance.	It is a non-reversible cycle.

Otto cycle vs Atkinson cycle

Atkinson Cycle	Otto cycle
Atkinson cycle uses slightly different valve timing diagram. The Inlet valve remains open till the start of compression stroke	Inlet valve opens at 5-100 before the Top Dead Center. This is to ensure that the inlet should open fully when piston reaches at TDC and fresh charge start entering to cylinder as early as possible after TDC.
Provides higher fuel economy as compared to Otto-cycle.	Provides Lower fuel economy as compared to Atkinson cycle.
Provides Lower peak power as compared to Otto- cycle.	Provides higher Peak power as compared to Atkinson cycle.
Mostly used in Hybrid vehicles where electric motor compensates the power deficiency.	Mostly used in 4-stroke and 2 – stroke SI engine where higher power is required

Otto cycle formula

The efficiency of Otto-cycle is given by the equation

Where r = compression ratio = 10

Otto cycle with polytropic process example

An SI engine has a compression ratio 8 while operating with low temperature of 300⁰C and a low pressure of 250 kPa. If Work o/p is 1000 kilo-Joule/kg, then compute highest efficiency. The compression and expansion takes place polytropically with polytropic index (n = 1.33).

Solution: The efficiency of Otto- cycle is given by the equation

Here γ = n

Why the Otto cycle is known as a constant volume cycle?

For this cycle, heat-addition and rejection happens at the fixed volume and the amount of work done is proportionate on heat addition and heat rejection rate, because of this reason Otto-cycle is termed as constant volume cycle.

What are the limitations of the Otto cycle?

• It has lower compression ratio thus it is Poor at moving heavy loads at low speed.
• Cannot withstand higher stresses and strains as compared with diesel engine.
• Overall fuel efficiency is lower than diesel cycle.

Are two stroke engines considered to be Otto cycle engines?

Two strokes engines work on both Otto-cycle as well as diesel cycle. The working of 2-stroke engine is given below:

1. Piston moves down and useful power is obtained. The downward motion of piston compresses the fresh charge stored in crankcase.
2. Near the end of expansion stroke, the piston will reveal the exhaust-port at first. Then the cylinder pressure will drop to atmospheric pressure as during that time combustion materialwill leave from the cylinder.
3. Further motion of piston reveals the transfer port allowing the slightly compressed charge in crank case to be entered the engine’s cylinder.
4. Projection in piston prevents the fresh charge from passdirectly to the exhaust port and scavenging the combustion materials.
5. When piston move up from bottom dead centre to top dead center and the transfer port closes at first then exhausts port will close and compression will happen. At the same time vacuum is created in crankcase and fresh charge enter into crankcase for next cycle.

Why is the Atkinson cycle more efficient even though it produces lower compression and pressure than the Otto cycle?

In Atkinson cycle, for isentropic expansion process in Otto cycle is further allowed to proceed and extend to lower cycle pressure in order to increase the work output and we know that efficiency increases for increase in work produced. That is, why the Atkinson cycle is more efficient even though it produces lower compression and pressure than the Otto cycle.

What is the compression ratio of the Otto cycle

Compression ratio of this cycle is elaborated  as

Where,

V_s = Swept volume of cylinder.

V_c = Clearance volume of the cylinder.

Generally in Otto cycle compression ratio is 6 – 10. It is limited to 10 because of knocking in the engine.

Otto cycle vs diesel cycle efficiency

The normal range of compression ratio for diesel cycle is 16-20 while in Otto cycle compression ratio is 6 – 10 and for more compression ratio used in diesel cycle, the efficiency of diesel cycle is greater than Otto cycle.

Case 1: For same compression ratio and exactly identical heat input, the relationship will be

[Q_in]_otto = [Q_in]_Diesel.

[Q_R]_otto< [Q_R]_Diesel.

In this case of same compression ratio and equal heat input it will be

Case 2: In this case of, same compression ratio and same heat-rejection, this relationship will be

[Q_in]_otto> [Q_in]_Diesel.

[Q_R]_otto= [Q_R]_Diesel.

In this case of, same compression ratio and same heat-rejection.

Case 3: In this case of, same Maximum Temperature and same heat-rejection.

[Q_R]_otto= [Q_R]_Diesel

[Q_in]_Diesel>[Q_in]_otto

For same Maximum Temperature and same heat rejection

Under which condition the efficiency of Brayton cycle and Otto cycle are going to be equal.

The efficiency of Otto cycle is given by the equation

Solution: The efficiency of Otto cycle is given by the equation

r = compression ratio = V₁/V₂

The efficiency of Brayton cycle is given by the equation

r = compression ratio = V₁/V₂

For same compression ratio of the Brayton and Otto cycle, their efficiency will be equal.

To know about Polytropic Process (click here)and Prandtl Number (Click here)

Content: Nusselt Number

What is Nusselt number | Nusselt number definition

“The Nusselt number is the ratio of convective to conductive heat transfer across a boundary.”

https://en.wikipedia.org/wiki/Nusselt_number

• The convection and conduction heat flows in-parallel to each other.
• The surface will be normal of the boundary surface, and vertical to the mean fluid-flow.

Nusselt number equation | Nusselt number formula

Average Nusselt Number can be formulated as:

Nu = Convective heat transfer / conductive heat transfer

Nu = h/(k/L_c)

Nu = hL_c/k

where h = convective heat transfer coefficient of the flow

 L = the characteristic length

 k = the thermal conductivity of the fluid.

The Local Nusselt Number is represented as

Nu = hx/k

x = distance from the boundary surface

Significance of Nusselt number.

Thisrelates in-between convective and conductive heat transfer for the similartypes of fluids.

It also helps in enhancing the convective heat transfer through a fluid layer relative to conductive heat transfer for the same fluid.

It is useful in determining the heat transfer coefficient of the fluid.

It helps to identify the factors which are providing the resistance to the heat transfer and helps in enhancing the factors which can improve the heat transfer process.

Nusselt number correlations.

In case of free-convection, the Nusselt number is represented as the function of Rayleigh number (Ra) and Prandtl Number (Pr), In simple representation

Nu = f (Ra, Pr).

In case of forced-convection, the Nusselt number is represented as the function of Reynold’s number (Re) and Prandtl Number (Pr), in simple way

Nu = f (Re, Pr)

Nusselt number for free convection.

For Free convection at vertical wall

For Ra_L<10⁸

For horizontal Plate

1. If top surface of hot body is in cold environment

Nu_L = 0.54Ra_L^1/4     for Rayleigh number in the range 10⁴<Ra_L< 10⁷

Nu_L = 0.15Ra_L^1/3for Rayleigh number in the range 10⁷<Ra_L< 10¹¹

1. If bottom surface of hot body is in contact with cold environment
2. Nu_L = 0.52Ra_L^1/5for Rayleigh number in the range 10⁵<Ra_L< 10¹⁰

Nusselt number correlations for forced convection.

For fully developed Laminar flow over flat plate

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

For Combined laminar and Turbulent boundary layer

Nu = [0.037Re_L^4/5 – 871] Pr^1/3

Nusselt number for laminar flow | Average Nusselt number flat plate

For fully developed Laminar flow over flat plate[Forced Convection]

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

For horizontal Plate [ Free Convection]

1. If top surface of hot body is in cold environment

Nu_L = 0.54Ra_L^1/4     for Rayleigh number in the range 10⁴<Ra_L< 10⁷

Nu_L = 0.15Ra_L^1/3     for Rayleigh number in the range 10⁷<Ra_L< 10¹¹

1. If bottom surface of hot body is in contact with cold environment
2. Nu_L = 0.52Ra_L^1/5for Rayleigh number in the range 10⁵<Ra_L< 10¹⁰

Nusselt number for laminar flow in pipe

For a circular pipe with diameter D with a fully developed region throughout the pipe, Re < 2300

Nu = hD/k

Where h = convective heat transfer coefficient of the flow

 D =Diameter of pipe

 k = the thermal conductivity of the fluid.

For a circular pipe with diameter D with a Transient flow throughout the pipe, 2300 < Re < 4000

Nusselt number for turbulent flow in pipe

Nusselt Number For a circular pipe with diameter D with a turbulent flow throughout the pipe Re > 4000

According to The Dittus-Boelter equation

Nu = 0.023 Re^0.8 Prⁿ

n = 0.3 for heating, n = 0.4 for cooling

Nusselt number in terms of Reynolds number

For fully developed Laminar flow over flat plate

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

For Combined laminar and Turbulent boundary layer

Nu = [0.037Re_L^4/5 – 871] Pr^1/3

Nusselt Number For a circular pipe with diameter D with a turbulent flow throughout the pipe Re > 4000

According to The Dittus-Boelter equation

Nu = 0.023 Re^0.8 Prⁿ

n = 0.3 for heating, n = 0.4 for cooling

Local Nusselt number

For fully developed Laminar flow over flat plate[Forced Convection]

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

Nusselt number correlations for natural convection

For Laminar flow over vertical plate (natural convection)Nux = 0.59 (Gr.Pr)0.25

Where Gr = Grashoff Number

Pr = Prandtl Number

g = acceleration due to gravity

β = fluid coefficient of thermal expansion

ΔT = Temperature difference

L = characteristic length

ν = kinematic viscosity

μ = dynamic viscosity

C_p = Specific heat at constant pressure

k = the thermal conductivity of the fluid.

For Turbulent Flow

Nu = 0.36 (Gr.Pr)^1/3

Nusselt number heat transfer coefficient

Average Nusselt Number can be formulated as:

Nu = Convective heat transfer / conductive heat transfer

Nu = h/(k/L_c)

Nu = hL_c/k

where h = convective heat transfer coefficient of the flow

 L = the characteristic length

 k = the thermal conductivity of the fluid.

Local Nusselt Number is given by

Nu = hx/k

x = distance from the boundary surface

For a circular pipe with diameter D,

Nu = hD/k

Where h = convective heat transfer coefficient of the flow

 D =Diameter of pipe

 k = the thermal conductivity of the fluid.

Nusselt number table | Nusselt number of air.

Biot number vs. Nusselt number

Both are dimensionless number used to find the convective heat transfer coefficient between wall or solid body and the fluid flowing over the body. They both are formulated as hL_c/k. However, Biot Number is used for solids and Nusselt number is used for fluids.

In Biot number formula hL_c/k for the thermal conductivity (k) of solid is taken into consideration, while in Nusselt Number the thermal conductivity (k) of fluid flowing over the solid is taken into consideration.

Biot number is useful in identifying whether the small body has homogenous temperature all around or not.

Nusselt number heat exchanger

For a circular pipe with diameter D with a fully developed region throughout the pipe, Re < 2300

Nu = hD/k

Where h = convective heat transfer coefficient of the flow

 D =Diameter of pipe

 k = the thermal conductivity of the fluid.

For a circular pipe with diameter D with a Transient flow throughout the pipe, 2300 < Re < 4000

Nusselt number for turbulent flow in pipe: Nusselt Number For a circular pipe with diameter D with a turbulent flow throughout the pipe Re > 4000

According to The Dittus-Boelter equation

Nu = 0.023 Re^0.8 Prⁿ

n = 0.3 for heating, n = 0.4 for cooling

Problems

Q.1)The non-dimensional fluid temp at vicinity of surface of a convectively-cool flat plate is specified as given below . Here y is computed vertical to the plate, L is the plate’s length, and a, b and c are constant. T_w and T_∞ are wall and ambient temp, correspondingly.

If the thermal conductivity (k)and the wall heat flux(q′′) then proof that, Nusselt number

Nu = q/Tw – T / (L/k) = b

Solution:

Tw – T (Tw – T) = a + b (y/L) + c (y/L) = 0

at y = 0

Nu = q (tw – T )(L/k) = b

Hence proved

Q.2) Water flowing through a tube having dia. of 25 mm at velocity of 1 m/sec. Thegiven properties of water are density ρ = 1000kg/m³, μ = 7.25*10^-4 N.s/m², k= 0.625 W/m. K, Pr = 4.85. and Nu = 0.023Re^0.8 Pr^0.4. Then calculate what will be convective heat transfer’s coefficient?

GATE ME-14-SET-4

Solution:

Re = p VD = 1000 x 1 x 25 x 10

(-3) (7.25)

Re = 34482.75

Pr = 4.85, Nu = 0.023Re^0.8 Pr^0.4,

Nu = 0.023*34482.758^0.8 * 4.85^0.4

Nu = 184.5466 = hD/k

h = 184.5466 / 0.625 (25 x 10 (-3)

FAQ

1. What is the difference between Biot number and Nusselt number?

Ans: Both are dimensionless number used to find the convective heat transfer coefficient between wall or solid body and the fluid flowing over the body. They both are formulated as hL_c/k. However, Biot Number is used for solids and Nusselt number is used for fluids.

In Biot number formula hL_c/k for the thermal conductivity (k) of solid is taken into consideration, while in Nusselt Number, the thermal conductivity (k) of fluid flowing over the solid is taken into consideration.

Biot number is useful in identifying whether the small body has homogenous temperature all around or not.

2. How do you find the average of a Nusselt number?

Ans: Average Nusselt Number can be formulated as:

Nu = Convective heat transfer / conductive heat transfer

Nu = h/(k/L_c)

Nu = hL_c/k

where h = convective heat transfer coefficient of the flow

 L = the characteristic length

 k = the thermal conductivity of the fluid.

Local Nusselt Number is given by

Nu = hx/k

x = distance from the boundary surface

3. how to calculate Nusselt number?

Ans: Average Nusselt Number can be formulated as:

Nu = Convective heat transfer / conductive heat transfer

Nu = h/(k/L_c)

Nu = hL_c/k

where h = convective heat transfer coefficient of the flow

 L = the characteristic length

 k = the thermal conductivity of the fluid.

Local Nusselt Number is given by

Nu = hx/k

x = distance from the boundary surface

For fully developed Laminar flow over flat plate[Forced Convection]

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

4. Can Nusselt number be negative?

Ans: Average Nusselt Number can be formulated as:

Nu = Convective heat transfer / conductive heat transfer

Nu = h/(k/L_c)

Nu = hL_c/k

where h = convective heat transfer coefficient of the flow

 L = the characteristic length

 k = the thermal conductivity of the fluid.

For all the properties being constant, heat transfer coefficient is directly proportional to Nu.

Thus, if heat transfer coefficient is negative then the Nusselt number can also be negative.

5. Nusselt number vs. Reynolds number

Ans: In forced convection, the Nusselt number is the function of Reynolds number and Prandtl Number

Nu = f (Re, Pr)

For a circular pipe with diameter D with a fully developed region throughout the pipe, Re < 2300

Nu = hD/k

Where h = convective heat transfer coefficient of the flow

 D =Diameter of pipe

 k = the thermal conductivity of the fluid.

For a circular pipe with diameter D with a Transient flow throughout the pipe, 2300 < Re < 4000

Nusselt number for turbulent flow in pipe

Nusselt Number For a circular pipe with diameter D with a turbulent flow throughout the pipe Re > 4000

According to The Dittus-Boelter equation

Nu = 0.023 Re^0.8 Prⁿ

n = 0.3 for heating, n = 0.4 for cooling

Nusselt number in terms of Reynolds number

For fully developed Laminar flow over flat plate

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

For Combined laminar and Turbulent boundary layer

Nu = [0.037Re_L^4/5 – 871] Pr^1/3

Nusselt Number For a circular pipe with diameter D with a turbulent flow throughout the pipe Re > 4000

According to The Dittus-Boelter equation

Nu = 0.023 Re^0.8 Prⁿ

n = 0.3 for heating, n = 0.4 for cooling

6. Calculate Nusselt number with Reynolds?

Ans: For fully developed Laminar flow over flat plate[Forced Convection]

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

For Combined laminar and Turbulent boundary layer

Nu = [0.037Re_L^4/5 – 871] Pr^1/3

7. What is physical significance of Nusselt number?

Ans: It gives the relation between convective heat transfer and conductive heat transfer for the same fluid.

It also helps in enhancing the convective heat transfer through a fluid layer relative to conductive heat transfer for the same fluid.

It is useful in determining the heat transfer coefficient of the fluid.

It helps to identify the factors which are providing the resistance to the heat transfer and helps in enhancing the factors which can improve the heat transfer process.

8. Why is a Nusselt number always greater than 1?

Ans: This is ratio,In the meantime actual heat transfer cannot become less than 1. Nusselt number is always greater than 1.

9. What is the difference between the Nusselt number and the Peclet number What is their physical significance?

Ans: The Nusselt number is the ratio of convective or actual heat-transfer to conductive heat transfer around a borderline, if convective heat transfer become prominent in the system than conductive heat transfer, Nusselt number will be high.

Whereas, product of Reynold’s number and Prandtl number is represented as Peclet Number. Asit become higher, this will signify high flow rates and flow momentum transfer generally.

10. What is an average Nusselt number How does it differ from a Nusselt number?

Ans: For fully developed Laminar flow over flat plate

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

11. What is the Nusselt number formula for free convection from fuel inside a closed cylinder tank?

Ans: Average Nusselt Number can be formulated as:

Nu = Convective heat transfer / conductive heat transfer

Nu = h/(k/L_c)

Nu = hL_c/k

where h = convective heat transfer coefficient of the flow

 L_c = the characteristic length

 k = the thermal conductivity of the fluid.

For horizontal cylindrical tank L_c = D

Thus, Nu = hD/k

12. Nusselt number for cylinder

Ans: Average Nusselt Number can be formulated as:

Nu = Convective heat transfer / conductive heat transfer

Nu = h/(k/L_c)

Nu = hL_c/k

where h = convective heat transfer coefficient of the flow

 L_c = the characteristic length

 k = the thermal conductivity of the fluid.

For horizontal cylindrical tank L_c = D

Thus, Nu = hD/k

For vertical Cylinder L_c = Length / height of the cylinder

Thus, Nu = hL/k

13. Nusselt number for flat plate

Ans: For horizontal Plate

1. If top surface of hot body is in cold environment

Nu_L = 0.54Ra_L^1/4     for Rayleigh number in the range 10⁴<Ra_L< 10⁷

Nu_L = 0.15Ra_L^1/3     for Rayleigh number in the range 10⁷<Ra_L< 10¹¹

1. If bottom surface of hot body is in contact with cold environment

Nu_L = 0.52Ra_L^1/5     for Rayleigh number in the range 10⁵<Ra_L< 10¹⁰

For fully developed Laminar flow over flat plate

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

For Combined laminar and Turbulent boundary layer

Nu = [0.037Re_L^4/5 – 871] Pr^1/3

14. Nusselt number for laminar flow

Ans:For fully developed Laminar flow over flat plate

Re < 5×10⁵, Local Nusselt number

Nu_L = 0.332 (Re_x)^1/2(Pr)^1/3

But For fully developed Laminar flow

Average Nusselt number = 2 * Local Nusselt number

Nu = 2*0.332 (Re_x)^1/2(Pr)^1/3

Nu = 0.664 (Re_x)^1/2(Pr)^1/3

For a circular pipe with diameter D with a fully developed region throughout the pipe, Re < 2300

Nu = hD/k

Where h = convective heat transfer coefficient of the flow

 D =Diameter of pipe

 k = the thermal conductivity of the fluid.

For a circular pipe with diameter D with a Transient flow throughout the pipe, 2300 < Re < 4000

To know about Polytropic Process (click here)and Prandtl Number (Click here)

Spring constant definition:

Spring constant is the measure of the stiffness of the spring. Springs having higher stiffness are more likely difficult to stretch. springs are elastic materials. when applied by external forces spring deform and after removal of the force, regains its original position. The deformation of the spring is a linear elastic deformation. Linear is the relationship curve between the force and the displacement.

Spring constant formula: 

F= -Kx 

Where, 

F= applied force,

K= spring constant 

x = displacement due to applied load from normal position.

Spring constant units: 

the spring-constant represented as K, and it’s unit is N/m.

How to find spring constant?

Spring constant equation: 

The Spring-constant is determined according to the Hooke’s law stated as below:

The applied force on the springs is directly proportional to the displacement of the spring from the equilibrium. 

 The proportionality constant is the spring’s constant. The spring force is in opposite direction of force. So, there is a negative sign between the relation of the force and the displacement.

F= -Kx 

Therefore,

K= -F/x(N/m)

Dimension of spring constant:

K=-[MT^-2]

Constant force spring:

Constant force spring is the spring that does not obey Hooke’s law. The spring has the force it exerts over its motion range is constant and does not vary by any means. Generally, these springs are constructed as springs rolled up such that the spring is relaxed when fully rolled up and after unrolling the restoring force takes place as the geometry remains constant as the spring unrolls. The constant force spring exerts the constant force for unrolling due to the change in radius of curvature is constant.

Applications of constant spring force:

• Brush springs for motors
• Constant force motor springs 
• Counterbalance springs for window
• Carriage returns springs of typewriters 
• Timers 
• Cable retractors 
• Movie cameras 
• Extension springs 

The constant spring force does not give constant force at all the time. Initially, it has a finite value and after the spring is deflected 1.25 times its diameter it reaches full load and maintains the constant force in the spring despite the deformation. These springs are made with metal strips and not with wires The springs are made up of materials like stainless steel, High carbon steel, etc. springs give tension in the linear direction.

The performance, corrosion elements, temperature affects the fatigue of such springs. They are more likely to have a lifespan of 2500 cycles to more than one million depending on the size and load applied.

Spring constant examples

Spring constant of a rubber band:

Rubber band acts like spring within certain limitations. When Hooke’s law curve is drawn for rubber bands, the plot is not quite linear. But if we stretch the band slowly it might follow Hooke’s law and have spring-constant value. Rubber band can stretch only its elastic limit that 

also depends on the size, length, and quality.

Spring constant values:

Spring constant value is determined using the Hooke’s law. As per the Hooke’s law, when spring is stretched, the force applied is directly proportional to the increase in length from the original position.

How to determine spring constant?

F=-Kx

K=-F/x

Spring constants of materials :

Spring constant for Steel =21000 kg/m³

Spring constant for Copper = 12000 kg/m³

How to find spring constant from graph ?

Spring constant graph:

Can the spring constant be negative?

This can not be negative.

Spring constant formula with mass:

T=

where,

T= period of spring

m=mass

k=spring constant

Effective spring constant:

Parallel: When two massless springs which obey Hooke’s law and connected through the thin vertical rods at the ends of the springs, connecting two ends of springs are said to be parallel connection.

The constant force direction is perpendicular to the force direction.

Spring constant K written as,

K=K1+K2

Series:

When springs are connected to each other in a series manner such that the total extension combination is the sum of total extension and spring’s constant combination all the springs.

The Force is applied at the end of the end spring. The force direction is in the reverse direction as the springs compressed.

Hooke’s law,

F1=k1x1

F2=k2x2

x 1+ x 2 =

Equivalent spring constant:

K =

Torsional spring constant:

A torsion spring is twisted along the axis of the spring.When it is twisted it exerts torque in opposite direction and is proportional to the angle of the twist.

A torsional bar is a straight bar that is subjected to twisting gives shear stress along the axis torque applied at its end.

Examples:

Helical torsional spring, torsion bar, torsion fiber

Applications:

clocks-clocks has spring coiled up together in a spiral, It is a form of helical torsional spring.

torsional spring constant formula | Torsion coefficient

Within elastic limit torsional springs obey Hook’s law as it twisted within elastic limit,

Torque represented as,

τ = -Kθ

τ = − κ θ

K is displacement called the torsional spring coefficient.

The -ve sign specifies that torque is acting in reverse to twist direction. 

The energy U, in Joules

U= ½*Kθ^2

Torsional balance:

Torsion balance is torsional pendulum. It works as a simple pendulum.

To measure the force, first, need to find out the spring’s constant. If the force is low, it’s difficult to measure the sparing constant. One needs to Measure the resonant vibration period of the balance.

The frequency depends on the Moment of Inertia and the elasticity of the material. So, the frequency is chosen accordingly.

Once the Inertia is calculated, springs constant is determined,

F=Kδ/L

Harmonic Oscillator:

Harmonic oscillator is a simple harmonic oscillator when undergoes deformation from the original equilibrium position experiences restoring force F is directly proportional to the displacement x.

Mathematically written as follows,

F= -Kx

Torsional Spring rate:

Torsional spring rate is the force of spring travelled around 360 degrees. This can be further calculated by the amount of force is divided by 360 degrees.

Factors affecting spring constant:

• Wire diameter: The diameter of the wire of the spring
• Coil diameter: The diameters of the coils, depending on the stiffness of the spring.
• Free length: Length of the spring from equilibrium at rest
• The number of active coils: The number of coils that compress or stretch.
• Material: Material of the spring used to manufacture.

Constant torque spring:

Constant torque spring is a type of spring that is a stressed constant force spring traveling between 2 spools. After the release of the compressed spring torque is calculated from the output spool as the spring returns back to its original equilibrium position in the storage spool

Spring constant range:

k = k’ δ’/δ,

K Varies from 

Minimum= 0.9N/m

Maximum=4.8N/m

Spring’s constant depends on the number of turns n.

Ideal spring constant:

The spring constant is the measure of the stiffness of the springs. The larger the value of k, the stiffer is the spring and it is difficult to stretch the spring. Any spring that obeys Hooke’s law equation is said to be an ideal spring.

Constant force spring assembly:

A Constant Force Spring is mounted on a drum by wrapping it around the drum. The spring has to be tightly wrapped. Then the free end of the spring is attached to the loading force such as in a counterbalance uses or vice-versa.

• The drum diameter should be larger than the inside diameter.
• Range: 10-20% drum diameter> Inside diameter.
• One and a half-wrapped spring should be on the drum at extreme extension.
• The strip will be unstable at the larger extensions so it is advisable to keep it smaller.
• Pulley diameter must be greater than the original diameter.

FAQs:

Why is spring constant important?

The spring-constant is important as it shows the basic material property. This gives exactly how much force is required to deform any spring of any material. The higher spring’s constant shows the material is stiffer and the lower spring’s constant shows the material is less stiff.

Can spring constant change?

Yes. spring-constant can change as per the force applied and the extension of the material.

Can spring constant be 0 ?

No. The spring-constant cannot be zero. If it is zero, the stiffness is zero.

Can spring constant has negative value?

No. the Spring-constant always has a positive value.

When are Young’s modulus and Hooke’s spring constant equal?

When the ratio of the length to that area of the spring is unity, then the young’s modulus and the spring’s constant value will be equal.

Spring constant is represented as, K=-F/x,

The above mentioned equation shows the relationship between springs constant and the extension of the spring for the same applied force

Why a spring is cut in half, its spring constant changes?

This is inversely proportional to the extension of the spring. when the spring is cut into half, the length of the spring reduces hence the spring’s constant will be doubled.

Does Newton’s third law fails with a spring ?

Answer : No

Spring constant problems:

Q1) A spring is stretched by 20cm and a 5kg load is added to it. Find the spring constant.

Given:

Mass m = 5kg.

Displacement x=20cm.

Solution:

1.Find out the force applied on the spring

F= m*x

  = 5*20*10^-2

  = 1N.

The load applied on the spring is 1N. So, the spring will apply an equal and opposite load of -1N.

2. Find out the spring constant

K= -F/x

   =-(-1/20*10^-2)

   = 5N/m

The constant of the spring is 5N/m.

Q2)A force of 25 KN is applied on the spring of spring constant of 15KN/m.Find out the displacement of the spring.

Given:

Applied force= 2.5KN

Spring-constant=15KN/m

Solution:

            1.Find out the displacement of the spring

            The spring will apply equal and opposite force of -2.5KN

             F=-Kx

X=-F/K

   = – 2.5/15

   = 0.167m

Hence the spring is displaced by 16.67cm.

Q3)A spring with a force constant of 5.2 N/m has a relaxed length of 2.45m and spring’s perpendicular length 3.57m. When a mass is attached to the end of the spring and permitted to rest. What is elastic potential energy stored in the spring?

Solution:

Given: 

Force constant= 2.45m

x = 2.45m

L= 3.57m

Force constant spring:

F= -Kx

The work was done due to stretching of the spring= Elastic potential energy of the spring.

W=Kx^2/2

Extension x = 3.57-2.45

                    =1.12

W=5.2*1.12^2/2

    =3.2614 J.

Q4) A massless spring with force constant k 400 N/m hangs vertically from the ceiling. A 0.2 kg block is attached to the end of the spring and released. The highest elastic strain energy kept in the spring is (g= 10m/s^2).

Given:

Force constant= 400N/m

m = 0.2kg

g= 10m/s^2

Solution:

Maximum elastic strain energy=1/2*K*x^2

=

=0.02J

Spring constant with multiple springs

A spring is cut into 4 equal parts and 2 are parallel What is the new effective spring constant of these parts?

The spring’s constants of the four springs is k1, k2, k3, k4 

respectively,

Parallel:

Equivalent spring’s constant (k5) = k1 + k2

Series;

Total equivalent springs constant of the system:

K=

If a spring constant of 20N /m and it is stretched by 5cm what is the force acting on the spring:

Given:

K=2 N/m.

x = 5cm.

According to Hooke’s law,

F= -Kx

  = – 20*5*10^-2

  =-1N

Spring force is in opposite direction

Hence spring force = 1N.                

An object with a weight of 5.13 kg placed on top of a spring compresses it by 25m What is the force constant of the spring How high will this object go when the spring releases its energy.

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Mass flow rate Definition

The mass flow rate  is the mass of a substance which passes per unit of time. In SI unit is kg /sec or and slug per second or pound per second in US customary units. The standard natation is (ṁ, pronounced as “m-dot”)”.

Mass flow rate Equation | Mass flow rate units | Mass flow rate symbol

It is denoted by ṁ, It is Formulated as,

In Hydrodynamics

Where,

ρ = Density of the fluid

A = Cross sectional Area

V = Velocity of flow of fluid

Q = Volume flow rate or discharge

It has unit kg/s, lb./min etc.

Mass flow rate conversion

Mass flow rate from volumetric flow rate

In hydrodynamic, the mass-flow rate can be derived from volume flow rate with help of Continuity Equation.

The Continuity equation is given by

Q=AV

Where,

A = Cross sectional Area

V = Velocity of flow of fluid

Multiplying the continuity equation with density of the fluid we get ,

Where,

ρ = Density of the fluid

Mass flow rate to velocity | It’s Relationship with each other

In Hydrodynamics

Where,

ρ = Density of the fluid

A = Cross sectional Area

V = Velocity of flow of fluid

Q = Volume flow rate or discharge

For an in-compressible fluid carrying  through a fixed cross section, the mass-flow rate is directly proportionate to the velocity of  fluid flown.

Reynolds number with mass flow rate | Their Generalized relation

The Reynolds number is given by the equation,

Where,

L_c = Characteristic length

V = Velocity of flow of fluid

ρ = Density of the fluid

μ = dynamic viscosity of the fluid

Multiply numerator and denominator by cross sectional Area A

But mass-flow rate is

Thus Reynolds Number becomes

Mass flow rate problems | Mass flow rate example

Q.1] A Turbine operates on a steady flow of air produces 1 kW of Power by expanding air from 300kPa, 350 K, 0.346 m_³/kg to 120 kPa. The inlet and outlet velocity are 30 m/s and 50 m/s respectively. The expansion follows the Law PV^1.4 = C. Determine the mass flow rate of air?

Solution:

According to Steady Flow energy equation

Q = 0, Z₁ = Z₂

PVⁿ = C

We get,

Mass-Flow rate is

Q.2] Air enters a device at 4 MPa and 300^oC with velocity of 150m/s. The inlet area is 10 cm² and Outlet area is 50 cm².Determine the mass flux if air exits at 0.4 MPa and 100^oC?

Ans: A₁ = 10 cm², P₁ = 4 MPa, T₁ = 573 K, V₁ = 150m/s, A₂ = 50 cm², P₂ = 0.4 MPa, T₂ = 373 K

Q.3] A perfect gas having specific heat at constant pressure as 1 kJ/kgK enters and leaves a gas turbine with same velocity. The temperature of the gas at turbine inlet and outlet are 1100, and 400 Kelvin respectively and The power generation is at the rate 4.6 Mega Watt and heat leakages is at the rate of 300 kilo-Joule/seconds through the turbine casing. Compute mass flow rate of the gas through the turbine.       (GATE-17-SET-2)

Solution: C_p = 1 kJ/kgK, V₁ = V₂, T₁ = 1100 K, T₂ = 400 K, Power = 4600 kW

Heat loss from turbine casing is 300 kJ/s = Q

According to Steady Flow energy equation

FAQ

Why is mass flow rate important?

Ans: Mass-flow rate is important in the wide range of field which include fluid dynamics, pharmacy, petrochemicals etc. It is important to ensure right fluid possessing desired properties is flowing to the required location. It is important for maintaining and controlling the quality of fluid flowing. Its accurate measurements ensure the safety of workers working in a hazardous and dangerous environment. It is also important for good machine performance and efficiency and environment.

Mass flow rate of water

Mass-flow rate is given by the equation

Density of water is 1000 kg/m³

Mass flow rate of air

Mass-flow rate is given by the equation

Density of air is 1 kg/m³

How to get mass flow rate from enthalpy?

Heat Transfer in fluid and thermodynamics is given by the following equation

Where Q = heat transfer, m = mass-flow rate, h = change in enthalpy For constant heat supplied or rejected, enthalpy is inversely proportionate to mass flow rate.

How to get mass flow rate from Velocity?

In hydrodynamic, the mass-flow rate can be derived from volume flow rate with help of Continuity Equation.

The Continuity equation is given by

Q = AV

Where,

A = Cross sectional Area

V = Velocity of flow of fluid

Multiplying the continuity equation with density of the fluid we get,

Can mass flow rate be negative

The magnitude of Mass flow rate cannot be negative. If we are provided the mass-flow rate with negative sign it generally indicates the direction of mass flux is reversed than the direction taken into consideration.

Mass flow rate for an ideal compressible gas

Air is assumed to be an Ideal compressible gas with C_p = 1 kJ/kg. K.

Mass flow rate is given by the equation

Density of air is 1 kg/m³

How can I find the mass flow of a refrigeration fluid R 134a and its temperatures in a domestic freezer How can I find them?

Assuming the Domestic freezer works on a vapor compression cycle, in order to find out mass-flow rate of the coolant R-134a we are required to find:

1. Net refrigeration capacity or effect – generally given for that particular model of freezer.
2. Compressor Inlet Pressure and Temperature
3. Compressor outlet Pressure and Temperature
4. Temperature and pressure at the inlet of evaporator
5. Temperature and pressure at the outlet of condenser
6. For P-h chart find enthalpy at all the above points.
7. Net Refrigeration effect = mass-flow rate * [h₁ – h₂]

What is the relationship between pressure and mass flow rate Does the mass flow rate increase if there’s a pressure increase and does the mass flow rate decrease if there’s a pressure decrease ?

Let,

L = length of pipe

V = Velocity of flow of fluid

μ = dynamic viscosity of the fluid

d = diameter of pipe

According to Hagen Poiseuille equation

Multiplying numerator and denominator by ρA

where, ν = kinematic viscosity = μ/ρ

Thus, as pressure difference increases, mass-flow rate increases and vice versa.

For a convergent nozzle if the exit pressure is less than the critical pressure then what will be the mass flow rate?

As per described situation, nozzle’s outlet velocity is

Mass-flow rate will be

Where

A₁, A₂ = Inlet and Outlet Area of nozzle

C₁, C₂ = Inlet and exit velocity of nozzle

P₁, P₂ = Inlet and Outlet Pressure

V₁, V₂ = Volume at Inlet and Outlet of nozzle

r = Pressure ratio =P₂/P₁

n = Index of expansion

Why is mass flow rate is ρVA but volumetric flow rate is AV?

In hydrodynamic, the mass flux can be derived from volume flow rate with help of Continuity Equation.

The Continuity equation is given by

Q =AV

Where,

A = Cross sectional Area

V = Velocity of flow of fluid

Multiplying the continuity equation with density of the fluid we get the mass flow rate,

Where,

ρ = Density of the fluid

How is the Coriolis principle used to measure mass flow?

A Coriolis mass flowmeter works on the principle of the Coriolis Effect and this are true mass meter because they measure the mass rate of flow directly rather than measuring the volumetric flow rate and converting it into the mass flow rate.

Coriolis meter operates linearly, In the meantime no adjustments are essential for changing fluid characteristic. It is independent of fluid characteristics.

Operating Principle: 

The fluid is allowed to flow through a U-shaped tube. An oscillation-based excitation force is utilized to the tube, causing it to oscillate. The vibration causes the fluid to induce twist or rotation to the pipe because of Coriolis acceleration. Coriolis acceleration is acting opposite to applied excitation force. The generated twist results in a time lag in flow between the entry and exit-side of the tube, and this Lag or phase difference is proportionate to the mass flow rate.

What is the relationship between mass flow rate and volume flow rate?

In hydrodynamic, the mass flow rate can be derived from volume flow rate with help of Continuity Equation.

The Continuity equation is given by

Q = AV

Where,

A = Cross sectional Area

V = Velocity of flow of fluid

Multiplying the continuity equation with density of the fluid we get,

Where,

ρ = Density of the fluid

What is the formula for finding mass flow rate in a water cooled condenser?

Let,

h₁ = enthalpy of water at inlet of the condenser

T₁ = Temperature of water at inlet of the condenser

h₂ = enthalpy of water at exit of the condenser

T₂ = Temperature of water at exit of the condenser

C_p = Specific heat of water at constant pressure

Power of the condenser,

How do you find mass flow with temperature and pressure?

Let,

L = length of pipe

V = Velocity of flow of fluid

μ = dynamic viscosity of the fluid

d = diameter of pipe

According to Hagen Poiseuille equation

Multiplying numerator and denominator by ρA

where, ν = kinematic viscosity = μ/ρ

Thus, as pressure difference increases, m increases.

According to Steady Flow energy equation

Why in choked flow we always control downstream pressure while the maximum mass flow rate is dependent on upstream pressure

It is impossible to regulate Choked mass flows by changing the downstream pressure. When sonic conditions reach the throat, Pressure disturbances caused due to regulated downstream pressure cannot propagate upstream. Thus, you cannot control the maximum flow rate by regulating the downstream backpressure for a choked flow.

What is the average fluid mass flow rate of water in pipes with diameter 10cm, velocity of flow is 20 m/s.

In Hydrodynamics

To know about Polytropic Process (click here)and Prandtl Number (Click here)

What is shear strain?

Shear strain is the ratio of change in dimensions to the original dimension due to shear stress and deformation perpendicular rather than parallel to it. Shear strain results from the use of 2 parallel and opposite forces working at the surfaces of an object.

Shear strain formula:

Shear strain=ΔxL0/L.

Shear Modulus:

“This is the constant of proportion and is well-defined by the ratio of stress to strain.”

Shear modulus is generally represented by S.

S=shear-stress. / shear-strain.

Shear strain units:

This is dimensionless quantity, so this is Unitless.

Shear strain symbol:

Shear strain symbol=γ or ε

Shear strain unit:

1, or radian

Shear strain from axial strain

Strain:

Applied loads or displacements lead to change in dimensions. For the uniaxial displacement, the axial strain defined basically as the ratio of the variation in length to the actual length.

Shear stress strain diagram

The three-dimensional strain components may also be represented as simple axial strains and shear-strains. The displacement vectors (u,v,w) acting along the axes (x,y,z) respectively.

The uniaxial strain in x-direction due to displacement gradient,

Similarly, the shear-strain in the y-direction due to displacement gradient is given by,

The shear-strain components are represented as strain matrix as following,

Three shear-strains are the strains represented in all planes in x,y,z directions as XY, YZ, XZ.

The strains are represented in the strain matrix-induced due to the stress:

Strain Measurement:

It is difficult to measure stress directly. So, the strain measurement can be done using electric resistance circuit gauges connected to it.

Strain gauge measurement | Shear strain from a strain gauge

The strain gauge measurement is used to determine the resistance of the wire foiled together to the conducting substrate. The wire resistance is R,

\\frac{\\Delta R}{R}= K.\\varepsilon

where K recognized as strain-factor

Alternatively,

\\varepsilon= deformation = strain

so, strain can be induced by using strain measurement.

Since the strains are low, the Wheatstone bridge needs to determine the resistances. The galvanometer reading has to be zero to find out the resistances R1, R2, R3, R4. More than one configuration can be used to measure the strain. A half wiring can be used and attached to the other gauges. There are one active meter and one dummy meter. The dummy gauge reduces the temperature effects by canceling out them. Such difference can lead to an improvement in the accuracy of the circuits.

Maximum shear strain equation:

Maximum normal strain (εmax.)

(εx+εy)/2+(((εx-εy)/2)2+(γxy/2)2)0.5.

Minimum normal strain (εmin).

(εx+εy)/2-(((εx-εy)/2)2+(γxy/2)2)0.5.

Maximum shear-strain (in-plane) ( γmax (in-plane)).

((εx-εy)2+(γxy)2)0.5

Principal angle (θp)

[atan(γxy/(εx-εy))]/2

Principal shear strain:

Principal Stress:

Shear stress is zero at an alignment then principal stress happens.

Principal Angle:

This is the angle of alignment in that principal stress will occur for a definite point.

Principal Strain:

This is the highest and least normal strain possible for an material at that specific point and shear-strain is zero at the angle where principal strain occurs.

The 3 stresses normal to principal shear plane are termed principal-stress, where as in a plane where shear-strain is zero is termed as principal-strain.

Pure shear strain:

principal stress and strain are zero.

What is shear strain energy ?

Strain energy due to shear stress | Shear strain energy theory:

Maximum shear strain energy | Distortion energy (Von Mises) theory

The failure of utmost ductile material could have been determined by the shear stress theories or Von Mises theory as the failure occurs at the shearing of the materials. This theory can be represented as

(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2=2σy2=constant

For  σ3 = 0,

The yield locus is an ellipse similar to sheer diagonal. In3D Stress system, this equation states the surface of a prism with circular cross section. More precisely a cylinder with its central axis along the line σ1 = σ2 = σ3.

The axis cut-thru the principal stress’s origin, and it is inclined at equal angle. when σ3 = 0,

The failure condition for ellipse formed by the intersection of the (σ1, σ2) plane with the inclined cylinder.

Shear strain energy per unit volume theory.

As per von Mises theory in 3D,the yield locus will be at the surface of the inclined cylinder. Points inside the cylinder show the safe points, whereas the points outside the cylinder show the failure conditions. The cylinder axis along σ1 = σ2 = σ3 line termed the hydrostatic stress line. It shows that the hydrostatic stress alone cannot give yield. It considers all the conditions altogether and shows that all cylinder points are safe.

Shear strain example problems

• metal cutting
• painting brush
• Chewing gum
• In river water case, river bed will experience the shear stress because of water flow situations.
• During screen-sliding circumstance.
• To Polishing a surface.
• To write on surface, will experience shear-strain.

The following image of rectangular hut shows the deformation of rectangular into parallelogram due to shear-strain.

The reason behind shear strain:

Shear stress is the applied force that will cause deformation of a material by slippage along a plane or a plane parallel to the stress imposed on the surface of the object.

Relationship between shear stress and strain

Shear strain vs shear stress | shear strain vs shear force

The shear strain is the deformation caused due to shear stress. Shear stress is the stress occurred due to shear forces in-between the object’s parallel surfaces.

Torsional shear strain

Torsional shear-strain τ = shear stress (N/m2, Pa) T = applied torque (Nm)

The shear-strain is calculated by the angle of twist, the length, and the distance along the radius.

γ = shear strain (radians)

Shear stress strain curve:

Shear stress acts along the surface or parallel to the surface and may cause 1 layer to slide on others. shear stress leads to deforming the rectangular object into the parallelogram.

Shear stress acts to change the dimension and angle of the object.

Shear stress= F/A

Shear-strain: The shear-strain is the deformation amount to a given line rather than parallel.

Shear strain gamma

Shear-strain gamma= delta x/L.

The relationship between the shear stress and shear-strain for a specific material is acknowledged as that material’s shear.

Shear stress strain curve Yield stress

Stress Strain curve Engineering and True

Variation of shear stress with rate of deformation

Shear stress-strain curves are a significant graphical measure.

Shear Stress vs Shear velocity

Shear stress vs shear rate for dilatant and pseudoplastic non-Newtonian fluids compared to Newtonian fluid.

Octahedral shear strain:

The octahedral shear stress/ strain gives the yield point of elastic material under a general stress state. The material displays yielding when the octahedral shear stress reach the extreme value of stress/strain. This is equivalent acknowledged as Von Mises yield criteria.

Shear strain rate :

The strain is the ratio of change in the length to its original length, so; shear-strain is a dimensionless quantity, so the strain-rate is in inversed time unit.

Shear strain formula in metal cutting:

Shear Plane Angle:

This is the angle between the horizontal plane and the shear plane, Significant for shear-strain applications in metal cutting.

The shear-strain rate at the shear plane, ϒAB is a function of cutting velocity and feed.

Difference between shear stress and Shear strain

Shear stress vs Shear strain:

Shear stress contemplates a block, which is subjected to a set of equal and opposite forces Q and this block recognized as bi-cycle-brake-block linked to wheel.There is a chance that one layer of the body slides on others, during shear stress initiation. If this failure is  not permitted, then a shear stress T will be formed.

Q – shear load shear stress (z) = area resisting shear A.

This shear stress will act on vertically to the area. The direct stresses will be at normal direction to the area of application though.

Considering bicycle brake block, the rectangular shaped blocks mightn’t be deformed after the baking force has been applied and block might change the shape in the form of strain.

The shear-strain is proportionate to the shear stress in the elastic range. The modulus of rigidity is represented as 

shear stress z shear strain y = – = constant = G

The constant G = the modulus of rigidity or shear modulus

Why do edge dislocations in crystal lattices introduce compressive tensile and shear lattice strains while screw dislocations only introduce shear strains ?

Because observational and pedagogical definition prescribes the coordinate frames in which this is true”, is probably the most accurate answer.

There are several approaches to visualizing the behavior of the strain fields around dislocations. The first approach is by direct observation; the second one borrows concepts from fracture mechanics. Both are equivalent.

• An edge dislocation directed to the Burgers vector. Although a screw-dislocation directed perpendicularly to it.
• The screw-dislocation ‘unzip’ the lattice as it travels thru it, creating a ‘screw’ or helical prearrangement of atom around the core.

Shear strain problems:

Problem : A rectangular block has area of 0.25m2. The height of the block is 10cm.A shearing force is applied to the top face of the block. And the displacement is 0.015mm.Find the stress, strain and shearing force. Modulus of rigidity= 2.5*10^10 N/m2

Given:

A= 0.25m2

H=10m

x=0.015mm

η =2.5*10^10N/m2.

Solution:

Shear strain = tan θ =X/H

                                   =0.015*10^-3/10*10^-2

                          tan θ =1.5*10^-3

Modulus of Rigidity = Shear stress/Shear strain

       2.5*10^10           = shear stress /0.0015

Shear stress               =2.5*10^10*1.5*10^-3

                                   =3.75*10^7 N/m2

Shear stress              =F/A

F                                = 3.75*10^7*0.25

                                  = 9.37*10^6 N.

Problem : A cube has side of 10 cm. The shearing force is applied on the upper side of the cube and the displacement is 0.75 cm by force of 0.25N. Calculate modulus of rigidity of the substance.

Given:

A= 10*10= 100 cm2

F=0.25N

H= 5cm

X=0.75cm

As we know,

Modulus of rigidity = Shear stress/Shear strain

Shear-strain is= X/H

                    =0.75/5

                    = 0.15

Shear Stress= F/A

                     = 0.25/100*10^-4

                     =25N.

Modulus of Rigidity = 25/0.15

                                   = 166.7 N/m^2

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Content

• Isothermal definition
• Isothermal expansion
• Isothermal compression
• Isothermal vs. adiabatic
• Isothermal calorimetry
• Isothermal amplification
• Isothermal nucleic acid amplification
• Isothermal transformation diagram
• Isothermal PV diagram
• Isothermal process example
• Isothermal work
• Isothermal layer
• Isothermal PCR
• Isothermal process equation
• Isothermal expansion of an ideal gas
• Isothermal reversible expansion
• Isothermal reaction
• Isothermal irreversible expansion
• Isothermal system
• Isothermal bulk modulus
• Isothermal internal energy
• Isothermal compressibility coefficient
• Isothermal heat transfer
• Isothermal atmosphere
• Isothermal surface
• Isothermal conditions
• Isothermal zone
• Isothermal lines
• Isothermal belt
• Isothermal vs isobaric
• Isothermal vs isentropic

Isothermal definition

An isothermal process is a thermodynamic process. In this isothermal process, the system’s temperature remains constant throughout the process. If we consider temperature is T. The temperature change is ΔT.

For the isothermal process, we can say that ΔT = 0

Isothermal expansion

Isothermal expansion is increasing volume with a constant temperature of the system.

Isothermal – temperature constant

Expansion – Increasing volume

Let’s consider the piston-cylinder arrangement for understanding if the piston moves from BDC (Bottom dead center) to TDC (Top dead center) with a constant temperature of the gas. This isothermal process is considered as Isothermal expansion.

Isothermal compression

Isothermal compression is decreasing volume with a constant temperature of the system.

Isothermal – temperature constant

Compression – decreasing volume

Let’s consider another condition if the piston is moving from TDC to BDC (Bottom dead center) with a constant temperature of the gas. This isothermal process is considered Isothermal compression.

Isothermal vs adiabatic

Isothermal means Constant Temperature.

Adiabatic means Constant heat energy.

Some conditions for an isothermal process are :

• The temperature should remain constant.
• The variation must be happening at a slow rate.
• Specific heat of the gas is infinite.

Some basic conditions for adiabatic are as below :

• No heat transfer happens in adiabatic.
• The variation must happen at a very speedy.
• The specific heat of gas is 0 (Zero).

Isothermal calorimetry

It is one technique to find thermodynamic parameters’ interaction in a chemical solution. Using isothermal calorimetry, one can find binding affinity, binding stoichiometry, and enthalpy changes between two or more molecules interactions.

Isothermal amplification

It is one of the techniques used for pathogen monitoring. In this techniques, the DNA is amplified with keeping sensitivity higher than benchmark polymerase chain reaction (PCR)

Isothermal nucleic acid amplification

Isothermal amplification of nucleic acids is a technique that is efficient and faster accumulating nucleic acid at the isothermal process. It is a simple and efficient process. Since then, around 1990, many isothermal amplification processes have been developed as alternatives to a polymerase chain reaction (PCR)

Isothermal transformation diagram

An isothermal transformation diagram is used to understand the kinetics of steel. It is also known as the time-temperature- transformation diagram.

It is associated with mechanical properties, microconstituents/microstructures, and heat treatments in carbon steels.

Isothermal PV diagram

Isothermal process example

Isothermal is a process in which the system’s temperature remains unchanged or constant.

We can take the example of a refrigerator and heat pump. Here, in both cases, the heat energy is removed and added, but the system’s temperature remains constant.

Examples: Refrigerator, heat pump

Isothermal work

We have used the PV diagram above paragraph. If we want to write work done formula for it. We should consider the area under the curve A-B-VA-VB. The Work done for this integral can be given as,

Here in the equation,

n is the number of moles

R is gas constant

T is the temperature in kelvin

Isothermal layer

An isothermal layer term is used in atmospheric science. It is defined as a vertical layer of air or gas with constant temperature throughout height. This situation is happening at the troposphere’s low level in various advection situations.

Isothermal PCR

The full form of PCR is a polymerase chain reaction. This reaction is used in isothermal amplification techniques to amplify DNA.

Isothermal process equation

If we consider universal gas law, then the equation is given as below,

PV = nRT

Now, here this is in isothermal process, so T = Constant,

PV = constant

The above equation holds good for a closed system containing ideal gas.

We have discussed the work done earlier. We can consider that equation for the isothermal process. As we know from figure Vb is the final volume, and Va is the initial volume.

Isothermal expansion of an ideal gas

• Isothermal – the temperature is constant.
• Expansion – the volume is increasing.

It means that isothermal expansion increases volume with a constant temperature of the system.

In this condition, the gas is doing work, so the work will be negative because the gas applies energy to increase in volume.

The change in internal energy is also zero ΔU = 0 (Ideal gas, Constant temperature)

Isothermal reversible expansion

This topic is covered in explaining the isothermal expansion of ideal gas.

Isothermal reaction

A chemical reaction occurring at one temperature, or we can say at a constant temperature, is an isothermal reaction. There no need for temperature change to continue reaction to end.

Isothermal irreversible expansion

An irreversible process is a real process we face in reality almost all the time. The system and its surrounding cannot be restored to their initial states.

Isothermal system

We have discussed the isothermal system in expansion and compression if we take piston-cylinder arrangement.

There are some assumptions for this system like,

• There is no friction between piston and cylinder
• There no heat or work loss from the system
• The internal energy of the system should be constant throughout the isothermal process.

If we supply heat at the bottom of the cylinder, then the piston will move from BDC to TDC, as shown in Figure. It is an isothermal expansion. Similarly, in isothermal compression reverse, as we have explained earlier. This complete system is isothermal.

Isothermal bulk modulus

Bulk modulus is reciprocal of compressibility.

Here, the term is the isothermal bulk modulus. It can be defined as the ratio of change in pressure to change in volume at a constant temperature. It is equal to P (pressure) if we solve the above equation.

Isothermal internal energy

We have discussed earlier that the constant temperature process’s internal energy remains constant.

Isothermal compressibility coefficient

The isothermal compressibility coefficient can be taken as the change of volume per unit change in pressure. It is also known as oil compressibility. It is widely used in resource estimation of oil or gas in petroleum study.

Isothermal heat transfer

The expansion and compression process at constant temperature work on the principle of zero degradation energy. If the temperature is constant, then internal energy change and enthalpy change are zero. So, heat transfer is the same as work transfer.

If we heat the gas in any cylinder, then the gas’s temperature will increase. We want a system at a constant temperature, so we have to put one sink (cold source) to reject gained temperature.

Suppose we consider a cylinder with a piston. The gas will expand in the cylinder, and the piston gives displacement work due to getting heated. The temperature will stay constant in this case also.

Isothermal atmosphere

It can be defined as the there is no change in temperature with height in the atmosphere, and the pressure is decreasing exponentially with moving upward. It is also known as exponential atmosphere. We can say that the atmosphere is in hydrostatic equilibrium.

In this type of atmosphere, we can calculate the thickness between two adjacent heights with the equation given below,

Where,

Z1 & Z2 are two different heights,

P1 & P2 are Pressures at Z1 & Z2, respectively,

R is gas constant for dry air,

T is the virtual temperature in K,

g is gravitational acceleration in m/s²

Isothermal surface

Suppose we consider any surface flat, circular, or curvature, etc. If all the points on that surface are at the same temperature, then we can say that the surface is isothermal.

Isothermal conditions

As my word, we know that the system’s temperature must stay constant in this isothermal process. To keep the temperature constant, the system is free to change other parameters like pressure, volume, etc. It is also possible during this process, the work-energy and heat energy can be changed, but the temperature remains the same.

Isothermal zone

This word is generally used in atmospheric science. It is a zone in the atmosphere where the relative temperature is constant at some kilometer height. Generally, it is in the lower part of the stratosphere. This zone provides convenient aircraft conditions because of its constant temperature, general access to clouds and rains, etc.

Isothermal lines

This word is used in geography. Suppose we draw a line on a map of the earth for connecting different places whose temperature is the same or near to the same. It is known as an isothermal line in general.

Here, each point reflects the particular temperature for reading taken in a period of time.

Isothermal belt

In 1858 Silas McDowell of Franklin, given this name for western North Carolina, Rutherford, and Polk countries. This term is used for a season in these zones when one can grow fruits, vegetables, etc., easily due to temperature consistency.

Isothermal vs isobaric

Isothermal – temperature constant

Isobaric – Pressure constant

Let’s compare both processes for work done. According to the figure, you can notice both processes. As we know, that work done is an area under the integral. In the figure, we can easily see that the isobaric process area is more so obviously, work done more in isobaric. There is some condition for it. The initial pressure and volume should be the same. This is not true because we never get work during isobaric in any of the thermodynamic cycles. This topic is logical.

The correct answer depends on the type of condition that volume is increased or decreased in the process.

Isothermal vs isentropic

Isothermal – temperature constant

Isentropic – Entropy constant

Let’s consider the compression process to understand it,

In isothermal compression, the piston is compressing gas very slowly. As much slowly to maintain the constant temperature of the system.

Whereas in the case of isentropic, there should be no heat transfer possible between the system and surrounding. The isentropic compression will occur without heat transfer with constant entropy.

The isentropic process is similar to adiabatic, where there is no heat transfer. The system for the isentropic process should be well insulated for heat loss. The isentropic compression process always gives more work output due to no heat loss.

FAQs

Is there heat transfer in the Isothermal process?

Answer: Yes,    now the question is why and how?

Let’s consider a piston-cylinder example to understand it,

If heat is supplied to the bottom of the cylinder. The temperature will be maintained constant, and the piston will move. Either expansion or compression process. The heat is transferred, but the system’s temperature will stay the same as it is. This is why during the Carnot cycle, heat is added at a constant temperature.

Why Isothermal process is very slow?

It is necessary that the Isothermal process occurs slowly. Now see, the heat transfer is possible by keeping the system’s temperature constant. It means there is a thermal equilibrium of the system with the body. The process’s timing is slow to keep this thermal equilibrium and constant temperature. The time required for effective heat transfer will be higher, making the process slow.

Isothermal process example problems

There are many applications in day-to-day life with a constant temperature. Some of them are explained as below,

• The temperature inside the refrigerator is maintained
• It is possible to melt the ice by keeping the temperature constant at 0°C
• The phase change process occurs at a constant temperature, evaporation, and condensation
• Heat pump which works opposite to refrigeration

What are some real-life examples of an Isothermal process?

There is a huge number of example can be possible for this question. Kindly refer above questions.

Any phase change process occurring at constant temperature is an example of an isothermal process.

Evaporation of water from sea and river,

Freezing of water and melting of ice.

Why does Isothermal process be more efficient than the adiabatic process?

Let’s consider the reversible process. If the process is expansion, then the isothermal process’s work is more than adiabatic. You can notice by a diagram. The work done is an area under the curve.

Suppose the process is compression, then opposite to the above sentence. The work done in the adiabatic process is more.

To judge this question depends on every condition. As per the above condition, the isothermal process is more efficient than the adiabatic.

What will be the specific heat for an Isothermal process an adiabatic process, and why?

The specific heat can be defined as the amount of heat is required to raise the temperature of a substance by 1 degree.

If the process is the constant temperature, the ΔT = 0, so the specific heat is undefined or infinite.

Cp = Infinite  (if temperature is constant)

For adiabatic process, the heat transfer is not possible , Q = 0

Cp = 0 (heat transfer is 0)

In an Isothermal process, the change in internal energy is 0 Why?

Internal energy is the function of the kinetic energy of the molecules.

The temperature indicates the average kinetic energy of molecules associated with the system.

If the temperature remains constant, then there is no change in kinetic energy. Hence, the internal energy remains constant. The change in internal energy is zero.

What is more efficient Isothermal compression or isentropic compression, And why?

The isentropic process occurs at constant entropy with no heat transfer. This process is always ideal and reversible. In the isentropic compression process, the system’s internal energy is increasing as there is no possibility of heat transfer between the system and surrounding.

In isothermal compression, the process occurs very slowly as the temperature and internal energy stay constant. There is heat transfer between the system and the surrounding.

That’s why the isentropic compression process is more efficient.

Does an Isothermal process have an enthalpy change?

We can understand it clearly by the equation of enthalpy.

Enthalpy H is given as below,

Change in enthalpy = change in internal energy + change in PV

For constant temperature process,

Change in internal energy = 0,

Change in PV = 0.

That’s why to Change in enthalpy= 0

Why is an adiabatic curve steeper than an Isothermal curve?

In the adiabatic process, the system’s temperature is increasing during compression. It is decreasing during expansion.  Due to this, this curve crosses the isothermal curve at a certain point in the diagram.

In isothermal, there is no change of temperature. The curve will not become steeper like adiabatic.

What would happen if I increase the volume of a system in an Isothermal process with external energy?

 Suppose you increase the volume of the system.  You want the system to be isothermal. You have to make another arrangement for maintaining temperature. The increasing of the volume decreases the pressure.

What is so special about the word “reversible” in an Isothermal or an adiabatic process?

The first law of thermodynamics states that both of the processes sketched on the PV diagram are reversible mean. The system will come to its initial stage to stay in equilibrium.

Why Isothermal and adiabatic in Carnot engine?

The Carnot cycle is the most efficient in thermodynamics. The reason behind it is all the process in the cycle is reversible.

Carnot tried to transfer energy between two sources at constant temperature (Isothermal).

He tried to maximize the expansion work and minimize the required compression. He selected an adiabatic process for it.

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What is a governor in a car?  

Speed governor | Engine governor

Whenever there is a “variation of load on the engine”, there will be change in the engine’s speed. For the maintaining the speed of an engine up to definite limit, a speed governor is utilized. There is a variation in the speed of the engine over a no of rotations because of engine’s load variations. The operation of a speed governor is more or less intermittent and these are critical for all types of engine because its adjustability on the supply of fuel as per requirement. A speed governor is also known as a governor or sped limiter.

Governor symbol

What are the two main components of a governor system?

Types of Governor

The Governor categorized in two types.

• Centrifugal Governor
• Inertia Governor

Centrifugal Governor

Centrifugal governor consists of a pair of governor balls attached to the arms, which are supported by the spindles as shown in the figure. This whole system is mounted on the shaft; this shaft is linked to the engine of the shaft thru a bevel gear mechanism. Under this assembly, a free to slide sleeve is attached to the shaft. A bell crank lever is connected to the sleeve. This lever connects the throttle valve and sleeve.

The activity of the governor be subject to the speed variation. The variations in the bell crank lever give the motion to the sleeve and eventually to the spindle and ball. The governor’s action is produced by the centrifugal effect of the masses of the balls.

When the speed of the engine increase, the balls intend to rotate with higher radius from the centered shaft position, this caused the sleeve to slide in the upward way on the spindle and these variations of the spindle will result in the closing action of the throttle-valve up to the mandatory limit thru the bell-crank lever mechanism. When the speed declines, the balls will rotate at a lesser radius, so the throttle valve opens accordingly.

It is a more common type of governor.

Type of Centrifugal Governors

The centrifugal governor is further classified as follows:

Inertia Governor

Inertia governor is based on the ‘Principle of Inertia of Matter’.

For Inertia Governor, more force acts with the centrifugal forces on the balls, whose position are decided by the angular acceleration and de-acceleration of the spindle in addition to the centrifugal force acts on the ball.

In the inertia type of governor, appropriate linkages and spring used for opening and closing the throttle valve according to the changes in the position of the ball.

In an inertia governor, when the acceleration or deceleration of an engine is very low, the additional inertia force practically becomes zero. In that case, the inertia governor becomes a centrifugal governor.

The inertia governor’s response is faster than that of the centrifugal governor.

Engine Governor

The throttle valve of an engine is operated by the governor called engine governor using a mechanism explained earlier.

Sensitiveness of a Governor

The movement of the sleeve essential for the minimal change in the speed of an engine is the measure of sensitiveness of a governor.

A governor’s sensitiveness is described as the ratio of the change in-between the highest and least speed to the mean equilibrium speed.

Thus,

Where,

N=mean speed

N1=minimum speed corresponding to full load conditions

N2=maximum speed corresponding to no-load conditions

Turbine Governor

A turbine governor is a component of a turbine control system that controls rotational speeds according to loading conditions.

A turbine governor provides on-line and start-up control for the generator, which drives the turbine.

Steam Turbine Governing

In a steam turbine, there is an inconsistent flow of steam. A steam turbine governing is a procedure of maintaining a constant rotation speed by controlling the steam turbine’s flow rate to the steam turbine.

Elevator Governor | Over-speed Governor

When the speed of an elevator crosses predetermined speed limits, a mechanism acts to control the system known as over speed governor.

A speed governor located in the elevator is a component of the safety system of the governor.

The position of the speed governor in the elevator is determined by the However, in many cases, it is installed in the machine room of the elevator.

Air Vane Governor

An air vane governor is a pneumatic type of governor.

In this system, airflow is used to regulate the throttle opening. This air vane of the governor is made up of plastic or metal. This governor also consists of the flywheel.

Speed Limiter for Car

A speed governor is used as a speed limiter for cars’ engines. It regulates the fuel supply of the car with varying load.

There are three types of the governor which are being used in automobiles:

·       Mechanical Governor

·       Hydraulic Governor

·       Pneumatic Governor

 

Woodward Governor

Wood ward governor is a well-known manufacturing company of governors.

Governor Switch | Governor Gear

Governor Switch, Governor Gear are parts of an evolved form of a speed governor.

Question and Answers related to Governor

What are the two main components of a governor system? 

The two main components of a governor are mechanical arrangement and hydraulic unit.

How does a mechanical governor work?

Centrifugal governor consists of a pair of governor balls attached to the arms, which are supported by the spindles as shown in the figure. This whole assembly is mounted on the shaft; this shaft is connected to the engine of the shaft using bevel gear. Under this assembly, a free to slide sleeve is attached to the shaft via a bell-cranks. This lever connects the throttle valve and sleeve.

As we already know, governor’s action is dependent on the speed variations. The changes in the bell crank lever give the motion to the sleeve and ultimately to the spindle and balls. The governor’s action is produced by the centrifugal effect of the masses of the balls.

During the engine’s speed increment, the ball will intend to rotate at a higher- radii from shaft’s position and caused the sleeve to sliding in the upward direction and this movement of the spindle consequences in the closing of the throttle-valves and if speed declines, these balls will rotate with less radii, so the throttle valve controlled accordingly.

What are the three types of governors?

There are three types of the governor which are being used in automobiles:

• Mechanical Governor
• Hydraulic Governor
• Pneumatic Governor

What is Governor Sensitivity?     

Governor sensitivity is known as the sensitiveness of a governor.

The movement of the sleeve for the minimal change in the speed of an engine is the measure of sensitiveness of a governor.

Where,

N=mean speed

N1=minimum speed corresponding to full load conditions

N2=maximum speed corresponding to no-load condition

Which governor is more sensitive?     

Proell governor is known as the most sensitive governor in the centrifugal type of governors.

Whereas Porter governor is more sensitive than Watt governor.

What are the applications of a governor?

1. A speed governor is used as a speed limiter for cars’ engines. It regulates the fuel supply of the car with varying load.
2. A governor is used in elevators. When the speed of an elevator crosses predetermined speed limits, a mechanism acts to control the system known as over speed governor.
3. A speed governor is used in different types of turbines. A turbine governor provides on-line and start-up control for the generator, which drives the turbine.

Which governor is used in cars? | What is a governor in a car?

A speed governor is used as a speed limiter for the engines of cars. It regulates the fuel supply of the car with varying load.

There are three types of the governor which are being used in automobiles:

1. Mechanical Governor
2. Hydraulic Governor
3. Pneumatic Governor

What is the range of governor?

The variance between the maximum and minimum speed of a governor is known as a governor’s range.

What is the meaning of speed governor?

Whenever there is a variation in the load on the engine, there is a variation in the speed of an engine and to maintain the engine’s speed at stated limit, a speed governor is employed.

Can you remove a speed governor?

Yes. A speed governor is inbuilt for a car from the company, but it can be removed if we want.

Is speed governor compulsory?

Yes. It has been made compulsory in many countries to have a speed governor.

How does a speed governor work?

The throttle valve of an engine is operated by the governor, when the load on the engine shaft increases, it’s speed will decline except the fuel supply is increased by the throttle valve opening. Similarly, if the load on the engine shaft declines, its speed will increase unless the fuel supply is reduced by closing the throttle-valve appropriately to slowdown engine’s to actual speed.

What is a Hartnell governor?

A Hartnell governor is a centrifugal type of governor in which it is loaded with a spring.  Additional spring is used to apply an extra force to the spring.

Where is Hartnell governor used?      

A Hartnell governor is used in regulating the speed of the engine.

Which lever is used in Hartnell governor?  

The lever used in Hartnell governor is a bell crank lever.

What are the types of speed governor?

Governors are broadly classified into two types.

1. Centrifugal Governor.
2. Inertia Governor.

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Content: Prandtl Number

Prandtl Number

“The Prandtl number (Pr) or Prandtl group is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity to thermal diffusivity.”

Prandtl Number formula

The Prandtl (Pr) number formula is given by

Pr = Momentum diffusitivity/Thermal diffusitivity

Pr = μC_p/k

Pr = ν/∝

Where:

μ = dynamic viscosity

C_p = Specific Heat of the fluid taken into consideration

k = Thermal Conductivity of the fluid

ν = Kinematic viscosity

α = Thermal diffusivity

ρ = Density of the fluid

Prandtl (Pr) number is independent of Length. It depends upon the property, Type and state of the fluid. It gives the relation between the viscosity and thermal conductivity.

Fluids having Prandtl (Pr) number in the Lower spectrum are free-flowing fluids and generally possess high thermal conductivity. They are excellent as heat conducting liquids in heat exchanger and similar applications. Liquid metals are brilliant in heat transfer.  As viscosity increases Prandtl (Pr) number increases and thus heat conduction capacity of fluid decreases.

Physical significance of Prandtl Number

During heat-transfer in-between wall and a flowing-fluid, heat is transferred from a high-temperature wall to the flowing fluid thru a momentum boundary-layer that comprises the bulk-fluid substance and a transitional and a thermal boundary-layer that comprises of stationary film. In the stagnant film, heat transfer occurs by conduction in the fluid. The importance of Prandtl (Pr) number of the flowing fluid is to taken into account as it relates momentum boundary-layer to the thermal one during heat-transfer through the fluid.

when Prandtl (Pr) number has Small values, Pr << 1, It represents that thermal diffusivity dominating over momentum diffusivity and liquid metal has lower Prandtl (Pr) number and heat diffuses significantly faster in that. Thermal boundary layer has higher thickness comparation of velocity-based boundary-layer in liquid-metal.

Similarly, for large values of Prandtl (Pr) number, Pr >> 1, the momentum diffusivity dominates over thermal diffusivity. oils have higher Prandtl (Pr) number and heat diffuses slowly in oils. Thermal boundary layer has Lower thickness relative to velocity boundary layer in oils.

For liquid mercury the heat conduction is more dominant in comparison to convection, Thus thermal diffusivity is dominant in Mercury. Though, for engine-oil, convection is highly effective in heat transfer from a high temperature area when compared to purely conduction case, thus, momentum diffusivity is significant parameter in Engine-oil.

Gases lie in the middle of this spectrum. Their Prandtl (Pr) number is about 1. Thermal boundary layer has equal thickness relative to velocity boundary layer.

The ratio of the thermal to momentum boundary layer over a flat plate is given by the following equation

δ_t/δ = Pr^-1/3 0.6<Pr<50

Magnetic Prandtl Number

Magnetic Prandtl Number is a dimensionless number which gives the relation between Momentum diffusivity and magnetic diffusivity. It is the ratio of viscous diffusion rate to the magnetic diffusion rate. It generally occurs in magnetohydrodynamics. It can also be evaluated as the ratio of magnetic Reynold’s Number to the Reynold’s Numbers.

Pr_m = ν/η

Pr_m = Re_m/Re

Where,

Rem is the magnetic Reynolds number

Re is the Reynolds number

ν is the viscous diffusion rate

η is the magnetic diffusion rate

Prandtl Number Heat Transfer

when Prandtl (Pr) number has Small values, Pr << 1, It represents that thermal diffusivity dominating over momentum diffusivity. Liquid metal has lower Prandtl (Pr) number and heat disseminates very quickly in Liquid metal and Thermal-boundary layer is much thicker in comparison to velocity-boundary layer in liquid-metal.

Similarly, for large values of Prandtl (Pr) number, Pr >> 1, the momentum diffusivity dominates over thermal diffusivity. oils have higher Prandtl (Pr) number and heat diffuses slowly in oils. Thermal boundary layer has Lower thickness relative to velocity boundary layer in oils.

For  liquid mercury the heat conduction is more dominant in comparison to convection, Thus thermal diffusivity is dominant in Mercury. Though, for engine’s oil, convection is highly effective in heat-transfer from a high temperature area when compared to purely conductive, thus, momentum diffusivity is significant in Engine’s oil.

Gases lie in the middle of this spectrum. Their Prandtl (Pr) number is about 1. Thermal boundary layer has equal thickness relative to velocity boundary layer.

The ratio of the thermal to momentum boundary layer over a flat plate is given by the equation

δ_t/δ = Pr^-1/3 0.6<Pr<50

Turbulent Prandtl Number

The turbulent Prandtl number Pr_t is a dimensionless term. It is the ratio of momentum eddy diffusivity to the heat transfer eddy diffusivity and utilized for the evaluation of heat transfer for turbulent boundary layer flow condition.

Does heat transfer coefficient depend on Prandtl number?

Heat Transfer coefficient is also calculated by means of Nusselt’s Number. This is represented by the ratio of Convective heat transfer to the conductive heat transfer.

For forced convection,

Nμ = hL_c/K

Where, 

h = the convective heat transfer coefficient

L_c = the characteristic length,

k = the thermal conductivity of the fluid.

Also, Nusselt Number is the function of Reynold’s Number and Prandtl (Pr) number. Thus, Change in Prandtl (Pr) number changes the Nusselt Number and thus heat transfer coefficient.

Does Prandtl number change with pressure?

Prandtl (Pr) number is assumed to be independent of pressure. Prandtl (Pr) number is a function of Temperature since μ,C_p are the function of Temperature but a very weak function of pressure.

Effect of Prandtl number on boundary layer | Effect of Prandtl number on heat transfer

when Prandtl (Pr) number has Small values, Pr << 1, It represents that thermal diffusivity dominating over momentum diffusivity. Liquid metals have lower Prandtl (Pr) number and heat diffuses very quickly in Liquid metals. Thermal boundary layer has higher thickness relative to velocity boundary layer in Liquid metals.

Similarly, for large values of Prandtl (Pr) number, Pr >> 1, the momentum diffusivity dominates over thermal diffusivity. oils have higher Prandtl (Pr) number and heat diffuses slowly in oils. Thermal boundary layer has Lower thickness relative to velocity boundary layer in oils.

For  liquid mercury the heat conduction is more dominant in comparison to convection, Thus thermal diffusivity is dominant in Mercury.

Gases lie in the middle of this spectrum. Their Prandtl (Pr) number is about 1. Thermal boundary layer has equal thickness relative to velocity boundary layer.

Prandtl number of Air

Prandtl (Pr) number for Air is given below in the table

Prandtl (Pr) number of Air at 1 atm pressure, temperature °C is given as:

Temperature	Pr
[°C]	Dimensionless
-100	0.734
-50	0.720
0	0.711
25	0.707
50	0.705
100	0.701
150	0.699
200	0.698
250	0.699
300	0.702

Prandtl number of Water at different Temperatures

Prandtl (Pr) number of Water in Liquid and vapor form at 1 atm Pressure is shown below:

Temperature	Pr number
[°C]	Dimensionless
0	13.6
5	11.2
10	9.46
20	6.99
25	6.13
30	5.43
50	3.56
75	2.39
100	1.76
100	1.03
125	0.996
150	0.978
175	0.965
200	0.958
250	0.947
300	0.939
350	0.932
400	0.926
500	0.916

Prandtl number of Ethylene glycol

Prandtl (Pr) number of Ethylene glycol is Pr = 40.36.

Prandtl number of Oil | Prandtl number of Engine Oil

Prandtl (Pr) number for oil lies between the range of 50-100,000

Prandtl (Pr) number of Engine Oil at 1 atm Pressure are given below:

Prandtl number Table

Temperature (K)	Pr number
260	144500
280	27200
300	6450
320	1990
340	795
360	395
380	230
400	155

Prandtl number of Hydrogen

Prandtl (Pr) number of Hydrogen at 1 atm Pressure and at 300 K is 0.701

Prandtl number of Gases | Prandtl Number of Argon, Krypton etc.

Prandtl number of Liquid Metals and other Liquids

Benzene Prandtl number

Prandtl (Pr) number of Benzene at 300 K is 7.79.

CO₂ Prandtl number

Prandtl (Pr) number of Hydrogen at 1 atm Pressure is 0.75

Prandtl number of Ethane

Prandtl (Pr) number of Ethane is 4.60 in Liquid form and 4.05 in gaseous form

Gasoline Prandtl number

Prandtl (Pr) number of Gasoline is 4.3

Glycerin Prandtl number

Prandtl (Pr) number of Glycerin lies between the range of 2000-100,000

Some Important FAQs

Q.1 How is Prandtl number calculated?

Ans:  Pr Number can be calculated by using the formula

Pr = μC_p/K

Where:

• μ = dynamic viscosity
• C_p = Specific Heat of the fluid taken into consideration
• k = Thermal Conductivity of the fluid

Q.2 What is the value of Prandtl number for liquid metals?

Ans: The Prandtl (Pr) number for Liquid metals is extremely Low. Pr<<<1. For example In liquid mercury has Prandtl (Pr) number = 0.03 which represents that, the heat conduction is more dominant in comparison to convection, Thus thermal diffusivity is dominant in Mercury.

Q.3 What is the Prandtl number of Water?

Ans: Prandtl (Pr) number of Water in Liquid and vapor form at 1 atm Pressure is shown below:

Temperature	Prandtl (Pr) number
[°C]	Dimensionless
0	13.6
5	11.2
10	9.46
20	6.99
25	6.13
30	5.43
50	3.56
75	2.39
100	1.76
100	1.03
125	0.996
150	0.978
175	0.965
200	0.958
250	0.947
300	0.939
350	0.932
400	0.926
500	0.916

Q.4 What does Prandtl number represent?

Ans: During the heat transfer amongst a wall-barrier and fluid, heat is transferred from a high-temp barrier to fluid through a momentum-boundary-layer. This includes fluids and a transitional and a thermal boundary-layer that comprises of film. In the stagnant film, heat transfer happens by fluid’s conduction on that time. The Pr number of the flowing fluid, is ratio which taken into account of momentum boundary layer to the thermal boundary layer.

Q.5 what is the Prandtl Number for Steam?

Ans: The Prandtl (Pr) number for steam at 500 C is 0.916.

Q.6 what is the Prandtl Number for Helium?

Ans: Prandtl (Pr) number of Helium is 0.71

Q.7 what is the Prandtl Number for Oxygen?

Ans: Prandtl (Pr) number of Oxygen is 0.70

Q.8 What is the Prandtl Number for Sodium?

Ans: Prandtl (Pr) number of Sodium is 0.01

Q.9 How is the Prandtl number related with kinematic viscosity and thermal diffusivity?

Ans: The Prandtl (Pr) number is well-defined as the ratio of momentum diffusivity to thermal diffusivity.

Its formula is given by:

The Pr Number formula is given by

Pr = Momentum diffusivity/ Thermal diffusivity

Pr = μC_p/K

Pr = μ/α

Where:

μ = dynamic viscosity

C_p = Specific Heat of the fluid taken into consideration

k = Thermal Conductivity of the fluid

ν = Kinematic viscosity

ν = μ/ρ

α = Thermal diffusivity

α = K/ρC_p

ρ = Density of the fluid

From the above formula we can say that Prandtl (Pr) Number is inversely proportional to Thermal diffusivity and directly proportional to Kinematic viscosity.

Q.10 Is there any fluid which has a Prandtl number in the range of 10 20 except water?

Ans: There are certain number of fluids that has Prandtl (Pr) Number in the range of 10-20. They are Listed below:

1. Acetic acid [Pr = 14.5] at 15C and [Pr = 10.5] at 100C
2. Water [Pr = 13.6] at 0C
3. n-Butyl Alcohol is [Pr = 11.5] at 100 C
4. Ethanol [Pr = 15.5] at 15C and [Pr = 10.1] at 100C
5. Nitro Benzene [Pr = 19.5] at 15C
6. Sulfuric acid at high concentration about 98% [Pr = 15] at 100C

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