Engineering

Content

• Thermal diffusivity definition
• Unit of Thermal diffusivity
• Thermal diffusivity formula
• Thermal diffusivity of water
• Thermal diffusivity of air
• Thermal diffusion
• Thermal diffusivity of aluminium
• Flash method Thermal diffusivity
• How to measure Thermal diffusivity
• Thermal conductivity and thermal diffusivity
• Thermal diffusivity measurement techniques
• Thermal diffusivity of asphalt
• Thermal diffusivity of rubber
• Thermal diffusivity values
• Thermal diffusivity symbol
• The highest Thermal diffusivity is of
• Thermal diffusivity of sand
• Thermal diffusivity heat transfer
• The Thermal diffusivities for gases are generally
• Thermal diffusion coefficient
• Thermal diffusion meaning
• Glass Thermal diffusivity
• stainless steel Thermal diffusivity
• Thermal diffusion ratio

Thermal diffusivity definition

Thermal diffusivity is defined as the ratio of conducted heat to heat stored in material per unit volume.

Unit of Thermal diffusivity

The unit of thermal diffusivity is given as m²/s

Thermal diffusivity formula

The equation of thermal diffusivity is given by,

α = k/ρ

Where,

α is thermal diffusivity,

k is thermal conductivity  (w/mK)

? is the density of the material (kg / m³)

Cp is the specific heat (J/ kg k)

Thermal diffusivity of water

The thermal diffusivity of water changes with temperature and pressure. If we consider atmospheric pressure condition, then the thermal conductivity values with temperature are given as below table.

Thermal diffusivity of air

The thermal diffusivity of air with temperature change is shown in the above table. Generally , the thermal diffusivity of gas is more than liquid in practice. we will study is more in next topic.

Thermal diffusion

Thermal diffusion of substance is the relative motion of the molecules due to temperature gradient.

Thermal diffusivity of aluminium

Thermal diffusivity of aluminium material is given as 9.7 * 10^-5 m²/s

Flash method Thermal diffusivity

The flash method is used to determine the thermal diffusivity of the material. The short duration radiant energy pulse is passed through the sample. The laser or light flash lamp source is used for radiant energy. The piece will absorb the emitted energy. The process is repeated for the sample. Due to this emitted radiation, there is a temperature increase of the material sample. The infrared temperature detector records this increase in temperature.

The duration of the measured signal is calculated. The thermal diffusivity will be found from the following equation.

α = 0.1388/l²(t²)

=

Where L is the sample thickness,

t/2 is the half time,

we can find thermal diffusivity, specific heat, and density using the flash method.

The schematic diagram of the flash method is shown in the figure below

How to measure Thermal diffusivity

The thermal diffusivity can be measured using the flash method as discussed above. in this method, the short energy pulse is radiated one end, and temperature rise is calculated on the other end.

Thermal conductivity and thermal diffusivity

To differentiate between thermal conductivity and thermal diffusivity, consider two materials having the same thermal conductivity but with different thermal diffusivity. Both will permit the same rate of heat flow in the steady-state condition. At the start of the heat transfer process, the material with higher thermal diffusivity will reach a steady-state first compared to other material since it retains less heat energy. Heat energy penetrates fast through this material, but after getting a steady-state, the rate of heat flow will be the same. Also, remember that the material having less thermal diffusivity takes more time to reach the steady state.

Thermal diffusivity measurement techniques

There are mainly three types of thermal diffusivity measurement techniques.

• Flash method
• Thermal wave interferometry
• Thermographic method

Thermal diffusivity of asphalt

The thermal diffusivity of the asphalt (Ah-70) is 0.123 mm²/s,

Asphalt (Ah-90) 0.128 mm²/s

Thermal diffusivity of rubber

The thermal diffusivity of the rubber material is in the range of 0.089-0.13 mm²/s

Thermal diffusivity values

Thermal diffusivity values for various materials are given in the table below. The values are changes with properties like temperature. These values are given for standard temperature and pressure.

Thermal diffusivity symbol

The symbol of thermal diffusivity is α

The highest Thermal diffusivity is of

The highest thermal diffusivity is of pure silver 165.63 mm² / s

Thermal diffusivity of sand

Thermal diffusivity of dry sand varied from 0.6 * 10^-7 to 7.0 * 10^-7 m²/s.

Thermal diffusivity heat transfer

There are three modes of heat transfer conduction, convection and radiation. Heat conduction is dependent on main two properties. One is thermal conductivity and thermal diffusivity. Thermal conductivity is well-known property, but thermal diffusivity is not well known. It defines the rate of heat transfer through a given medium.

The rate of heat transfer is faster is the thermal diffusivity is higher. Thermal diffusivity is balancing between the medium of heat transfer and heat storage.

The Thermal diffusivities for gases are generally

The thermal diffusivities of gases substance are found more than liquid substance

Thermal diffusion coefficient

It is one of physical parameter which describes the dependency of mass diffusion flow of the mixture. In other words, the thermal diffusion coefficient is the ratio of a temperature gradient to the absolute temperature.

Thermal diffusion meaning

Thermal diffusion of substance is the relative motion of the molecules due to temperature gradient.

Glass Thermal diffusivity

The thermal diffusivity of glass is 0.34 * 10^-6 m²/s  at normal atmospheric condition.

Stainless steel Thermal diffusivity

The thermal diffusivity of stainless steel at 100 °C is 4.55 *10^-6 m²/s

Thermal diffusion ratio

The thermal diffusion ratio is the ratio of the thermal diffusion coefficient to the concentration coefficient.

FAQs

Thermal diffusivity of gas vs liquid

The thermal diffusivities of gases substance are found more than liquid substance

Which material has the highest Thermal diffusivity

The most increased thermal diffusivity is of pure silver 165.63 mm² / s

Application of Thermal diffusivity

The conduction heat transfer in any apparatus requires the study of thermal diffusivity. The industries are using the analysis of thermal diffusivity to optimize the heat transfer rate.

If we take a particular example, then insulation is one example. In insulation, the thermal diffusivity of the material is minimum so that it can resist maximum heat flow.

We are using computers, laptops and other electronic gadgets. Do you know what the method to extract heat from devices is? Yes, it’s Heat sinks.

Heat sink requires higher thermal diffusivity to transfer faster heat from any gadgets.

An increase in heat transfer in any electronics degrades its performance. The higher thermal diffusivity material should be used to improve its performance in that case.

Thermal diffusivity of concrete

The thermal diffusivity of the concrete is 0.75 *10 ^-6 m²/s

What is the physical significance of thermal diffusivity?

Thermal diffusivity can be defined as the ratio of the thermal conductivity of the substance to the heat storage capacity of the substance.

The ratio defines the generated heat gets diffused out at a specific rate. The higher value of thermal diffusivity indicates that the time required for heat diffusion is less. The study of the equation of thermal diffusivity can be possible by the higher value of thermal conductivity or the lower value of heat capacity.

Thermal diffusivity is helpful for more intransient heat transfers. In steady-state heat transfer, the thermal conductivity is enough to study.

Why is the thermal diffusivity of gas greater than liquid even though the thermal conductivity of the liquid is greater than gases?

Thermal diffusivity means the ability of a material to transfer heat and store the heat at an unsteady state. A faster heat transfer can be possible if the thermal diffusivity is higher. The lower thermal diffusivity of material means the storage of heat in it.

Gas possesses low volumetric heat capacity because of low density. Due to low volumetric heat capacity, the value of thermal diffusivity is high.

Liquid possesses a high heat capacity compare to gas; hence thermal diffusivity is lower in the liquid.

What is the order of thermal diffusivity for solid, liquid, and gas?

The order of thermal diffusivity in solid, liquid and gas as shown below,

Gas > Liquid > Solid

What is the difference between momentum diffusion and thermal diffusion?

Momentum diffusion

It can be considered the kinematic viscosity of the fluid, i.e. the ability of the fluid to flow the momentum. Momentum diffusion is occurred by shear stress in a fluid. Shear stress causes a random and any direction movement of molecules.

Thermal diffusion

It can be defined as thermal conductivity divided by the multiplication of density and specific heat capacity (when the pressure is constant). It measures the heat transfer rate for a given material from the hot side to the cool side. It is predictive analogous to whether a given material is “cool to the touch.”

How is the Prandtl number related to kinematic viscosity and thermal diffusivity?

The Prandtl Number is dimensionless. It can be given as the ratio of momentum diffusivity (it is kinematic viscosity as explained above) to thermal diffusivity.

It can be formulated in equation as,

Pr = v/α

Pr = Prandtl number

V= momentum diffusivity ( m²/s )

α = Thermal diffusivity ( m²/ s )

MCQs

Thermal diffusivity is the _________

(a) Dimensionless parameter     (b) Function of heat       (c) Physical property of the material

(d) All of the above

Thermal diffusivity of a material is __________________?

(a) directly proportional with thermal conductivity (k)

(b) inversely proportional with the density of a material

(c) inversely proportional with specific heat

(d) all of the above

(e) none of the above

Find the wrong statement: Specific heat of a material ______________.

(a) Constant for a material           (b) Heat capacity per unit mass

(c) Extrinsic property                      (d) Has units as J/kg-K.

What is a unit of thermal diffusivity?

(a) m/h

(b) m²/h

(c) m/hk

(d) m²/hk

Thermal diffusivity of solid is less than liquid.

(a) True

(b) False

Thermal diffusivity is higher in…….

(a) Rubber

(b) Lead

(c) Iron

(d) Concrete

Thermal diffusivity is lower in……

(a) Rubber

(b) Lead

(c) Aluminum

(d) Iron

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Principal Stress Definition:

Principal Stress is the maximum and minimum stresses derived from normal stress at an angle on a plane where shear stress is zero.

How to calculate principal stress ?

Principal stress equation | Principal stress formula:
Maximum and Minimum principal stress equations:

Principal stress derivation | Determine the principal planes and the principal stresses

Normal stresses:

Differentiate,

“p” represents the principal plane.

There are two principal stresses,
one at angle
and other at
Maximum and Minimum principal stresses:

R=

substitute in equation 1:

substitute value of R

Maximum and minimum normal stresses are the principal stresses:

The state of Stress:

The principal stress is the reference co-ordinate axes to the representation of the stress matrix and this stress components are the significance of the state of stress could be represented as,

Stress tensor:

Principal stresses from stress tensor and stress invariants |principal stress invariants

There are three principal planes at any stressed body, with normal vectors n, called principal directions where the stress vector is in the same direction as normal vector n with no shear stresses and these components depend on the alignment of the co-ordinate system.

A Stress vector parallel to the normal unit vector n is specified as,

Where,
\\lambda represents constant of proportionality.

The principal stress vectors represented as,

The magnitude of three principal stresses gives three linear equations.
The determinant of the coefficient matrix is equal to zero and represented as,

Principal stresses are the form of normal stresses, and the stress vector in the coordinate system is represented in the matrix form as follows:

I1, I2, I3 are the stress invariants of the principal stresses,
The stress invariants are dependent on the principal stresses and are calculated as follows,

The principal stresses equation for stress invariants:

Principal stress trajectories | Principal directions of stress

Stress trajectories show the principal stress directions and their varying magnitude of the principal stresses.

Von mises stress vs principal stress

Von mises principal stress equation

Von Mises is the theoretical measure of the stress yield failure criterion in ductile materials.
The positive or negative sign depends on the principal stresses.
Principal stresses Boundary conditions:

Theories of failure give the yield stresses of the components subjected to multiaxial loading. Further, when it is compared with the yield point of the components shows the margin of the safety of the component.

Maximum principal stress is considered for brittle elements such as casting components (i.e., clutch housing, gearbox, etc.)
Von-mises stress theory is based on shear strain energy theory is suggested for ductile materials like aluminum, steel components.

Why von mises stress is recommended for ductile and Principal Stress for brittle materials?

Failure of brittle materials used to uni-axial test is along a plane vertical to the axis of loading. So, the failure is because of normal stress generally. Out of all theories of failure, principal stress theory is based on normal Stress. Hence for brittle materials, principal stress theory is recommended,

Ductile materials fail at 45 degrees inclined at the plane of loading. So, the failure is due to shear stress. Out of all theories of failure shear strain energy or von-mises theory and maximum shear stress theory is based on shear stress. By comparison, von mises gives better results. Hence for ductile materials, von mises theory is recommended.

Different types of stress

Absolute principal Stress | Effective principal Stress:

The principal stresses are based on maximum Stress and minimum Stress. So, the range of the Stress is between the maximum and minimum Stress, (stress range is limited and less) and might lead to higher fatigue life. So, it is important to find out the effective principal Stress that gives the maximum value out of the two over the given period of time.

What is Maximum Normal Stress theory?

This states that brittle failure occurs when the maximum principal Stress exceeds the compression or the tensile strength of the material. Suppose that a factor of safety n is considered in the design. The safe design conditions require that.

Maximum principal stress equation

Where σ1, σ2, σ3 are three principal stresses, maximum, minimum, and intermediate, in the three directions, Sut and Suc are the ultimate tensile strength and the ultimate compressive strength, respectively.

To avoid brittle failure, the principal stresses at any point in a structure should lie within the square failure envelope based on the maximum normal stress theory.

Maximum principal stress theory |Maximum principal stress definition

consider two-dimensional stress state and the corresponding principal stresses such as σ1 >σ2 >σ3
Where σ3=0, σ2 may be compressive or tensile depending on the loading conditions where σ2 may be less or greater than σ3.

According to maximum principal stress theory, failure will occur when
σ1 or σ2 =σy or σt
The conditions are represented graphically with coordinates σ1,σ2. If the state of Stress with coordinates (σ1,σ2 ) falls outside of the rectangular region, failure will occur as per the maximum principal stress theory.

Mohr’s circle principal stresses

Explain the Mohr’s circles for the three-dimensional state of stress:

• Consider a plane with a reference point as P. Sigma is represented as normal Stress and tau by the shearing stress on the same plane.
• Take another plane with reference point Q representing sigma and tau as normal stress and shear stress, respectively. Different planes are passing through point p, different values of principal and shear stress.
• For each plane n, a point Q with coordinates as shear stress and principal Stress can be located.
• Determine the normal and shear stresses for point Q in all possible directions of n.
• Obtain three principal stresses as maximum principal Stress, minimum principal Stress, and intermediate principal Stress and represent them in ascending order of the values of the stresses.
• Draw three circles with diameters as the difference between the principal stresses.

• The shaded area region is the Mohr’s circle plane region.
• The circles represent the Mohr’s circles.
• (σ1-σ3) and the associated normal stress is (σ1+σ3)
• There are three normal stresses, so are three shear stresses.
• The principal shear planes are the planes where shear stresses act and principal normal stress acts at a plane where shear stress is ‘0’ and shear stress act at a plane where normal principal stress is zero. The principal shear stress act at 45° to the normal planes.

The shear stresses are denoted by
And the principal stresses are denoted by

Third principal stress

3^rd principal Stress is relative to the maximum compressive stress due to the loading conditions.

3D principal stress examples:

For three-dimensional case, all three planes have zero shear stresses, and these planes are mutually perpendicular, and normal stresses have maximum and minimum stress values and these are the normal stresses that represent the principal maximal and minimal stress.

These principal stresses are denoted by,
σ1,σ2, σ3.
Example:
3D Stress in hub-a a steel shaft is force-fitted into the hub.
3D Stress in machine component.

Principal deviatoric stress:

Principal deviatoric stresses are obtained by subtracting mean Stress from each principal Stress.

Intermediate principal Stress:

The principal Stress, which is neither maximum nor minimum, is called intermediate Stress.

Principal stress angle | Orientation of principal stress: θP

Orientation of the principal Stress is computed by equating shear stress to zero in x-y direction at the principle plane rotated through an angle theta. Solve θ to get θP, the principal stress angle.

Important Frequently Asked Questions (FAQs):

Maximum principal stress theory is applicable for which material?

Answer: Brittle materials.

What are the 3 principal stresses? | What is maximum and minimum principal stress ?

Maximum principal Stress | Major principal stress: Most tensile (σ1)
Minimum principal Stress | Minor Principal Stress: Most compressive (σ3)
Intermediate principal Stress (σ2)

Principal Stress vs normal Stress:

Normal Stress is the force applied to the body per unit area. Principal Stress is the stress applied to the body having zero shear stress principal Stress is in the form of normal Stress giving maximum and minimum stresses on the principal plane.

Principal Stress vs Bending Stress:

Bending Stress is the Stress that occurs in the body due to the application of a large amount of load that causes the object to bend.

Principal Stress vs axial Stress:

Axial Stress and principal stress are the parts of normal stress.

What is the significance of principal Stress?

Principal Stress shows the maximum and minimum normal stress. Maximum normal Stress shows the component’s ability to sustain the maximum amount of force.

What are the principal stresses in a shaft with torque applied ?

The shear stress due to torque has maximum magnitude at the outer fiber. The bending stress is due to horizontal loads (horizontal gear forces, belt, or chain forces) that induce bending stresses which are maximum at the outer fibers.

Why shear stress is zero on principal plane ?

The normal Stress is maximum or minimum, and shear stress is zero.

when shear stress=0,

Important Principal stress problems:

1) A rectangular stress vector having shear stress in XY direction of 60Mpa and normal tensile Stresses of 40Mpa.How to find principal stresses ?

Solution:
Given:
Principal stresses are calculated as,

σ1=100Mpa

σ2=-20Mpa

2)What are the coordinates of the center of Mohr’s circle for an element subjected to two mutually perpendicular stresses, one tensile of magnitude 80MPa and other compressive of magnitude 50MPa?

σx = 80 MPa,
σy = -50 MPa
Co-ordinates of center of Mohr’s circle =[ ½( σx + σy),0]
= [(30/2),0]
= (15,0)

3)A body was subjected to two mutually perpendicular stresses of -4MPa and 20MPa, respectively. Calculate the shear stress on the plane of the shear.

σx+σy /2= -4+20/2 = 8Mpa
Radius= σ1-σ2/2 = 20-(-4)/2 = 12
where σx,σy are principal stresses
at pure shear stress,σn=0
shear stress= squareroot12^2-8^2= 8.94Mpa.

4) Application of principal stress | Find the principal stresses for the following cases.

i)σx=30 Mpa, σy=0, \\tau=15Mpa.

solution:

σ1=36.21Mpa

σ2=-6.21Mpa
ii)σx=0,σy=80MMpa, \\tau=60Mpa.

σ1=97Mpa

σ2=12.92Mpa

iii)\\tau=10Mpa, σx=50Mpa,σy=50Mpa.

σ1=60Mpa

σ2=40Mpa

5) The maximum principal Stress is given 100 Mpa, and the Minimum principal Stress is 50 MPa. Calculate the maximum shear stress and the orientation of the principal plane using Mohr’s circle.

Given:
Maximum principal stress=100Mpa(compressive)
Minimum principal stress=50 Mpa(compressive)
Solution:
Maximum shear stress is the radius of the Mohr’s circle, then we can write as follows.

R=

2θ = 90, from the maximum principal stress direction.
So, the orientation at that point is θ = 45 from the maximum principal stress direction.

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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.

For More related articles click here    

 

         

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)