Cutting Force Analysis - Merchant's Circle

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Forces experienced by a tool during cutting is detrimental in design of mechanical structure of cutting machine, predicting power consumption, determining the tool life and increasing the productivity. In this video lecture we will analyse how to predict cutting forces using Merchant's circle analysis.

Summary of above lecture is described in following sections

Why Cutting Force Analysis is Important ?

Most of the time cutting force acting on a tool is measured experimentally. But it is also important to predict quantity of cutting force and how different cutting parameters are affecting cutting force even before setting up the machining operation due to following reasons.

  1. In order to design of mechanical structure of cutting machine which will withstand cutting force and thrust force effectively.
  2. To determine power consumption during machining process. This will help in selecting suitable motor drive.
  3. To predict tool life.
  4. To increase productivity

Cutting Terminology

The following figure describes important terminologies used in cutting force analysis.

Fig.1 A tool under orthogonal cutting operation
The figure shows a case where tool velocity (V) and cutting edge are perpendicular to each other,this is known as orthogonal cutting. Here we are analyzing cutting force for an orthogonal cutting operation. Cutting force (Fc) is the force parallel to cutting tool velocity. Rake angle of cutting tool is represented by 'alpha'

Forces Acting on the chip

If you make a free body analysis of the chip, forces acting on the chip would be as follows.

Fig.2 Forces acting on the chip on tool side and shear plane side
At cutting tool side due to motion of chip against tool there will be a frictional force and a normal force to support that. At material side thickness of the metal increases while it flows from uncut to cut portion. This thickness increase is due to inter planar slip between different metal layers. There should be a shear force (Fs) to support this phenomenon. According to shear plane theory this metal layer slip happens at single plane called shear plane. So shear force acts on shear plane. Angle of shear plane can approximately determined using shear plane theory analysis.It is as follows
Shear force on shear plane can be determined using shear strain rate and properties of material. A normal force (Fn) is also present perpendicular to shear plane. The resultant force (R) at cutting tool side and metal side should balance each other in order to make the chip in equilibrium. Direction of resultant force, R is determined as shown in Figure 2.

Merchant's Circle Analysis

Steps involved in Merchant's circle analysis is as follows. Since we know angle of resultant force at tool side, draw a line parallel to this. On one end of this line draw shear force(Fs), magnitude and direction of which is known. Now draw a line perpendicular to shear force line, it will meet resultant force line at one point. You can draw a circle assuming the intersected line as diameter of the circle, this is known as Merchant's circle. It is shown in figure below.

Fig.3 Construction of Merchant's circle
In order to determine cutting force (Fc) one can draw a line parallel to tool motion in Merchant's circle, starting from end of diameter. The chord so obtained will give magnitude of cutting force. If you draw a line perpendicular to Fc that will give thrust force acting on the tool (Ft). So resultant force, R at tool side also can be considered as a summation of cutting force and and thrust force. The diagram so obtained is shown in following figure.
Fig.4 Determination of cutting force and thrust force using Merchant's circle diagram

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Kalina Cycle Power Plant

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Kalina cycle based power plants are latest development in power plant technology. Owing to its unique features which try to resemble Carnot cycle, Kalina cycles work on thermal efficiency range of 40-60 %. In this video lecture you will come to know what is Kalina cycle and why it is a promising technology.

Summary of above lecture along with latest developments Kalina cycle power plants are described below.

Working fluid - Mixture of 2 fluids

What makes Kalina cycle different from conventional Rankine cycle of power production is its choice of working fluid. Kalina cycle uses mixture of 2 fluids as working fluid, most commonly used is ammonia and water mixture. To get answer why Kalina uses a mixture as its working fluid, have a look at T-s diagrams of ordinary Rankine cycle and Kalina cycle.

Fig.1 Comparison of Rankine and Kalina cycles
The major difference of Kalina cycle from Rankine cycle is that in Kalina heat addition and heat rejection happen at varying temperature even during phase change, since the fluid is a mixture. But in Rankine heat addition and heat rejection happen at uniform temperature during phase change. This is the one thing which makes all the difference in performance of Kalina cycle.

Comparison with Carnot Engine - Reason for high efficiency

In a Carnot engine heat addition and rejection happen at uniform temperature.

Fig.2 In a Carnot engine heat addition and rejection happen at uniform temperature
Efficiency of such an engine can easily be proved as
So it is clear that if heat rejection temperature (Tc) decreases or heat absorption temperature (Tb) increases thermal efficiency of Carnot engine will increase.Same analysis can be done for Rankine and Kalina cycles, using average temperature of heat addition and rejection as reference temperatures. This is marked in dotted lines in following figures.
Fig.3 Average heat addition and rejection temperatures of Kalina cycle is much wider than a Rankine cycle
So it is clear from the diagrams that Kalina cycle has got lower average heat rejection temperature (Tc) and higher average heat addition temperature (Tb) compared to Rankine cycle. It will obviously lead to high thermal efficiency. This forms theoretical background of Kalina cycle, but in order to implement it we have to overcome some practical hurdles.

Difficulty at Condenser - Use of Separator

Kalina cycle uses high concentration ammonia mixture (around 70% ammonia) at steam turbine part, but such a mixture has got very low condensing temperature.

Fig.4 Phase diagram of Ammonia-Water mixture
Means you have to supply a very low temperature cooling water at condenser for this purpose. Production of such low temperature cooling water is not economical. You can observe from Fig.4 that condensing temperature of ammonia-water mixture increases drastically with decrease in ammonia concentration. So in a Kalina cycle power plant, we will decrease ammonia concentration at condenser side. An equipment called separator will produce 2 streams of fluid from condenser outlet, one with high concentration and other with low concentration (30% ammonia). Low concentration ammonia mixture will get mixed with exist fluid at turbine and will produce a medium concentration (40% ammonia) ammonia mixture. This mixture will have fairly high condensing temperature and can be condensed with supply of ordinary cooling water. This is shown in following figure.
Fig.5 Use of recuperator in producing low concentration ammonia mixture at condenser
Concentration of fluid is brought back to original state by mixing high concentration ammonia stream from separator with fluid exit at condenser.

Use of Recuperator

It is clear from T-s diagram of Kalina cycle that temperature at exit of steam turbine (point 4) is greater than temperature at inlet of boiler (point 2). So there exists a chance of heating up boiler liquid by virtue of this high temperature steam turbine output. This is accomplished with help of a heat exchanger called recuperator. This is shown in following figure.

Fig.6 Increase in further thermal efficiency with help of a recuperator
Thus with use of recuperator one need not supply same amount of heat at boiler side as supplied in previous case, this will further increase efficiency of Kalina cycle power plant. But this opportunity of heat transfer is not there in Rankine cycle based power plant. You can notice that in Rankine cycle temperature at point 4 is always less than temperature at point 2, thus there is no chance of heat transfer from steam turbine outlet to boiler inlet.

Advancements in Kalina Cycle Power Plants

Instead of ammonia-water mixture industries have started implementing organic mixture based Kalina cycles in order to harness maximum from given condition. Some examples are mixture of R22 & R114 and mixture of Hexamethyldisiloxane & Decamethyltetrasiloxane.

Thanks to its unique feature of varying thermo-physical properties by varying mixture concentration at different parts of cycle, Kalina cycle power plants are widely used in Geothermal stations and waste heat recovery units. They can easily match to any source (heat addition) and sink (heat rejection) condition by varying mixture concentration in the cycle.

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How does a Thermal Power Plant Work ?

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The purpose of this video is to give you a conceptual introduction on working of thermal power plants. Here working of a thermal power plant is explained in a logical step by step manner.

A detailed analysis of thermal power plants based on it's thermodynamic cycle is given below.

Thermal Power Plants

Thermal power plants use water as working fluid. Nuclear and coal based power plants fall under this category. The way energy from fuel gets transformed into electricity forms the working of a power plant. In a thermal power plant a steam turbine is rotated with help of high pressure and high temperature steam and this rotation is transferred to a generator to produce electricity.

Fig.1 Power is produced in thermal power plants by rotating steam turbine

Energy absorption from steam

When turbine blades get rotated by high pressure high temperature steam, the steam loses its energy. This in turn will result in a low pressure and low temperature steam at the outlet of the turbine. Here steam is expanded till saturation point is reached. Since there is no heat addition or removal from the steam, ideally entropy of the steam remains same. This change is depicted in the following p-v and T-s diagrams. If we can bring this low pressure, low temperature steam back to its original state, then we can produce electricity continuously.

Fig.2 Pressure and temperature drop of steam when turbine absorbs energy from it

Use of Condenser

Compressing a fluid which is in gaseous state requires a huge amount of energy,so before compressing the fluid it should be converted into liquid state. A condenser is used for this purpose, which rejects heat to the surrounding and converts steam into liquid. Ideally there will not be any pressure change during this heat rejection process, since the fluid is free to expand in a condenser. Changes in fluid are shown in the p-v and T-s diagram below.

Fig.3 Use of condenser in order to transform vapor into liquid state


At exit of the condenser fluid is in liquid state, so we can use a pump to raise the pressure.During this process the volume and temperature (2-3 deg.C rise)of fluid hardly changes, since it is in liquid state. Now the fluid has regained its original pressure.

Fig.4 Compressor pumps the fluid to its original pressure

Heat Addition in Boiler & Rankine Cycle

Here external heat is added to the fluid in order to bring fluid back to its original temperature. This heat is added through a heat exchanger called a boiler. Here the pressure of the fluid remains the same, since it is free to expand in heat exchanger tubes. Temperature rises and liquid gets transformed to vapor and regains its original temperature. This completes the thermodynamic cycle of a thermal power plant, called Rankine Cycle. This cycle can be repeated and continuous power production is possible.

Fig.5 Heat addition at boiler brings the fluid to its original temperature

Condenser Heat Rejection - Cooling Tower

In order to reject heat from the condenser a colder liquid should make contact with it. In a thermal power plant continuous supply of cold liquid is produced with the help of a cooling tower. Cold fluid from the cooling tower absorbs heat from a condenser and gets heated, this heat is rejected to the atmosphere via natural convection with the help of a cooling tower.

Boiler furnace for Heat Addition

Heat is added to the boiler with help of a boiler furnace. Here fuel reacts with air and produces heat. In a thermal power plant, the fuel can be either coal or nuclear. When coal is used as a fuel it produces a lot of pollutants which have to be removed before ejecting to the surroundings. This is done using a series of steps, the most important of them is an electro static precipitator (ESP) which removes ash particles from the exhaust. Now much cleaner exhaust is ejected into the atmosphere via a stack.

Fig.6 Main accessories of Rankine cycle - Cooling tower, Boiler furnace, ESP & Chimney

Optimizing a Thermal plant performance

There are various flow parameters which have to be fine-tuned in order to get optimum performance from a thermal power plant.Lowering the condenser temperature or raising the average boiler temperature will result in a high efficiency power plant cycle according to the 2nd law of thermodynamics (Carnot efficiency),most of the performance improving technologies are working on this idea. Some latest trends are listed below.

  1. Expanding Turbine After Saturation
  2. Expanding the steam in the turbine even after reaching the saturation point may be a dangerous affair. As the steam goes below saturation, wetness of the steam increases. These condensed water droplets collide with the turbine blades rotating at a high speed, thus it can cause extreme tip erosion to the blades. Turbine blade tip erosion is shown in figure below. But as you expand more you will be able to absorb more energy from the steam, thus increasing power plant efficiency. Up to 15% wetness level is considered to be safe for steam turbine operation. So most of the steam turbine will expand up to this point in order to extract maximum energy from the fluid. This is shown in figure below.

    Fig.7 Expanding turbine below saturation point in order to gain maximum power from steam

  3. Raising average boiler temperature
  4. If you can increase the average heat addition temperature of the boiler, that will result in a power plant with higher efficiency. One way to do this is to increase the compressor pressure. This will shift the saturation point of the fluid to a higher level, thus providing higher average temperature of heat addition. This is shown in the figure below. The blue line represents change in the cycle after raising the compressor pressure.

    Fig.8 Raising compressor pressure in order achieve higher average boiler temperature

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Principal stress, Principal plane & Mohr's circle analysis

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Concepts of principal stress and plane form backbone of material stress analysis. Purpose of this video lecture is to give you a good introduction to concept of Principal stress, Principal plane and Mohr’s circle analysis.
Summary of the video lecture is given below

Summary of Lecture

  • Engineers most often wants to determine maximum normal stress induced at a given point for their design purpose. But there can be infinite number of planes passing through a point, and normal stress on each plane will be different from one another.
  • There will be one plane on which normal stress value is maximum, this plane is known as Principal plane ( more precisely maximum principal plane) and normal stress on this plane is known as principal stress (more precisely maximum principal stress).
  • Similarly there will be one more plane on which normal stress value is minimum, this is also a principal plane (minimum principal plane) and normal stress on this plane is known as Principal stress (minimum principal stress).
  • 2 Dimensional Stress Analysis – Stress acting on a 2D element is shown in figure below
  • Fig.1 Stress boundary conditions on a 2 dimensional element
  • Mohr’s circle method is the most easy and convenient way to do stress analysis
  • The procedure to draw Mohr’s circle for above case is explained below
  • Step1 – Draw normal and shear axes with positive axes as shown
    Fig.2 Normal and shear axes of a Mohr circle
    Step2 – Mark normal stress values with sign convention, tensile stress is positive and compression stress is negative
    Fig.3 Marking normal stress values on normal axis
    Step3 - Draw shear stress values starting from already marked normal stress points.
    Fig.4 Drawing shear stress values
    Step4 - Connect end of shear stress lines
    Fig.4 Connecting end of shear stress lines
    Step5 - Draw Mohr’s circle assuming the connection line as diameter of the circle
    Fig.5 Mohr circle construction
    Step6 – Stress Analysis on Mohr circle - To get normal and shear stress values at any plane theta, take angle 2theta in Mohr circle starting from diagonal of the circle and locate a peripheral point as as shown. Shear stress value will be Y axis value and normal stress value will be X axis value.
    Fig.6 Determination on normal and shear stress using Mohr cirlce
  • 3 Dimensional Stress Analysis – Stress boundary condition of a 3 dimensional case is shown in left side of  Fig.7. There will be 3 normal stress values induced in a 3 dimensional case, this is shown in right size of the figure.
  • Fig.7 Stress boundary conditions in a 3 dimensional  body and normal stress values induced in it
  • There is no graphical method for 3 Dimensional stress analysis, instead we have to use analytical method for this. Values of Principal stress in a 3 dimensional systems are given by solution of following equation.
Where values of stress invariants I1,I2 and I3 are given by

Application of Principal Stresses

Values of principal stresses at a given point is vital design information. Material failure theories extensively use this data to predict whether the design will withstand given load at a specified location.

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