What is carnot cycle

The Carnot cycle is a thermodynamic cycle that can be invented by French Physicist Sadi Carnot in 1824.

What is the Carnot cycle?

In the Carnot cycle, two reversible isothermal processes and two reversible adiabatic processes are done in a heat engine.

Process of Carnot cycle :

The Carnot cycle consist of four processes following below :

  • A reversible isothermal gas expansion process. From the diagram shown below the ideal gas in the system absorbs some amount of heat from the heat source at a high temperature and then expands thus the work was done on surroundings. 
  • A reversible adiabatic gas expansion process. In this process, the system is thermally insulated. The gas expands continuously and does work on surrounding, which causes the system to cool to a lower temperature. 
  • A reversible isothermal gas compression process. In this process, work has done surrounding gas and cause a loss of heat.
  • A reversible adiabatic gas compression process. In this process, surrounding continue to do work to a gas, which causes the temperature to rise back to high temperature attain in the first process. 
Carnot cycle

The efficiency of the Carnot cycle :

It can be defined as the ratio of the energy output to the energy input. Here energy output is work done and energy input is heat addition.  

From the calculation,

Q1 = Heat addition = R T1 ln v2 / v1

Wnet = R ln v2 / v1 (T1 – T2 )

Ŋcycle = Wnet / Q1  =  T1 – T2 / T1

The large back work is a big drawback of this cycle. 

Assumption of the Carnot cycle :

  • No friction at all between the piston and cylinder and also other moving parts of the engine, thus there is no heat generated and lost due to friction.
  • There is no transfer of heat with the external atmosphere because the engine is completely insulated.
  • There is also no exchange of heat between various parts of the engine.

Rankine cycle vs Carnot cycle

Carnot cycle is two reversible isothermal and two reversible adiabatic process are done in a heat engine. Rankine cycle is a thermodynamic cycle of heat engine that converts heat into mechanical work undergoing the phase change. Let us have a deep insight into the difference between the Rankine cycle and the Carnot cycle. 

Main difference :

The pressure of working fluid is raised from the pressure of condenser to the boiler pressure in Carnot cycle, whereas in the Rankine cycle, the saturated liquid is pumped to the boiler with a pump. 

Difference between Carnot and Rankine cycle :

  • The heat addition process is isothermal in case of Carnot cycle, whereas isobaric in Rankine cycle. 
  • Heat addition and heat rejection are done by keeping the temperature constant in the Carnot cycle while both are done by keeping the pressure constant in the Rankine cycle. 
  • Carnot cycle is a theoretical cycle while the Rankine cycle is a practical cycle. 
  • Rankine cycle like other cycles leads to entropy generation due to the heat transfer process, but Carnot cycle doesn’t.
  • The efficiency of the Rankine cycle is less than a Carnot cycle.
  • Carnot cycle uses air as the working substance while the Rankine cycle uses water as a working substance. 
  • Carnot cycle is an ideal cycle for heat engine while the Rankine cycle is ideal for the vapour power cycle. 

What is rankine cycle

The Rankine cycle is the fundamental operating cycle of all power plants where an operating fluid is continuously evaporated and condensed. 

What is the Rankine cycle?

For each process in the vapour power cycle, it is possible to assume a hypothetical or ideal process which represents the basic intended operation and involves no extraneous effects. 

For the steam boiler, this would be a reversible constant pressure heating process of water to form steam.
For the turbine, the ideal process would be a reversible adiabatic expansion of steam. 
For the condenser, it would be a reversible constant pressure heat rejection as the steam condenses until it becomes saturated liquid. 
For the pump, the ideal process would be the reversible adiabatic compression of this liquid ending at the initial pressure. 

When all these four processes are ideal, the cycle is an ideal cycle, called the Rankine cycle.

Rankine cycle

The efficiency of the Rankine cycle :

Efficiency is the ratio between energy output to energy input. Here work done is energy output and energy input is heat addition. 

Net work is done = Work done in turbine + Work done in pump 

Net heat transfer = Heat produced in boiler + Heat rejected in the condenser 

ŋ = WT - WP / QS 

Difference between brayton and rankine cycle

What is the Brayton cycle?

Brayton cycle is a jet engine, the air is sucked, compressed and then released into the atmosphere, thus making an open cycle.

What is the Rankine cycle?

Rankine cycle is a steam engine, the water is boiled, evaporated, used for work and then condensed for re-use, thus making it a closed cycle. 

Main difference :

The working fluid undergoes a phase change in the Rankine cycle whereas in Brayton cycle there is no phase change the working fluid always remains in the gaseous phase.

Let us have a deep insight into the Brayton cycle vs Rankine cycle. 

Difference :

  • Brayton cycle consists of two reversible isobaric processes and Rankine cycle consist of two reversible adiabatic processes.
  • Both the pump and the steam turbine in the case of the Rankine cycle, and the compressor and the gas turbine in the case of the Brayton cycle operate through the same pressure difference. 
  • The average specific weight of air handled by the compressor is less than the same of gas in the gas turbine in Brayton cycle so the work done by the gas turbine is more than the work input to the compressor while in the case of Rankine cycle, the specific weight of water in the pump is much less than the steam expanding in the steam turbine, therefore, steam power plants are more popular than gas turbine plants for electricity generation.
  • Brayton cycle operates between a higher pressure ratio than the Rankine cycle for the same capacity.

What is thermocouple

A thermocouple circuit made up from joining two wires of two dissimilar metals so due to the different effect net e.m.f is generated in the circuit which depends on the difference in temperature between the hot and the cold junctions.

This e.m.f can be measured by a microvoltmeter to a high degree of accuracy. The choice of selecting metal depends on the temperature range to be investigated, and copper constantan, chrome-alumni and platinum-rhodium are mostly used combinations. 

Advantages of a thermocouple are that it comes to thermal equilibrium with the system, whose temperature is to be measured, quite rapidly, because its mass is small.

Different effect to generate net e.m.f is :
  • Seebeck effect 
  • Peltier effect
  • Thomson effect 

Electrical resistance thermometer

Electrical resistance thermometer

In the resistance thermometer, the change in resistance of a metal wire due to its change in temperature is the thermometric property. The wire, frequently platinum, may be incorporated in a Wheatstone bridge circuit. The platinum resistance thermometer measures temperature to a high degree of accuracy and sensitivity, which makes it suitable as a standard for the calibration of other thermometers.

In a restricted range, the following quadratic equation is often used :

R = R0 ( 1 + At + Bt2 )


R0 = Resistance of platinum wire when it is surrounded by melting ice and A and B are constants.

Gas thermometer

To measure temperature, a reference body is used, and a certain physical characteristic of this body which changes with temperature is selected. 

The change in the selected characteristics is an indication of the change in temperature and selected characteristics are the thermometric property and reference body which is used is called thermometer.

All thermometers are working examples of the zeroth law of thermodynamics

Significance :

They used to calibrate other thermometers. 

Working principle and construction :

Constant pressure gas thermometer :

A small amount of gas is enclosed in bulb B which is in communication via the capillary tube C with one limb of the mercury manometer M. The other limb of the mercury manometer is open to the atmosphere and can be moved vertically to adjust the mercury levels so that the mercury just touches lip L of the capillary. The pressure in the bulb is used as a thermometric property and it is given by :

p = p0 + ρM Zg
p0 = atmospheric pressure 
ρM = density of mercury

Gas thermometer

When the bulb is brought in contact with the system whose temperature is to be measured, the bulb, in course of time, comes in thermal equilibrium with the system.

The gas in the bulb expands, on being heated, pushing the mercury downward. The flexible limb of the manometer is then adjusted so that the mercury again touches the lip L. The difference in mercury level Z is recorded and the pressure p of the gas in the bulb is estimated. Since the volume of the trapped gas is constant, from the ideal gas equation, 

ΔT = V / R * Δp

The temperature increase is proportional to the pressure increase. 

Constant volume gas thermometer :

In a constant-pressure gas thermometer, the mercury levels have to be adjusted to keep Z constant, and the volume of gas V, which could vary with the temperature of the system, becomes the thermometric property. 
ΔT = V / R * ΔV

The temperature increase is proportional to the observed volume increase.

Constant volume gas thermometer is mostly use, since it is simpler in construction and easier to operate. 

Ideal gas

From the experimental observations p-v-T behaviour of the gases given by,

pṽ = ṜT

Where Ṝ is the universal gas constant value of  Ṝ is 8.3141 J / mol K
ṽ is the molar specific volume m3/gmol

Dividing upper equation by the molecular weight µ.

pv = RT

Where v is specific weight m3/gmol
R is the characteristic gas constant

We also get this the equation in terms of total volume V of gas, 

PV = nṜT
PV = mRT
Where n is the number of moles and m is the mass of the gas. An equation can be written for two states of the gas is 

P1V/ T1 = P2V2 / T2

All equation is called the ideal gas equation of state. At very low pressure or density, all gases and vapours approach ideal gas behaviour.

Celsius temperature scale

The Celsius temperature scale employs a degree of the same magnitude as that of the ideal gas scale, but its zero points are shifted, so that the Celsius temperature of the triple point of water 0.01 degree Celsius or 0.010C. 

If t denotes the Celsius temperature, then 

t = T - 273.150 

Thus the Celsius temperature ts at which steam condenses at 1-atmosphere pressure 

ts = Ts - 273.150 

    = 373.15 - 273.15 = 100.00 0C

Similar measurements for ice points show this temperature on the Celsius scale to be 
0.00 0C.

Measurement of temperature

The temperature of a system determines that the system is in thermal equilibrium with other system or not? 

If a body is at 85C, it will be 85C, whether measured by mercury in glass thermometer, resistance thermometer or constant volume gas thermometer

If X is the thermometric property, let us arbitrarily choose for the temperature common to the thermometer and to all systems in thermal equilibrium with it the following linear function of X :

Θ(X)  = aX


a = arbitrary constant 

Two temperatures on the linear X scale are to each other as the ratio of the corresponding X.

Zeroth law of thermodynamics

Zeroth law of thermodynamics is the basis of temperature measurement. The property which distinguishes thermodynamics from other sciences is temperature. 

One might say that temperature bears as important relation to thermodynamics as the force does to statics or velocity does to dynamics. 

When two bodies maintain at different temperatures are brought into contact, after some time they attain a common temperature and are then said to exist in thermal equilibrium.

Zeroth law of thermodynamics :

When a body A is in thermal equilibrium with a body B, and also separately with a body C, then B and C will be in thermal equilibrium with each other. 

Notes :

In order to obtain a quantitative measure of temperature, a reference body is used, and a certain physical characteristic of this body which changes with temperature is selected. 
The change in the selected characteristics may be taken as an indication of the change in temperature. The selected characteristics are called thermometric property, and the reference body which is used in the determination of temperature is called thermometer. 

There are five different kinds of thermometer used are following below :
  1. Constant volume gas thermometer
  2. Constant pressure gas thermometer 
  3. Electrical resistance thermometer
  4. Thermocouple 
  5. Mercury in glass thermometer 

Specific heat and latent heat

What is the specific heat?

Specific heat is the amount of heat required to raise a unit mass of the substance through a unit rises in temperature. 

The symbol c will be used for specific heat. 

C = Q / m * Δt J / kg k 

Q = The amount of heat transfer ( J )
m = Mass of the substance ( kg )
Δt = The rise in temperature ( K ) 

Since heat is not a property, so the specific heat is qualified with the process through which exchange of heat is made. 

The product of mass and specific heat ( mc ) is called the heat capacity of the substance. 

For gases, 
If the process is at constant pressure ( cp ).
If the process is at constant volume ( cv ).

For solids and liquids, 

The specific heat does not depend on the process. 

What is latent heat?

The amount of heat required to cause a phase change in a unit mass of a substance at constant pressure and temperature.

There are three phases in which matter can exist: solid, liquid, and vapour or gas.

The latent heat of fusion is the amount of heat transferred to melt a unit mass of solid into a liquid. 


to freeze unit mass of liquid to solid. 

The latent heat of vaporization is the quantity of heat required to vaporize unit mass of liquid into vapour. 


to condense unit mass of vapour into liquid. 

The latent heat of sublimation is the amount of heat transferred to convert unit mass of solid to vapour. 

Welding distortion

One of the major problems found with weldments is called distortion. Distortion is caused mainly because of the shrinkage that takes place in weldments. The shrinkage taking place is a weldment depends upon the geometry and type of the weld. 

There are three types of distortions possible in welding :

  • Transverse shrinkage occurring perpendicular to the weld line.
  • Longitudinal shrinkage occurring parallel to the weld line, which is very small of the order of about 0.1 % of the weld length and hence can be neglected.
  • Angular change as a rotation about the weld line.
Now we can check the methods for reducing all three types of shrinkage :

The methods for reducing the transverse shrinkage are :
  • Decrease the total weight of the weld metal. 
  • Increase the metal deposited in the first pass.
Some other factors which influence the transverse shrinkage are :
  • Root opening.
  • Joint design.
  • Electrode diameter.
  • Degree of constraint.
The methods for reducing the angular shrinkage are :
  • Fillet welds in a structure are affected by the way the structure is designed and the type of restraint provided.
Control of distortion :

As we saw above, distortions are inevitable in a welding operation. It is, therefore, necessary to find ways by which these can be minimized to prepare to satisfactory weldment.
  • One of the important ways to control the distortions is a good design of the product with a minimum number of joints.
  • Proper design of the joint helps in reducing the magnitude of the problem.
In spite of this, there still will be distortion and therefore during the assembly process care should be taken to see that the distortions are controlled. 

There are two possibilities one is to present the members to compensate for this distortion another one is to assemble parts correctly and then apply a proper restraint to minimize the distortion during the welding process. 

The most generally preferred method in the industry is using restraint. There are many ways used for restraining such as clamps, fixtures and even tack welds. Through this method reduces the distortion, it causes high residual stresses, which may lead to cracking. Hence it is necessary to carefully apply restraint without causing too high a magnitude of harmful residual stresses. 

Another method available for reducing the distortions is the preheating of the members of the weldment such that the heat of welding would be properly balanced. 

Applications of friction welding

The quality of weld obtained is very high so that the friction welding has been widely accepted in the aerospace industry as well as the automobile industry for the welding of critical parts. 

It is frequently being used instead of upset welding applications where one of the components to be joined has axial symmetry.

Applications of friction welding :

  • The main applications of this process are welding of studs to plates of any thickness. 
  • For welding of tubes and shafts. 
  • Gears, axle tube, valves, driveline all these components of automobile industries are made by this process. 
  • It is used to replace forging or casting assembly.
  • Hydraulic piston rod, truck roller bushes are also joined by this process.
  • It is also used in electrical industries for welding copper and aluminium equipment. 
  • Gear levers, drill bits connecting rod also a made by this process. 
  • Almost any metal that can be hot forged and is unsuitable for a dry bearing application can be friction welded. 
  • Production of marine engine valves, the valves so produced are as good as or superior to those produced by forging.

Friction welding

It is a solid-state welding process that generates heat through mechanical friction between workpieces in relative motion to one another with the use of lateral force and fuses the materials. 
Friction welding is not a fusion welding process because no melting occurs. 

Working principle :

In this process of welding, the heat required for welding is obtained by the friction between the ends of the two parts to be joined. One of the parts to be joined is rotated at a high speed around 3000 RPM and the other part is axially aligned with the second one and pressed tightly against it.

Also, the friction between the two parts raises the temperature of both the ends. After that, the rotation of the part is stopped abruptly and the pressure on the fixed part is increased so that the joining takes place. 

Friction welding

Machine set up :

The machine for friction welding is similar to a centre lathe. Through a centre lathe could be used for smaller sized jobs, the bigger ones required a special welding machine because in a lathe machine power available would not be sufficient. The power requirements of friction welding may be between 25 kVA to 175 kVA, which is far beyond that of the many general-purpose centre lathes. 

Major parameter :

The major parameters in friction welding are the rotational speed and the axial pressure applied. The axial pressure applied depends on the strength and hardness of the metals being joined. 

The pressure may range from 40 Mpa for low carbon steels to as high as 450 Mpa for alloy steels.

The rotational speed may also change the requirement of the pressure. It may be of the order of 1500 to 3000 RPM.  

The other variable that needs to be closely controlled is the time of contact between the two parts. The total welding time that is taken in the friction welding is between 2 to 30 seconds. 

Advantages of friction welding :
  • The major advantage of friction welding is the ease with which the joining can take place. 
  • Edge cleaning is not a problem since the oxides and contaminants present would easily be removed during the initial rubbing. 
  • The heat generated is small and well below the melting temperature, there will be no distortion and warping. 
  • The quality of the weld achieved is very high and it is economical in operation. 
  • No skilled operator required since it completely automatic in operation. 
Because of the above advantages, the quality of weld obtained is very high so that the friction welding has been widely accepted in the aerospace industry as well as the automobile industry for the welding of critical parts. 

Disadvantages of friction welding :
  • This welding process mostly used only for round bars of some cross-section.
  • Non-forgeable materiel can not be weld.
  • Preparation of workpiece is more critical.
  • High machine setup cost. 
  • Joint design is limited. 
  • It can only be used for smaller parts of machines, big parts are not compatible with it. 

Classification of gears

Gears can be classified according to the relative positions of their shaft axes as follows :

Parallel shafts :

In the parallel shaft, the manner of contact and uniform rotary motion between two parallel shafts is equivalent to the rolling of two cylinders. 
The following are the main types of gears to join parallel shafts :

  • Spur gears :
In this type of gear, the teeth are straight and parallel to the axes and thus are not subjected to the axial trust due to load. 

  • Spur rack and pinion :
Spur rack is a special case of a spur gear where it is made of infinite diameter so that a pitch surface is a plane. 

The spur rack and pinion combination convert rotary motion into translatory motion or vice-versa. 

This type of gear usually used in a lathe in which rack transmits motion to the saddle.

  • Helical gears or Helical spur gears :
In this type of gear, teeth are curved, each being helical in shape. Two mating gears have the same helix angle but have teeth of opposite hands. 

In the helical gear, at the beginning of engagement contact occurs only at the point of the leading edge of the curved teeth. As the gear rotates, the contact extends along with a diagonal line across the teeth. Thus, the load application is gradual which results in low impact stresses and it can be used for higher velocities than the spur gears. 

  • Double helical gears or Herringbone gears :
A double-helical gear is equivalent to a pair of helical gears secured together, one having a right-hand helix an the other a left-hand helix. 

Axial thrust which occurs in case of single helical gear is eliminated in double helical gears. 

If the left and the right inclinations of a double helical gear meet at a common apex and there is no groove in between, the gear is known as herringbone gear. 

Intersecting shafts :

Kinematically, the motion between two intersecting shafts is equivalent to the rolling of two cones, assuming no slipping. This type of gear is known as bevel gears.

  • Straight bevel gears :
The teeth are straight, radial to the point of intersection of the shaft axes vary in cross-section throughout their length. 

Usually, they are used to connect shafts at right angles which runs at low speeds. Gears of the same size and connecting two shafts at right angles to each other are known as mitre gears. 

  • Spiral bevel gears :
When the teeth of bevel gears are inclined at an angle to the face of the bevel, they are known as spiral bevels or helical bevels. 

They are smoother in action and quieter than straight tooth bevels as there is gradual load application and low impact stresses. 

  • Zerol bevel gears :
Spiral bevel gears with curved teeth but with a zero degree spiral angle are known as zerol bevel gears. 

They are quieter in action than the straight bevel type as the teeth are curved. 

Skew shafts :

In case of parallel and intersecting shafts, a uniform rotary motion is possible by pure rolling contact. But in case of skew shafts, this is not possible.

  • Crossed helical gears :
The use of crossed helical or spiral gears is limited to light loads. By a suitable choice of helix angle for the mating gears, the two shafts can be set at any angle. 

  • Worm gears :
Worm gear is special case of spiral gear in which the larger wheel, usually has a hollow or concave shape such that a portion of the pitch diameter of the other gear is enveloped on it. 

The smaller of the two wheels is called the worm which also has a larger spiral angle. 

  • Hypoid gears :
Hypoid gears are approximations of hyerboloids though they look like spiral gears. 

The hypoid pinion is larger and stronger than a spiral bevel pinion. 

A hypoid pair has a quite and smooth action. 

Hypoid gear

Involute tooth profile

What is involute?

The locus of the point on a straight line which rolls without slipping on the circumference of a circle is called the involute.

In other words, it is the path traced out by the end of a piece of the taut cord being unwound from the circumference of a circle. The circle on which the straight line rolls or from which the cord is unwound is known as the base circle.

Formation of involute tooth :

Involute tooth profile

From the above figure, an involute generated by a line rolling over the circumference of a base circle with centre at O. At the start, the tracing point is at A. As, the line rolls on the circumference of the circle, the path ABC traced out by the point A is the involute. 

D can be regarded as the instantaneous centre of rotation of B, the motion of B is perpendicular to BD. Since BD is tangent to the base circle, the normal to the involute is a tangent to the base circle.  

A short length EF of the involute drawn from A can be utilized to make the profile of an involute tooth. The other side HJ of the tooth has been taken from the involute drawn from G in the reverse direction. The profile of an involute tooth is made up of a single curve, and teeth, usually, are termed as single curve teeth. 

Notes :

Because of ease of standardization and manufacture, and low cost of production, the use of involute teeth has become universal by entirely superseding the cycloidal shape. Only one cutter or tool is necessary to manufacture a complete set of interchangeable gears. The cutter is in the form of a rack as all gears will gear with their corresponding rack. The cutters of this form can be made to a higher degree of accuracy as the teeth of an involute rack are straight. 

Key points :
  • Points of contact lie on the line of action which is the common tangent to the two base circles. 
  • The contact is made when the tip of a tooth of the driven wheel touches the flank of a tooth of the driving wheel and the contact is broken when the tip of the driving wheel touches the flank of the driven wheel. 
  • If the direction of angular movement of the wheels is reversed, the points of contact will lie on the other common tangent to the base circles. 
  • Initial contact occurs where the addendum circle of the driven wheel intersects the line of action. 
  • Final contact occurs at a point where the addendum circle of the driver intersects the line of action. 
  • For a pair of involute gears, the velocity ratio is inversely proportional to the pitch circle diameters as well as base circle diameters. 

Cycloidal tooth profile

What is cycloid?

The cycloid is defined as the locus of a point on the circumference of a circle that rolls without slipping on a fixed straight line.
In this type, the faces of the teeth are epicycloids and the flanks are the hypocycloids. 

Now one question arises in your mind that what is epicycloids and hypocycloids? Let we check 

What is epicycloid?
An epicycloid is the locus of a point on the circumference of a circle that rolls without slipping on the circumference of another circle. 

What is hypocycloid?
A hypocycloid is the locus of a point on the circumference of a circle that rolls without slipping inside the circumference of another circle. 

Formation of cycloidal tooth :

cycloidal tooth

A circle O1 rolls inside another circle APB is called pitch circle. At the start, the point of contact of the two circles is at A. As the circle O1 rolls inside the pitch circle, the locus of the point A on the circle O1 traces the path ALP which is called hypocycloid. A small portion of this curve near the pith circle is used for the flank of the tooth.  

The line joining the generating point A with the point of contact of the two circles is normal to the hypocycloid when the circle O1 touches the pitch circle at D, the point A is at C and CD is normal to the hypocycloid ALP. 

In the same way, if the circle O2 rolls outside the pitch circle, starting from P, an epicycloid PFB is obtained.

A small portion of the curve near the pitch circle is used for the face of the tooth also. 

Some key points :
  • The path of approach is equal to the arc of approach. 
  • The path of contact is equal to the arc of contact. 
  • In case of cycloidal teeth, the pressure angle varies from the maximum at the beginning of the engagement to zero when the point of contact coincides with pitch point and then again increased to a maximum in the reverse direction. 
  • Since cycloidal teeth are made up of the two curves, it is very difficult to produce accurate profiles. This has rendered this system obsolete. 

Forms of tooth

Types of tooth :

Two curves of any shape that fulfil the law of gearing can be used as the profiles of teeth. 

In other words, an arbitrary shape of one of the mating teeth can be taken and applying the law of gearing the shape of the other can be determined. Such gear is said to have conjugate teeth. 

However, it will be very difficult to manufacture such gears and the cost will be high. It will be very difficult to replace them with the available gears. Thus, arises the need to standardize gear tooth. 

Common forms of teeth that also satisfy the law of gearing are the following :

Contact ratio of gears

Arc of contact :

The distance travelled by a point on either pitch circle of the two wheels during the period of contact of a pair of teeth is called the arc of contact. 

It is the length of the pitch circle traversed by a point on it during the mating of a pair of teeth. 

Thus, all teeth laying in between the arc of contact will be meshing with the teeth on the other wheel. 

Number of teeth in contact n = Arc of contact / Circular pitch 

As we mentioned above the ratio of the arc of contact to the circular pitch is also the contact ratio, the number of teeth is also expressed in terms of contact ratio. 

For continuous transmission of motion, at least one tooth of one wheel must be in contact with another tooth of the second wheel. Therefore, n must be greater than unity.

If n lies between 1 and 2, the number of teeth in contact at any time will not be less than one and never more than two. 

If n is between 2 and 3, it is never less than pairs of teeth and not more than three pairs, and so on. 

If n is 1.6, one pair of teeth are always in contact whereas two pairs of teeth are in contact for 60% of the time.