Theory for Production

T l W T l Lif &T l W T l Lif & M hi biliM hi biliToolWear,ToolLife&ToolWear,ToolLife&MachinabilityMachinability

BySKMondalBySKMondal

ToolFailureTool failure is two types1. Slowdeath: The gradual or progressive wearingg p g gaway of rake face (crater wear) or flank (flank wear) ofthe cutting tool or both.g2.Suddendeath:FailuresleadingtoprematureendofthetoolThe suddendeath type of tool failure is difficult topredict. Tool failure mechanisms include plasticpredict. Tool failure mechanisms include plasticdeformation, brittle fracture, fatigue fracture or edgechipping. However it is difficult to predict which ofpp g pthese processes will dominate and when tool failurewill occur.

ToolWear( ) Fl kW(a) FlankWear(b) CraterWear( )(c) Chipping off of the cutting edge

lToolWear FlankWear:(Wearland)ReasonAb i b h d i l d i l i i h kAbrasion by hard particles and inclusions in the workpiece.Shearing off the micro welds between tool and workmaterialmaterial.Abrasion by fragments of builtupedge ploughing

i h l f f h lagainst the clearance face of the tool.At low speed flank wear predominates.p pIf MRR increased flank wear increased.

FlankWear:(Wearland)EffectFl k di l ff h di iFlank wear directly affect the component dimensionsproduced.Flank wear is usually the most common determinant oftool lifetool life.

FlankWear:(Wearland)StagesFl k W i h f iFlankWear occurs in three stages of varying wear rates

FlankWear:(Wearland)PrimarywearTh i h h h i d i i klThe region where the sharp cutting edge is quicklybroken down and a finite wear land is established.

Secondary wearyThe region where the wear progresses at a uniform rate.

FlankWear:(Wearland)Tertiary wearTh i h d llThe region where wear progresses at a graduallyincreasing rate.In the tertiary region the wear of the cutting tool hasbecome sensitive to increased tool temperature due tobecome sensitive to increased tool temperature due tohigh wear land.R i di i d d b f h hiRegrinding is recommended before they enter thisregion.

For IES, GATE, PSUs Page 2 of 49 Bhopal -2014

ToollifecriteriaISO(A i id h ffl k (VB)i h (Acertainwidthofflankwear(VB)isthemostcommoncriterion)Uniformwear:0.3mmaveragedoverallpastLocalizedwear:0 5mmonanyindividualpastLocalizedwear:0.5mmonanyindividualpast

CraterwearMore common in ductile materials which produce

hcontinuous chip.

C h k fCrater wear occurs on the rake face.

At hi h d t d i tAt very high speed crater wear predominates

For crater wear temperature is main culprit and toolFor crater wear temperature is main culprit and tool

defuse into the chip material & tool temperature isdefuse into the chip material & tool temperature is

maximum at some distance from the tool tip.

CraterwearContd..Crater depth exhibits linear increase with time.It increases with MRRIt increases with MRR.

Crater wear has little or no influence on cutting forcesCrater wear has little or no influence on cutting forces,work piece tolerance or surface finish.

WearMechanism1. Abrasionwear

2. Adhesionwear

3 Diffusionwear3. Diffusionwear

4. Chemicaloroxidationwear

Whychippingofforfinecracksd l d h ddevelopedatthecuttingedge

Tool material is too brittle

Weak design of tool, such as high positive rake angle

As a result of crack that is already in the toolAs a result of crack that is already in the tool

E i i h k l di f h lExcessive static or shock loading of the tool.

NotchWearNotch wear on the trailing edge is to a great extent an

id ti h i i h th ttioxidation wear mechanism occurring where the cutting

edge leaves the machined workpiece material in the feededge leaves the machined workpiece material in the feed

direction.

But abrasion and adhesion wear in a combined effect can

contribute to the formation of one or several notches.

Listtheimportantpropertiesofcuttingtoolt i l d l i h h i i t tmaterialsandexplainwhyeachisimportant.

Hardness at high temperatures this provides longerHardness at high temperatures this provides longerlife of the cutting tool and allows higher cutting speeds.Toughness to provide the structural strength neededto resist impacts and cutting forcesto resist impacts and cutting forcesWear resistance to prolong usage before replacementdoesnt chemically react another wear factorFormable/manufacturable can be manufactured in aFormable/manufacturable can be manufactured in auseful geometry

ToolLifeCriteriaTool life criteria can be defined as a predeterminednumerical value of any type of tool deterioration whichnumerical value of any type of tool deterioration whichcan be measured.

Some of thewaysActualcuttingtimetofailure.VolumeofmetalremovedVolumeofmetalremoved.Numberofpartsproduced.p pCuttingspeedforagiventimeLengthofworkmachined.

TaylorsToolLifeEquationbasedonFlankWearCausesCausesSlidingofthetoolalongthemachinedsurfaceTemperaturerise

nVT CnVT C=Where,V=cuttingspeed(m/min)T Time(min)T=Time(min)n=exponentdependsontoolmaterialC=constantbasedontoolandworkmaterialandcuttingcondition.For IES, GATE, PSUs Page 3 of 49 Bhopal -2014

ValuesofExponentnn = 0.08 to 0.2 for HSS tool= 0.1 to 0.15 for Cast Alloys

f bid l= 0.2 to 0.4 for carbide tool[IAS1999; IES2006][IAS 1999; IES 2006]

= 0.5 to 0.7 for ceramic tool5 7[NTPC2003]

ExtendedorModifiedTaylorsequation

i.e Cuttingspeedhasthegreatereffectfollowedbyfeedg p g yanddepthofcutrespectively.

l fToolLifeCurve

1.HSS 2.Carbide 3.Ceramic

Cuttingspeedusedfordifferenttoolmaterials

HSS(min)30m/min

lFormula=no oV T C

Optimum tool life for minimumcost

1T T if T , & givento c c t mm

C n C CC n

= +

1 if & given

m

tt m

C n C CC

=

g

Optimum tool life for Maximum Productivity

t mmC n

p y(minimum production time)

1 1T To cn

n =

Units:Tc min(Toolchangingtime)c g gCt Rs./servicingorreplacement(Toolingcost)cost)Cm Rs/min(Machiningcost)V m/min(Cuttingspeed)

Toolingcost(Ct)=toolregrindcost+tooldepreciationperservice/replacement

Machiningcost(C ) labour cost+overheadcostperMachiningcost(Cm)=labour cost+overheadcostpermin

MinimumCostVsProductionRate

V V Vmax.production max.profit min. costV >V >V

MachinabilityDefinitionM hi bili b i l d fi d bili fMachinability can be tentatively defined as ability ofbeing machined and more reasonably as ease ofmachining.

h f h h hSuch ease of machining or machining charactersof any toolwork pair is to be judged by:y p j g y

Tool wear or tool lifeMagnitude of the cutting forcesSurface finishSurface finishMagnitude of cutting temperatureg g pChip forms.

MachinabilityContd.M hi bilit ill b hi h h tti fMachinability will be high when cutting forces,temperature, surfaces roughness and tool wear are less,t l lif i l d hi id ll if d h ttool life is long and chips are ideally uniform and short.

The addition of sulphur lead and tellurium to nonThe addition of sulphur, lead and tellurium to nonferrous and steel improves machinability.S l h i dd d t t l l if th i ffi i tSulphur is added to steel only if there is sufficientmanganese in it. Sulphur forms manganese sulphidehi h i t i l t d h d t i t lwhich exists as an isolated phase and act as internal

lubrication and chip breaker.If insufficient manganese is there a low melting ironsulphide will formed around the austenite grainboundary. Such steel is very weak and brittle.

FreeCuttingsteelsAddition of lead in low carbon resulphurised steels andalso in aluminium copper and their alloys help reducealso in aluminium, copper and their alloys help reducetheir s. The dispersed lead particles act as discontinuityand solid lubricants and thus improve machinability byand solid lubricants and thus improve machinability byreducing friction, cutting forces and temperature, tool

d BUE f iwear and BUE formation.It contains less than 0.35% lead by weight .35 y gA free cutting steel containsC % Si % M % P % S % Pb %C0.07%, Si0.03%, Mn0.9%, P0.04%, S0.22%, Pb0.15%

MachinabilityIndexOrMachinabilityRating

The machinability index KM is defined byKM = V6 /V6 RKM = V60/V60R

Where V60 is the cutting speed for the target materialh l lif f 6 i V i h f hthat ensures tool life of 60 min, V60R is the same for thereference material.If KM > 1, the machinability of the target material isbetter that this of the reference material and vice versabetter that this of the reference material, and vice versa


RoleofmicrostructureonMachinabilityC i l d l l fCoarsemicrostructureleadstolesservalueofs.

Therefore,s canbedesirablyreducedbyP h lik li f lProperheattreatmentlikeannealingofsteelsControlledadditionofmaterialslikesulphur (S),leadp ( ),(Pb),Tellerium etcleadingtofreecuttingofsoftductilemetalsandalloysmetalsandalloys.

Brittlematerialsarerelativelymoremachinable.

ff f l k l ( )Effects of tool rake angle(s) onmachinabilitymachinability

AsRakeangleincreasesmachinabilityincreases.

Buttoomuchincreaseinrakeweakensthecuttingedge.

EffectsofCuttingEdgeangle(s)onmachinabilitymachinability

Th i ti i th tti d l d t ff tThe variation in the cutting edge angles does not affect

cutting force or specific energy requirement for cuttingcutting force or specific energy requirement for cutting.

Increase in SCEA and reduction in ECEA improvesIncrease in SCEA and reduction in ECEA improves

surface finish sizeably in continuous chip formation

hence Machinability.

Effectsofclearanceangleonmachinability

Inadequate clearance angle reduces tool life and surfacefinish by tool work rubbing, and again too largeclearance reduces the tool strength and tool life hencegmachinability.

EffectsofNoseRadiusonmachinabilityProper tool nose radiusing improves machinability tosome extent throughsome extent throughincrease in tool life by increasing mechanical strength

d d i h l iand reducing temperature at the tool tipreduction of surface roughness, hmaxg , max

2fmax 8

fhR

=max 8R

SurfaceRoughnessIdeal Surface ( Zero nose radius)

fPeak to valley roughness (h) =tan cot

fSCEA ECEA+

and (Ra) = ( )4 4h f

SCEA ECEA=a

Practical Surface ( with nose radius = R)( )4 4 tan cotSCEA ECEA+

2 2

ah and R8 18 3f fR R

= =a8 18 3R RChange in feed (f) is more important than a change in nose radiusg ( ) p g(R) and depth of cut has no effect on surface roughness.

Cutting fluidCuttingfluidThe cutting fluid acts primarily as a coolant and

dl l b i t d i th f i ti ff t tsecondly as a lubricant, reducing the friction effects atthe toolchip interface and the workblank regions.Cast Iron: Machined dry or compressed air, Soluble oilfor high speed machining and grindingfor high speed machining and grindingBrass: Machined dry or straight mineral oil with ori h EPAwithout EPA.

Aluminium: Machined dry or kerosene oil mixed withymineral oil or soluble oilStainless steel and Heat resistant alloy: HighStainless steel and Heat resistant alloy: Highperformance soluble oil or neat oil with high

i i h hl i d EP ddi iconcentration with chlorinated EP additive.

IAS 2009 MainIAS2009MainWhatareextremepressurelubricants?Whatareextreme pressurelubricants?

[3 marks]Wh hi h d bbi iWhere high pressures and rubbing action areencountered, hydrodynamic lubrication cannot be

i i d E P (EP) ddi i bmaintained; so Extreme Pressure (EP) additives must beadded to the lubricant. EP lubrication is provided by a

b f h i l h bnumber of chemical components such as boron,phosphorus, sulfur, chlorine, or combination of these.Th d i d b h hi hThe compounds are activated by the higher temperatureresulting from extreme pressure. As the temperaturei EP l l b i d lrises, EP molecules become reactive and releasederivatives such as iron chloride or iron sulfide andf lid i iforms a solid protective coating.


MetalFormingMetalFormingSh tM t lO tiSheetMetalOperationP d M llPowderMetallurgy


FourImportantformingtechniquesare:Rolling

Forgingg g

ExtrusionExtrusion

D iDrawing

TerminologyTerminologySemifinished productI i h fi lid f f lIngot: is the first solid form of steel.Bloom: is the product of first breakdown of ingot has squarep g qcross section 6 x 6 in. or largerBillet: is hot rolled from a bloom and is square 1 5 in on aBillet: is hot rolled from a bloom and is square, 1.5 in. on aside or larger.Sl b i th h t ll d i t bl t lSlab: is the hot rolled ingot or bloom rectangular crosssection 10 in. or more wide and 1.5 in. or more thick.

slabIngot Bloom Billet

TerminologyMill product

Plate is the product with thickness > 5 mm

Sheet is the product with thickness < 5 mm and width > 600

mmmm

Strip is the product with a thickness < 5 mm and width

St i H d iStrainHardeningStrainhardening(coldWorking)Strainhardening(coldWorking)

no K =

Strainrateeffect(hotWorking)

o

( g)

mo C =o

VelocityPlaten1 vdhWhere heightousInstantane

VelocityPlaten1===

hv

dtdh

h

ll b lMalleabilityMalleability is the property of a material whereby it can

b h d h ld b h llbe shaped when cold by hammering or rolling.

A ll bl i l i bl f d i l iA malleable material is capable of undergoing plastic

deformation without fracturedeformation without fracture.

A malleable material should be plastic but it is notA malleable material should be plastic but it is not

essential to be so strong.g

Lead, soft steel, wrought iron, copper and aluminium are

some materials in order of diminishing malleability.

ColdWorkingColdWorkinggW ki b l li i Workingbelowrecrystalizationtemp.

d f ld kAdvantagesofColdWorking1. Better accuracy, closer tolerances

2. Better surface finish

3. Strain hardening increases strength and hardness

4. Grain flow during deformation can cause desirable

directional properties in productdirectional properties in product

5 No heating of work required (less total energy)5. No heating of work required (less total energy)

d f ld kDisadvantagesofColdWorking1. Equipmentofhigherforcesandpowerrequired

S f f t ti k i tb f f l d2. Surfacesofstartingworkpiecemustbefreeofscaleand

dirt

3. Ductilityandstrainhardeninglimittheamountofforming

thatcanbedone

4. Insomeoperations,metalmustbeannealedtoallow

furtherdeformationfurtherdeformation

5. Somemetalsaresimplynotductileenoughtobecold5 p y g

worked.

HotWorkingHotWorking

WorkingaboverecrystalizationtempWorkingaboverecrystalizationtemp.

AdvantagesofHotWorking1. The porosity of the metal is largely eliminated.2 The grain structure of the metal is refined2. The grain structure of the metal is refined.3. The impurities like slag are squeezed into fibers anddi ib d h h h ldistributed throughout the metal.4. The mechanical properties such as toughness,4 p p g ,percentage elongation, percentage reduction in area, andresistance to shock and vibration are improved due toresistance to shock and vibration are improved due tothe refinement of grains.

DisadvantagesofHotWorking1. It requires expensive tools.2 It produces poor surface finish due to the rapid2. It produces poor surface finish, due to the rapidoxidation and scale formation on the metal surface.

D h f fi i h l l3. Due to the poor surface finish, close tolerancecannot be maintained.

Mi St t l Ch i H tMicroStructuralChangesinaHotWorking Process (Rolling)WorkingProcess(Rolling)


AnnealingAnnealing relieves the stresses from cold working threestages: recovery recrystallization and grain growth

g

stages: recovery, recrystallization and grain growth.During recovery, physical properties of the coldworked

i l d i h b bl h imaterial are restored without any observable change inmicrostructure.

WarmFormingDeformation produced at temperatures intermediate to

h d ld f k fhot and cold forming is known as warm forming.

C d ld f i i d l d iCompared to cold forming, it reduces loads, increase

material ductilitymaterial ductility.

Compared to hot forming it produce less scaling andCompared to hot forming, it produce less scaling and

decarburization, better dimensional precision andp

smoother surfaces.

h lIsothermalFormingDuring hot forming, cooler surfaces surround a hotter

interior and the variations in strength can result in noninterior, and the variations in strength can result in non

uniform deformation and cracking of the surface.

For temp.sensitive materials deformation is performed

under isothermal conditions.

Th di li b h d h k iThe dies or tooling must be heated to the workpiece

temperature, sacrificing die life for product quality.p , g p q y

Close tolerances, low residual stresses and uniform metal

flow.

RollingRollingg


RollingDefinition: The process of plastically deforming metal

b b llby passing it between rolls.

M id l d hi h d i d l lMost widely used, high production and close tolerance.

F i ti b t th ll d th t l fFriction between the rolls and the metal surface

produces high compressive stressproduces high compressive stress.

Hotworking (unless mentioned cold rolling.Hot working (unless mentioned cold rolling.

Metal will undergo biaxial compression.g p

HotRollingDone above the recrystallization temp.

Results fine grained structure.

S f lit d fi l di i l tSurface quality and final dimensions are less accurate.

Breakdown of ingots into blooms and billets is done byBreakdown of ingots into blooms and billets is done by

hotrolling. This is followed by further hotrolling intog y g

plate, sheet, rod, bar, pipe, rail.

Hot rolling is terminated when the temp. falls to about

(50 to 100C) above the recrystallization temp.For IES, GATE, PSUs Page 9 of 49 Bhopal -2014

ColdRollingDone below the recrystallization temp..

Products are sheet, strip, foil etc. with good surface

fi i h d i d h i l h i h lfinish and increased mechanical strength with close

product dimensionsproduct dimensions.

Performed on fourhigh or clustertype rolling millsPerformed on four high or cluster type rolling mills.

(Due to high force and power)( g p )

RingRollingRing rolls are used for tube rolling, ring rolling.

As the rolls squeeze and rotate, the wall thickness is

d d d h di f h i ireduced and the diameter of the ring increases.

Sh d ll b d t d id i t fShaped rolls can be used to produce a wide variety of

crosssection profilescross section profiles.

Ring rolls are made of spheroidized graphite bainitic andRing rolls are made of spheroidized graphite bainitic and

pearlitic matrix or alloy cast steel base.

SheetrollingIn sheet rolling we are only attempting to reduce the

h k f lcross section thickness of a material.

RollForming RollBendingA continuous form of threepoint bending is roll

bending, where plates, sheets, and rolled shapes can

be bent to a desired curvature on forming rolls.

Upper roll being adjustable to control the degree of

tcurvature.


Shaperolling PackrollingPack rolling involves hot rolling multiple sheets of

l h l f lmaterial at once, such as aluminium foil.

A hi f id fil h i ldiA thin surface oxide film prevents their welding.

ThreadrollingUsed to produce threads in substantial quantities.

This is a coldforming process in which the threads are

f d b lli h d bl k b h d d diformed by rolling a thread blank between hardened dies

that cause the metal to flow radially into the desiredthat cause the metal to flow radially into the desired

shape.p

No metal is removed, greater strength, smoother, harder,g g

and more wearresistant surface than cut threads.

Threadrollingcontd.Major diameter is always greater than the diameter of the

bl k (blank (

Bl k di i li l l h h i h di fBlank diameter is little larger than the pitch diameter of

the threadthe thread.

Restricted to ductile materialsRestricted to ductile materials.

ManufactureofgearsbyrollingThe straight and helical teeth of disc or rod type external

l f ll d d d d lsteel gears of small to medium diameter and module are

generated by cold rollinggenerated by cold rolling.

High accuracy and surface integrityHigh accuracy and surface integrity.

Employed for high productivity and high quality (costlyEmployed for high productivity and high quality. (costly

machine))

Larger size gears are formed by hot rolling and then

finished by machining.

Fig.Productionofteethofspurgearsbyrolling

llRollpiercing It is a variation of rolling called roll piercing.The billet or round stock is rolled between two rolls,,both of them rotating in the same direction with theiraxes at an angle of 4.5 to 6.5 degree.axes at an angle of 4.5 to 6.5 degree.These rolls have a central cylindrical portion with thesides tapering slightly There are two small side rollssides tapering slightly. There are two small side rolls,which help in guiding the metal.Because of the angle at which the roll meets the metal,it gets in addition to a rotary motion, an additionalaxial advance, which brings the metal into the rolls.This crossrolling action makes the metal friable at thegcentre which is then easily pierced and given acylindrical shape by the centralpiercing mandrel.cylindrical shape by the central piercing mandrel.


PlanetarymillConsist of a pair of heavy backing rolls surrounded by a largeConsist of a pair of heavy backing rolls surrounded by a largenumber of planetary rolls.Each planetary roll gives an almost constant reduction to theEach planetary roll gives an almost constant reduction to theslab as it sweeps out a circular path between the backing rollsand the slab.As each pair of planetary rolls ceases to have contact with thework piece, another pair of rolls makes contact and repeath d ithat reduction.The overall reduction is the summation of a series of smalld ti b h i f ll Th f th l t illreductions by each pair of rolls. Therefore, the planetary mill

can reduce a slab directly to strip in one pass through themillmill.The operation requires feed rolls to introduce the slab intothe mill, and a pair of planishing rolls on the exit to improvethe mill, and a pair of planishing rolls on the exit to improvethe surface finish.

Camber

Camber can be used to correct the roll deflection (at onlyone value of the roll force).

LubricationforRollingHot rolling of ferrous metals is done without a lubricant.

Hot rolling of nonferrous metals a wide variety of

d d il l i d f id dcompounded oils, emulsions and fatty acids are used.

C ld lli l b i t t l bl il lCold rolling lubricants are watersoluble oils, low

viscosity lubricants such as mineral oils emulsionsviscosity lubricants, such as mineral oils, emulsions,

paraffin and fatty acids.p y

DefectsinRollingf hDefects Whatis Cause

Surface Scale, rust, Inclusions andSurfaceDefects

Scale, rust,scratches, pits,cracks

Inclusions andimpurities in thematerialscracks materials

Wavy edges Strip is thinner Due to roll bendingalong its edgesthan at its centre.

edges elongates moreand buckle.

Alligatoring Edge breaks Nonuniformdeformationdeformation

GeometryofRollingProcess DraftT l d i d f k i lliTotalreductionordrafttakeninrolling.

h h h 2 (R R ) D (1 ) 0 fh = h - h = 2 (R - R cos ) = D (1 - cos ) Usually,thereductioninbloomingmillsisabout100y, gmmandinslabbing mills,about50to60mm.

MaximumDraftPossibleMaximumDraftPossible

( ) 2h R( ) 2maxh = R

TorqueandPowerTh i i i ll i fThe power is spent principally in four ways1) The energy needed to deform the metal.) gy2) The energy needed to overcome the frictional force.) Th l i h i i d i i3) The power lost in the pinions and powertransmissionsystem.4) Electrical losses in the various motors and generators.

Remarks: Losses in the windup reel and uncoiler mustpalso be considered.


TorqueandPower

Will bWill bediscusseddiscussedin class

[ForIESConventionalOnly][ForIESConventionalOnly]

AssumptionsinRollingR ll i h i id li d1. Rolls are straight, rigid cylinders.

2. Strip is wide compared with its thickness, so that nop p ,widening of strip occurs (plane strain conditions).

3 The arc of contact is circular with a radius greater than3. The arc of contact is circular with a radius greater thanthe radius of the roll.

4. The material is rigid perfectly plastic (constant yieldstrength).st e gt ).

5. The coefficient of friction is constant over the toolk i t fwork interface.

StressEquilibriumofanElementinRolling

Considering the thickness of the element perpendicular tothe plane of paper to be unity We get equilibriumthe plane of paper to be unity, We get equilibriumequation in xdirection as,- h + ( +d ) (h + dh) - 2pR d sin x x x

x+ 2 R d cos = 0

xFor sliding friction, = p Simplifying and neglecting d d t i d 1 t

( )second order terms, sin and cos 1, we get

xd h

=

( ) 2 ( )x pRd

=

'0 0

23x

p = =

( ) ( )'03

2d h p pR = ( ) ( )0 2h p pRdd p

( )'0 '0

1 2d ph pRd

=

0

' d p ph

+ ( ) ( )'1 2d h pR = 0 ' '0 0

hd

+

( ) ( )01 2h pRd =

'0

'

Due to cold rolling, increases as h decreases,'0thus nearly a constant and itsderivative zero. h

d

( )( )

'0/ 2

d p Rd

= ( )

( )

'0

2

/p h

=

( )( )

2

'

2 1 cos

/f fh h R h R

d

= + +

( )( ) ( )

02'

/ 2/

d p R dh Rp

=

+

( )0/Integrating both side

fh Rp +

( )'Integrating both side

2ln / R dp 2 ( )R d I II say ( )0ln /f

ph

=+ 2 2

( )f

d I II sayR h R

=+

22Rd 2Rd 2d hI ln

h h / R Rh R = = = = + f

2f

h h / R Rh RhNow h / R

+

= +Now h / R R

d h

= +

d hor 2d R

R

=

2f

2RII dh R

=+ f

22 d

h / R = 2

f

1

h / R R R

+

1

f f

R R2 .tan .h h

=

( ) = + ' 10

h R Rln p / ln 2 .tan . lnCR h h

( )

f f

' H

R h hhC =

H0p C eR

=

1

f f

R Rwhere H 2 .tan .h h =

f fh hNow at entry ,

=

=0Hence H H with replaced by in above equation

At exit 0== '0

At exit 0Therefor p

=

oH' o0

hIn theentryzone, p C. eR 0

y , pR

R= oH

o

Rand C .eh

( )= 0

o

H H'0

hp . e00

p . eh

I th it

In the exit zoneh

=

' H0

f

hp .eh f

At theneutral po int aboveequationswill givesameresults


Theory for Production

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