Mechanical Engineers Data Handbook Episode 5 pps

APPLIED MECHANICS 93 2.9.5 Rolling bearings The term ‘rolling bearing’ refers to both ball and roller bearings.. Ball bearings of the journal type are used for transverse loads but wil

90 MECHANICAL ENGINEER’S DATA HANDBOOK

The power output of a rotary machine may be

measured by means of a friction brake The forces are

measured by spring balances or load cells Other types

of dynamometer include fluid brakes and electric

generators

Torque absorbed T = r ( F , - F 2 )

Power P = 2 n N T

The full analysis of heavily loaded plain bearings is

extremely complex For so called ‘lightly-loaded bear-

ings’ the calculation of power loss is simple for both

journal and thrust bearings

Important factors are, load capacity, length to diameter ratio, and allowable pressure on bearing material

Information is also given on rolling bearings

2.9 I Lightly loaded plain bearings

2.9.2 Load capacity for plain bearings

Automobile and aircraft engine main bearings

Automobile and aircraft engine crankpin bearings

Marine steam turbine main bearings

Marine steam turbine crankpin bearings

Land steam turbine main bearings

Generators and motors

1 s-4

2 4 0.5-4 0.3-1 O 0.4-2.0 0.54.7 0.54.7 2-2.5

0.5-1.75 0.5-1.50 1.0-1.5 1.0-1.5

1 .0-2.0 1.0-2.5

1 M 0 1.5-2.0

1 &2.0 1.5-2.0

Bearing load W Projected area-LD

This assumes a uniform pressure; actually the maxi-

mum pressure is considerably higher

92 MECHANICAL ENGINEER'S DATA HANDBOOK

2.9.3 Bearing materials

Metals

Material

Brinell Thin shaft Load capacity, p Maximum

Tin base babbitt

Lead base babbitt

3 W O 20-30 60-80 40-70

25 45-50

< 150

< 150 200-250 200-250

200 300-400

300 200-300

3 w 0 0

5.5-10.3 5.5-8 O 8.0-10.3 10.3-15 10.3-16.5

2 30 20-30

capacity, p temperature velocity, u Maximum pu

Electric motors, generators, etc Ground Broached or reamed 0.020 + 0.01 5

General machinery, continuous

N =revolutions per minute, D= diameter (mm)

APPLIED MECHANICS 93

2.9.5 Rolling bearings

The term ‘rolling bearing’ refers to both ball and roller

bearings Ball bearings of the journal type are used for

transverse loads but will take a considerable axial

load They may also be used for thrust bearings

Rollers are used for journal bearings but will not take

axial load Taper roller bearings will take axial thrust

as well as transverse load

Advantages of rolling bearings

(1) Coefficient of friction is low compared with plain

bearings especially at low speeds This results in

lower power loss

(2) Wear is negligible if lubrication is correct

(3) They are much shorter than plain bearings and

take up less axial space

2.9.6 Types of rolling bearings

The following table lists the most common types of

rolling bearings

(4) Because of extremely small clearance they permit more accurate location; important for gears for example

( 5 ) Self-aligning types permit angular deflection of the

shaft and misalignment

Disadvantages of rolling bearings

(1) The outside diameter is large

(2) The noise is greater than for plain bearings, especially at high speeds

(3) There is greater need of cleanliness when fitted to achieve correct life

(4) They cannot always be fitted, e.g on crankshafts

( 5 ) They are more expensive for small quantities but relatively cheap when produced in large quanti- ties

(6) Failure may be catastrophic

Ball journal Used for radial load but will take one third load

axially Deep grooved type now used extensively

Light, medium and heavy duty types available

Angular contact

hall journal

Takes a larger axial load in one direction Must be used in pairs if load in either direction

Self-aligning The outer race has a spherical surface mounted in a

ball, single row ring which allows for a few degrees of shaft

misalignment

94 MECHANICAL ENGINEER’S DATA HANDBOOK

Double row Used for larger loads without increase in outer

ball journal diameter

Roller journal For high radial loads but no axial load Allows axial

Needle rollers These run directly on the shaft with or without cages

Occupy small space

Shields, seals Shields on one or both sides prevent ingress of dirt

Seals allow packing with grease for life A groove allows fitting of a circlip for location in bore

and grooves

APPLIED MECHANICS 95

2.9.1 Service factor for rolling bearings

The bearing load should be multiplied by the following

factor when selecting a bearing

2.9.8 Coefficient of friction for bearings

Plain bearings - boundary lubrication Rolling bearings

P

Mixed film (boundary plus

Dry (metal to metal) 0 2 M 4 0

Plain journal bearings - oil bath lubrication

Gears are toothed wheels which transmit motion and

power between rotating shafts by means of success-

ively engaging teeth They give a constant velocity

ratio and different types are available to suit different

relative positions of the axes of the shafts (see table) Most teeth are of the ‘involute’ type The nomen- clature for spur gears is given in the figures

96 MECHANICAL ENGINEER’S DATA HANDBOOK

I_ Centre distance

s

/

2 IO I Classification of gears

axis

intersecting not intersecting

2.10.2 Metric gear teeth

D

T Metric module m=- (in millimetres)

where: D=pitch circle diameter, T=number of teeth

The preferred values of module are: 1, 1.25, 1.5,2,2.5,

3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40 and 50

ItD

T Circular pitch p = - = r m

The figure shows the metric tooth form for a ‘rack’ (Le

a gear with infinite diameter)

APPLIED MECHANICS 97

Design of gears

The design of gears is complex and it is recommended

that British Standards (or other similar sources) be

F, = tangential component of tooth force

F , = separating component of tooth force

r#J =pressure angle of teeth

D , =pitch circle diameter of driver gear

D, =pitch circle diameter of driven gear

N , =speed of driver gear

N , =speed of driven gear

n, =number of teeth in driver gear

n2 =number of teeth in driven gear

P =power

T = torque

9 =efficiency

Tangential force on gears F, = F cos r#J

Separating force on gears F , = F , tan q5

Torque on driver gear T I =- FIDI

D

2 Output power P , = 2 n N 2 F , ~ q

Po

Efficiency q = -

pi

Rack and pinion drive

For a pinion, pitch circle diameter D speed N and torque T :

2.10.4 Helical spur gears

In this case there is an additional component of force

Fa in the axial direction

98 MECHANICAL ENGINEER'S DATA HANDBOOK

Axial force Fa = F, tan a

Double helical gears

To eliminate the axial thrust, gears have two sections

with helices of opposite hand These are also called

4 =pressure angle of teeth

B = pinion pitch cone angle

Tangential force on gears = F,

Separating force F, = F, tan (b

Pinion thrust F, = F, sin B

Gear thrust F, = F, cos fl

Spiral bevel gear

Let:

a =spiral angle of pinion

r i g h t bevel gear

,,' p,

APPLIED MECHANICS 99 2.10.6 Worm gears

The worm gear is basically a screw (the worm)

engaging with a nut (the gear) The gear is, in effect, a

partial nut whose length is wrapped around in a circle

Let :

b,, = normal pressure angle

u =worm helix angle

n, = number of threads or starts on worm

n, = number of teeth in gear

D, = worm pitch circle diameter

D, = gear pitch circle diameter

cos 4, sin u + p cos u

Separating force on each component F,= ,F,

100 MECHANICAL ENGINEER’S DATA HANDBOOK Coefficient of friction for worm gears

Velocity (m s - * )

Hard steel worm/phosphor bronze wheel 0.06 0.05 0.035 0.023 0.017 0.014

2.10.7 Epicyclic gears

The main advantage of an epicyclic gear train is that

the input and output shafts are coaxial The basic type

consists of a ‘sun gear’ several ‘planet gears’ and a ‘ring

gear’ which has internal teeth Various ratios can be

obtained, depending on which member is held station-

ary

Let :

N = speed

n = number of teeth Note that a negative result indicates rotation reversal

Ratio of output to input speed for various types

Thermodynamics and heat transfer

3 I I H e a t capacity

Heat capacity is the amount of heat required to raise

the temperature of a body or quantity of substance by

1 K The symbol is C (units joules per kelvin, J K - I )

Heat supplied Q = C ( t 2 - t l )

where: t , and t , are the initial and final temperatures

3 I 2 Specific heat capacity

This is the heat to raise 1 kg of substance by 1 K The

symbol is c (units joules per kilogram per kelvin,

Jkg-' K-')

Q = m c ( t , - t , )

where: m=mass

3.1.3 Latent heat

This is the quantity of heat required to change the state

of 1 kg of substance For example:

Solid to liquid: specific heat of melting; h,, (J kg- ')

Liquid to gas: specific heat ofevaporation, h,, (J kg- * )

3 I 4 Mixing of fluids

If m1 kg of fluid 1 at temperature t , is mixed with m, kg

of fluid 2 at temperature t,, then

Final mass m = m l + m , at a temperature

For a so-called 'perfect gas':

Boyle's law: pv = constant for a constant

temperature T

V

T

Charles' law: -=constant for a constant pressure p

where: p =pressure, V = volume, T=absolute

3.2.2 Universal gas constant

If R is multiplied by M the molecular weight of the gas, then :

Universal gas constant R,= MR=8.3143

kJ kg-' K - ' (for all perfect gases)

APPLIED MECHANICS 101

THERMODYNAMICS A N D HEAT TRANSFER 103

3.2.3 Specific heat relationships

There are two particular values of specific heat: that at

constant volume c,, and that at constant pressure cp

Ratio of specific heats y = -1!

This is the energy of a gas by virtue of its temperature

u =cVT (specific internal energy)

U =mc,T (total internal energy)

Change in internal energy:

where: h = specific enthalpy, H = total enthalpy

and it can be shown that

Non-pow energy equation

Gain in internal energy =Heat supplied - Work done

uz-ul=Q- W

where: W = p d v

Steady p o w energy equation

j12

This includes kinetic energy and enthalpy:

or, if the kinetic energy is small (which is usually the case)

h, - h l =Q- W (neglecting height differences)

3.2.7 Entropy

Entropy, when plotted versus temperature, gives a curve under which the area is heat The symbol for entropy is s and the units are kilojoules per kilogram per kelvin (kJkg-'K-')

3.2.8 Exergy and anergy

In a heat engine process from state 1 with surroundings

at state 2 exergy is that part of the total enthalpy drop available for work production

I

104 MECHANICAL ENGINEER’S DATA HANDBOOK

Exergy c f , = ( H , - H , ) - T , ( S , - S , ) Constant temperature (isothermal)

That part of the total enthalpy not available is called In this case:

pv =constant the ‘anergy’

THERMODYNAMICS AND HEAT TRANSFER 10.5

P = P A + P B + P ~ + +Pi

Z(miRi)

Apparent gas constant R = -

m

Apparent molecular weight M = R,/R

where: R,= universal gas constant

106 MECHANICAL ENGINEER’S DATA HANDBOOK

A substance may exist as a solid, liquid, vapour or gas

A mixture of liquid (usually in the form of very small

drops) and dry vapour is known as a ‘wet vapour’

When all the liquid has just been converted to vapour

the substance is referred to as ‘saturated vapour’ or

‘dry saturated vapour’ Further heating produces what

is known as ‘superheated vapour’ and the temperature

u,=specific volume of liquid (m3 kg-’)

u,=specific volume of saturated vapour (m3 kg-’)

u = specific internal energy (kJ kg- I )

u, = specific internal energy of liquid (kJ kg- ’)

ug= specific internal energy of vapour (kJ kg-’)

u,, = specific internal energy change from liquid to

vapour (kJkg-’)

h =specific enthalpy (kJ kg - I )

h, = specific enthalpy of liquid, kJ/kg

h, = specific enthalpy of vapour, kJ/kg

h,, = specific enthalpy change from liquid to vapour

(latent heat) kJ/kg

s = specific entropy, kJ/kg K

sf = specific entropy of liquid, kJ/kg K

sg = specific entropy of vapour, kJ/kg K

sfg = specific entropy change from liquid to vapour,

Specific volume of wet vapour u, = uf( 1 - x) + XD,==XU,

(since u, is small)

Specific internal energy of wet vapour

u, = Uf + x(u, - Uf) = Uf + XUfs

rise (at constant pressure) required to do this is known

as the ‘degree of superheat’ The method of determin- ing the properties of vapours is given, and is to be used

in conjunction with vapour tables, the most compre- hensive of which are for water vapour Processes are shown on the temperature-entropy and en- thalpy-entropy diagrams

Specific enthalpy of wet vapour specific entropy of wet vapour

Superheated vapour Tables (e.g for water) give values of u, u, h, and s for a particular pressure and a range of temperatures above the saturation tempera- ture t, For steam above 70 bar use u=h-pu

the enthalpy of the liquid at saturation temperature,

h,, is the enthalpy corresponding to the latent heat,

THERMODYNAMICS A N D HEAT TRANSFER 107

108 MECHANICAL ENGINEER'S DATA HANDBOOK

3.4.2 Latent heats and boiling points

Latent heat of evaporation (kJkg-') at atmospheric pressure

Latent heat of fusion (kJkg-') at atmospheric pressure

THERMODYNAMICS AND HEAT TRANSFER 109 Boiling point ("C) at atmospheric pressure

General properties of air (at 300K, 1 bar)

Mean molecular weight

Specific heat at constant pressure

Specific heat at constant volume

Ratio of specific heats

k = 0.02614 W m- K - '

a = 2203 m2 s -

y = 1.40

P,=0.711

110 MECHANICAL ENGINEER'S DATA HANDBOOK 3.4.4 Specific heat capacities

Specific heat capacity of solids and liquids

Seawater Silica

Si 1 icon

Silver Tin Titanium Tungsten Turpentine Uranium Vanadium Water Water, heavy Wood (typical) Zinc

1.676 2.100 2.140 2.140 0.796 0.133 2.010 0.880 0.796 3.940 0.800 0.737 0.236 0.220 0.523 0.142 1.760 0.116 0.482 4.196 4.221 2.0 to

~

0.7 18 1.663 0.3136 1.51 0.6573 0.7449 0.383 1.4947 3.1568 10.1965 0.583 1.7124 0.7436 0.708 0.6586 1.507 0.5150

1.4 1.32 1.668 1.11 1.29 1.398 1.33 1.18 1.659 1.405 1.40 1.30 1.40 1.31 1.394 1.12 1.25

~~

0.2871 0.528 0.2081 0.17 0.1889 0.2968 0.128 0.2765 2.077 4.124 0.230 0.5183 0.2968 0.220 0.2598 0.1886 0.1298

~

28.96 15.75

16

28 37.8

32

44

64

THERMODYNAMICS A N D HEAT TRANSFER 1 1 1

Nozzles are used in steam and gas turbines, in rocket

motors, in jet engines and in many other applications

Two types of nozzle are considered: the ‘convergent

nozzle’, where the flow is subsonic; and the ‘conver- gent divergent nozzle’, for supersonic flow

Symbols used:

p = inlet pressure

p , =outlet pressure

p , =critical pressure at throat

u I = inlet specific volume

u2 =outlet specific volume

h=mass flow rate

Critical pressure ratio r, = -

Outlet area A,=7

Outlet pressure p 2 equal to or less than p c ,