IMechE Engineers' Databook

Section 2

Units

2.1 The Greek Alphabet

The Greek alphabet is used extensively to denote engineering quantities. Each letter can have various meanings, depending on the context in which it is used.

Table 2.1 The Greek alphabet.

2.2 Units Systems

Unfortunately, the world of mechanical engineering has not yet achieved uniformity in the system of units it uses. The oldest system is that of British Imperial units - still used in many parts of the world, including the USA. The CGS (or MKS) system is a metric system, still used in some European countries, but is gradually being superseded by the Systeme International (SI) system. Whilst the SI system is understood (more or less) universally, you will still encounter units from the others.

2.2.1 The SI System

The strength of the SI system is its coherence. There are four mechanical and two electrical base units, from which all other quantities are derived. The mechanical ones are:

Length: metre (m) Mass: kilogram (kg) Time: second (s) Temperature: Kelvin (K)

Remember, other units are derived from these; e.g. the Newton (N) is defined as N = kg m/s2.

2.2.2 SI prefixes

As a rule, prefixes are applied to the basic SI unit, except for weight, where the prefix is used with the unit gram (g), not the basic SI unit kilogram (kg). Prefixes are not used for units of angular measurement (degrees, radians), time (seconds), or temperature (°C or K).

Prefixes should be chosen in such a way that the numerical value of a unit lies between 0.1 and 1000.

Table 2.2 SI prefixes.

2.2.3 Conversions

Units often need to be converted. The least confusing way to do this is by expressing equality:

For example: to convert 600 mm H2O to Pascals (Pa)

Using 1 mm H2O = 9.80665 Pa

Add denominators as

Solve for x

Hence 600 mm H2O = 5883.99 Pa

Setting out calculations in this way can help avoid confusion, particularly when they involve large numbers and/or several sequential stages of conversion.

2.3 Units and Conversions

2.3.1 Force

The SI unit is the Newton (N) - it is a derived unit.

Table 2.3 Force (F)

Note: Strictly, all the units in the table except the Newton (N) represent weight equivalents of mass, and so depend on g. The true SI unit of force is the Newton (N) which is equivalent to 1 kg m/s2.

2.3.2 Weight

The true weight of a body is a measure of the gravitational attraction of the earth on it. Since this attraction is a force, the weight of a body is correctly expressed in Newtons (N).

Mass is measured in kilogram (kg)

Force (N) = mass (kg) × g (m/s2)

1 kg = 2.20462 lbf

1000 kg = 1 tonne (metric) = 0.9842 tons (imperial)

1 ton (US) = 2000 lb = 907.185 kg

Table 2.4 Density (?)

2.3.3 Pressure

The SI unit is the Pascal (Pa).

Figure 2.1

1 Pa = 1 N/m2

1 Pa = 1.45038 × 10-4 lbf/in2 (i.e. psi)

In practice, pressures are measured in MPa, bar, atmospheres, torr or the height of a liquid column, depending on the application.

Figure 2.2

Figure 2.3

Table 2.5 Pressure (P)

And for liquid columns:

1 mm Hg = 13.59 mm H2O = 133.3224 Pa = 1.333224 mbar

1 mm H2O = 9.80665 Pa

1 torr = 133.3224 Pa

For conversion of liquid column pressures; 1 in = 25.4 mm.

2.3.4 Temperature

The SI unit is degrees Kelvin (K). The most commonly used unit is degrees Celsius (°C).

Absolute zero is defined as 0 K or - 273.15 °C, the point at which a perfect gas has zero volume.

The imperial unit of temperature is degrees Fahrenheit (°F).

°C = 5/9 (°F-32)

°F = 9/5 (°C) + 32

Figure 2.4

Figure 2.5

2.3.5 Heat Energy

The SI unit for heat energy (in fact all forms of energy) is the Joule (J).

Table 2.6 Heat energy.

Specific energy is measured in Joules per kilogram (J/kg).

1 J/kg = 0.429923 × 10-3 Btu/lb

Specific heat capacity is measured in Joules per kilogram Kelvin (J/kg K).

1 J/kg K = 0.238846 × 10-3 Btu/lb °F

1 kcal/kg K = 4186.8 J/kg K

Heat flowrate is also defined as power, with the SI unit of Watts (W).

1 W = 3.41214 Btu/h = 0.238846 cal/s

2.3.6 Power

The Watt is a small quantity of power, so kW is normally used.

Table 2.7 Power (P)

Figure 2.6

Figure 2.7 Comparative power outputs

2.3.7 Flow

The SI unit of volume flowrate is m3/s.

1 m3/s = 219.969 UK gall/s = 1000 litres/s

1 m3/h = 2.77778 × 10-4 m3/s

1 UK gall/min = 7.57682 × 10-5 m3/s

1 UK gall = 4.546 litres

Figure 2.8

The SI unit of mass flowrate is kg/s.

1 kg/s = 2.20462 lb/s = 3.54314 ton (imp)/h

1 US gall = 3.785 litres

2.3.8 Torque

The SI unit of torque is the Newton metre (N.m). You may also see this referred to as 'moment of force'.

Figure 2.9

1 N.m = 0.737 lbf ft (i.e. 'foot pounds')

1 kgfm = 9.81 N.m

2.3.9 Stress

Stress is measured in Pascals - the same SI unit used for pressure, although it is a different physical quantity. 1 Pa is an impractical small unit so MPa is normally used.

1 MPa = 1 MN/m2 = 1 N/mm2

1 kgf/mm2 = 9.80665 MPa

Figure 2.10

2.3.10 Linear Velocity (Speed)

The SI unit is metres per second (m/s).

Table 2.8 Velocity (v)

2.3.11 Acceleration

The SI unit of acceleration is metres per second squared (m/s2).

1 m/s2 = 3.28084 ft/s2

Standard gravity (g) is normally taken as 9.81 m/s2.

2.3.12 Angular Velocity

The SI unit is radians per second (rad/s).

1 rad/s = 0.159155 rev/s = 57.2958 degree/s

The radian is the SI unit used for plane angles.

Figure 2.11

A complete circle is 2p radians

A quarter-circle (90°) is p/2 or 1.57 radians

1 degree = p/180 radians

2.3.13 Volume and Capacity

The SI unit is cubic metres (m3), but many imperial units are still in use.

1 m3 = 35.3147 ft3 = 61 023.7 in3

2.3.14 Area

The SI unit is square metres (m2) but many imperial units are still in use.

Table 2.9 Area (A)

Other metric units of area:

Japan: 1 tsubo = 3.306 m2 1 se = 0.9917 a 1 ho-ri = 15.42 km2 Russia: 1 kwadr. archin = 0.5058 m2 1 kwadr. saschen = 4.5522 m2 1 dessjatine = 1.0925 ha 1 kwadr. werst = 1.138 km2

Figure 2.12 Making sense of microns (µ)

Figure 2.13

2.3.15 Viscosity

Dynamic viscosity (µ) is measured in the SI system in Pascal seconds (Pa s).

1 Pa s = 1 N s/m2 = 1 kg/m s

A common unit from another units system is the centipoise (cP), or standard imperial units may be used:

Table 2.10 Dynamic viscosity (µ)

Kinematic viscosity (?) is a function of dynamic viscosity. Kinematic viscosity = dynamic viscosity/density, i.e. ? = µ/? The SI unit is m2/s. Other imperial and CGS units are also used.

1 m2/s = 10.7639 ft2/s = 5.58001 × 106 in2/h

1 Stoke (St) = 100 centistokes (cSt) = 10-4 m2/s

2.4 Consistency of Units

Within any system of units, the consistency of units forms a 'quick check' of the validity of equations. The units must match on both sides.

Example:

To check kinematic viscosity (?)

Replacing N with kgm/s2

Cancelling gives

OK, units match.

2.4.1 Foolproof Conversions: Using Unity Brackets

When converting between units it is easy to make mistakes by dividing by a conversion factor instead of multiplying, or vice versa. The best way to avoid this is by using the technique of unity brackets.

A unity bracket is a term consisting of a numerator and denominator in different units which...