Aeronautical Engineer Data Book Episode 2 doc

2.3 Temperature 2.3.6 Heat and work The basic unit for heat ‘energy’ is the British thermal unit BTU.. 2.3.8 Flow The basic unit of volume flowrate is US gallon/min in SI it is litres/s

11 Fundamental dimensions and units

10

bar

1

MN 2 m

1 bar 1.013 bar

1.1097 kg/cm

5 N/m

5 Pa

2 O

Rules of thumb: An apple ‘weighs’ about 1.5 newtons

A meganewton is equivalent to about 100 tonnes

An average car weighs about 15 kN

Fig 2.1 Pressure relationships

KSI

21000

2 1.0197

2 0.9807

210.0

20.1

2

0.09807 2

10.197

214.223

2 0.06895

20.0703

psi

Bar Kg/cm 2

N/mm 2

(MPa)

2 14.503

Fig 2.2 Pressure conversions

12 Aeronautical Engineer’s Data Book

0 K

Volume

Fig 2.3 Temperature

2.3.6 Heat and work

The basic unit for heat ‘energy’ is the British thermal unit (BTU)

Specific heat ‘energy’ is measured in BTU/lb (in SI it is joules per kilogram (J/kg))

Table 2.6 shows common conversions

Specific heat is measured in BTU/lb °F (or in

SI, joules per kilogram kelvin (J/kg K))

1 BTU/lb °F = 4186.798 J/kg K

1 kcal/kg K = 4186.8 J/kg K

Heat flowrate is also defined as power, with the unit of BTU/h (or in SI, in watts (W))

1 BTU/h = 0.07 cal/s = 0.293 W

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

2.3.7 Power

BTU/h or horsepower (hp) are normally used

or, in SI, kilowatts (kW) See Table 2.7

2.3.8 Flow

The basic unit of volume flowrate is US gallon/min (in SI it is litres/s)

1 US gallon = 4 quarts = 128 US fluid ounces

13 Fundamental dimensions and units

1 US gallon = 0.8 British imperial gallons = 3.78833 litres

˚F

2500

2000

1500

1000

900

700

500

400

˚C

˚F

140

100

300

250

210

90

200

80

70

60

50

40

30

20

10

0

–10

190

180

170

160

150

140

130

120

110

100

90

80

70

60

50

40 +30 +20

–20

+10

0 0 –10

–30

–40

–50 –60 –70 –80 –90 –100

–20 –30 –40 –50 –60 –80 –100 –120 –140

˚C

–120

–140

–160

–180

Temperature

conversions

˚C

900

700

500

400

300

200

150

˚F

–160

–180

–250

–300

–350

–400

Table 2.5 Pressure (p)

Table 2.6 Heat

15

Table 2.7 Power (P)

Table 2.8 Velocity (v)

16 Aeronautical Engineer’s Data Book

2.3.9 Torque

The basic unit of torque is the foot pound (ft.lbf) (in SI it is the newton metre (N m)) You may also see this referred to as ‘moment of force’ (see Figure 2.5)

1 ft.lbf= 1.357 N m

1 kgf.m = 9.81 N m

2.3.10 Stress

for pressure although it is a different physical quantity In SI the basic unit is the pascal (Pa)

1 Pa is an impractically by small unit so MPa is normally used (see Figure 2.6)

2.3.11 Linear velocity (speed)

The basic unit of linear velocity (speed) is feet per second (in SI it is m/s) In aeronautics, the most common non-SI unit is the knot, which is equivalent to 1 nautical mile (1853.2 m) per hour See Table 2.8

2.3.12 Acceleration

The basic unit of acceleration is feet per second

Standard gravity (g) is normally taken as

2.3.13 Angular velocity

The basic unit is radians per second (rad/s)

1 rad/s = 0.159155 rev/s = 57.2958 degree/s The radian is also the SI unit used for plane angles

17 Fundamental dimensions and units

Force (N )

Radius (

r)

Torque = Nr

Fig 2.5 Torque

Area 1 m 2

1 MN

Fig 2.6 Stress

2 π radians

θ

Fig 2.7 Angular measure

18

Table 2.9 Area (A)

19 Fundamental dimensions and units

2.3.14 Length and area

Comparative lengths in USCS and SI units are:

1 ft = 0.3048 m

1 in = 25.4 mm

1 statute mile = 1609.3 m

1 nautical mile = 1853.2 m

Table 2.9

Small dimensions are measured in ‘micro­ measurements’ (see Figure 2.8)

The microinch (

Oil filter mesh

450 µin

Diameter of a hair: 2000 µin Smoke

particle

120 µin

A smooth-machined ‘mating’

–32 µin

1 micron ( µm) = 39.37µin

A fine ‘lapped’

with peaks within 1 µin

surface with peaks 16

surface

Fig 2.8 Micromeasurements

2.3.15 Viscosity

(Pa s)

A common unit of viscosity is the centipoise (cP) See Table 2.10

20 Aeronautical Engineer’s Data Book

Table 2.10 Dynamic viscosity ( )

1 lb (force)-s 1 4.788 4.788 4.882 per ft 2 2 10 4 2 10 2

1 centipoise 2.089 1 10 –2 1.020

1 poise 2.089 100 1 1.020

1 N-s per m 2 0.2048 9.807 98.07 1

2 10 3

viscosity

Kinematic viscosity = dynamic viscosity/

Saybolt Seconds Universal (SSU) are also used

< SSU < 100 seconds

> 100 seconds

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:

dynamic viscosity (µ)

333 = µ 2 1/#

3 = 3 2 3

OK, units match

� � � � �

33

21 Fundamental dimensions and units

2.5 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 has a value of unity

as are

33 or 33 or 33

Remember that, as the value of the term inside the bracket is unity, it has no effect on any term that it multiplies

Example:

0.16 lb

in3 Step 2: Apply the ‘weight’ unity bracket:

# = 3 3

Step 3: Then apply the ‘dimension’ unity brackets (cubed):

3

m

33 �

22 Aeronautical Engineer’s Data Book

Step 4: Expand and cancel*:

3

m

*Take care to use the correct algebraic rules for the expansion, e.g

Unity brackets can be used for all unit conver­ sions provided you follow the rules for algebra correctly

2.6 Imperial–metric conversions

See Table 2.11

2.7 Dimensional analysis

2.7.1 Dimensional analysis (DA) – what is it?

DA is a technique based on the idea that one physical quantity is related to others in a precise mathematical way

It is used in aeronautics for:

• Checking the validity of equations

• Finding the arrangement of variables in a formula

• Helping to tackle problems that do not possess a compete theoretical solution – particularly those involving fluid mechanics

2.7.2 Primary and secondary quantities

Primary quantities are quantities which are absolutely independent of each other They are:

23 Fundamental dimensions and units

Table 2.11 Imperial-metric conversions

Fraction Decimal Millimetre Fraction Decimal Millimetre

1/64 0.01562 0.39687 33/64 0.51562 13.09687 1/32 0.03125 0.79375 17/32 0.53125 13.49375 3/64 0.04687 1.19062 35/64 0.54687 13.89062 1/16 0.06250 1.58750 9/16 0.56250 14.28750 5/64 0.07812 1.98437 37/64 0.57812 14.68437 3/32 0.09375 2.38125 19/32 0.59375 15.08125 7/64 0.10937 2.77812 39/64 0.60937 15.47812 1/8 0.12500 3.17500 5/8 0.62500 15.87500 9/64 0.14062 3.57187 41/64 0.64062 16.27187 5/32 0.15625 3.96875 21/32 0.65625 16.66875 11/64 0.17187 4.36562 43/64 0.67187 17.06562 3/16 0.18750 4.76250 11/16 0.68750 17.46250 13/64 0.20312 5.15937 45/64 0.70312 17.85937 7/32 0.21875 5.55625 23/32 0.71875 18.25625 15/64 0.23437 5.95312 47/64 0.73437 18.65312 1/4 0.25000 6.35000 3/4 0.75000 19.05000 17/64 0.26562 6.74687 49/64 0.76562 19.44687 9/32 0.28125 7.14375 25/32 0.78125 19.84375 19/64 0.29687 5.54062 51/64 0.79687 20.24062 15/16 0.31250 7.93750 13/16 0.81250 20.63750 21/64 0.32812 8.33437 53/64 0.82812 21.03437 11/32 0.34375 8.73125 27/32 0.84375 21.43125 23/64 0.35937 9.12812 55/64 0.85937 21.82812 3/8 0.37500 9.52500 7/8 0.87500 22.22500 25/64 0.39062 9.92187 57/64 0.89062 22.62187 13/32 0.40625 10.31875 29/32 0.90625 23.01875 27/64 0.42187 10.71562 59/64 0.92187 23.41562 7/16 0.43750 11.11250 15/16 0.93750 23.81250 29/64 0.45312 11.50937 61/64 0.95312 24.20937 15/32 0.46875 11.90625 31/12 0.96875 24.60625 31/64 0.48437 12.30312 63/64 0.98437 25.00312 1/2 0.50000 12.70000 1 1.00000 25.40000

M Mass

L Length

T Time

For example, velocity (v) is represented by

length divided by time, and this is shown by:

‘the dimension of’

Table 2.12 shows the most commonly used quantities

24 Aeronautical Engineer’s Data Book

Table 2.12 Dimensional analysis quantities

Angular velocity (

Angular acceleration (

Density (

Kinematic viscosity (

M

L

T

L L L

3

3

4

T T T

–1 –2 –1

ML

–3

Hence velocity is called a secondary quantity because it can be expressed in terms of primary quantities

2.7.3 An example of deriving formulae using DA

To find the frequencies (n) of eddies behind a

cylinder situated in a free stream of fluid, we

can assume that n is related in some way to the diameter (d) of the cylinder, the speed (V) of

Introducing a numerical constant Y and some

possible exponentials gives:

c

n = Y{d a ,V b ,# , d}

Y is a dimensionless constant so, in dimensional

analysis terms, this equation becomes, after substituting primary dimensions:

� �

� 3 �

�

3

25 Fundamental dimensions and units

T–1 = L a (LT–1)b (ML–3)c (L2T–1)

In order for the equation to balance:

Solving for a, b, c in terms of d gives:

a = –1 –d

b = 1 –d

Giving

n = d (–1 –d) V (1 –d) #0 d

Rearranging gives:

nd/V = (Vd/ )X

Note how dimensional analysis can give the

‘form’ of the formula but not the numerical

value of the undetermined constant X which, in

this case, is a compound constant containing the

original constant Y and the unknown index d

2.8 Essential mathematics

2.8.1 Basic algebra

m+n

a

a m 2 a n

(a m)n = a

1

a = a

a o

(a n b m)p = a np b

a n a n

3 = 3

b n

b

n�ab = � n�a�2 n� �

3

n�a\b = n

�b

��

3

26 Aeronautical Engineer’s Data Book

2.8.2 Logarithms

N

log(ab) = log a + log b

a

b

�

n

loga

loge N = 2.3026 log10

2.8.3 Quadratic equations

x = 33

2a

b

a c

d

+

+ q)2

�

27 Fundamental dimensions and units

value of the function occurs when (x + q) = 0 and its value is r

of the function occurs when (x + q) = 0 and its value is r

2.8.4 Cubic equations

3

3

1

3

x = y – p

where

3

3

On setting

3)1/2]1/3

S = [–b + (b2

3)1/2]1/3

T = [–b – (b2+ a

the three roots are

x1= S + T –3313p

3

2

1

3

2

1

x3= – p

For real coefficients

+ a3< 0

there are alternative expressions:

3

3

3

x

3 3

3

3

1

3 ( + 4π) – p 3313

b

2.8.5 Complex numbers

33

3

c

the complex number z = x + iy consists of the real part x and the imaginary part iy

z = x – iy is the conjugate of the complex

number z = x + iy

28 Aeronautical Engineer’s Data Book

If x + iy = a + ib then x = a and y = b

(a + ib) + (c + id) = (a + c) = i(b + d) (a + ib) – (c + id) = (a – c) = i(b + d) (a + ib)(c + id) = (ac – bd) + i(ad + bc)

a + ib ac + bd bc – ad

33 = 3c + id c + d 2 32 + i 33c2+ d 2

Every complex number may be written in polar form Thus

x + iy = r(cos  + i sin ) = r

r is called the modulus of z and this may be

written r = |z|

�2

r = �x2+ y�

 is called the argument and this may be written

 = arg z

y

x

sin 2)

r

z1\z2=

r1

r2

2.8.6 Standard series

Binomial series n(n – 1)

2!

n(n – 1)(n – 2)

3!

+ (x2< a2)

The number of terms becomes inifinite when n

is negative or fractional

3 � � 3 � �

29 Fundamental dimensions and units

(b2 x2< a2)

Exponential series

x x

Logarithmic series

ln x = (x – 1) – 33 2(x – 1)2 + 33 3(x – 1)3 – (0

x – 1 x – 1 2 3

1 + �x > 332�

5

x – 1 1 x – 1 3 1 x – 1

ln x = 2[33 33x + 1 3 x + 1�33� + 335 x + 1 �33�

+ (x positive)

x x x

Trigonometric series

x x x

x x x

3

x 2x5 17x7 62x9

2

π + �x2< 334 �

5

sin–1 x = x + 33 33 + 33 + 33 + 33 33

+ (x2 < 1)

30 Aeronautical Engineer’s Data Book

tan–1 x = x – 33 x3+ 3 1 3 x5– 33 x7 + (x2 5 1)

2.8.7 Vector algebra

Vectors have direction and magnitude and

satisfy the triangle rule for addition Quantities

such as velocity, force, and straight-line displacements may be represented by vectors Three-dimensional vectors are used to repre­

Vector Addition

by

Product of a vector V by a scalar quantity s

sV = (sa)i + (sb)j + (sc)k

where sV has the same direction as V, and its magnitude is s times the magnitude of V

Scalar product of two vectors, V 1 ·V 2

Vector product of two vectors, V1 2 V2

Derivatives of vectors

33 (A · B) = A · 33 + B · 33

dt dt

dt de

dt