FLUID MECHANICS: AMME2261 summary DON’T FORGET TUTE QUESTIONS AVAILABLE THURSDAY FOR THE NEXT TUTE!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Week 1

Contents Module 1: Introduction to fluid properties ............................................................................................ 5

Definition of fluids: ............................................................................................................................. 5

Methods of analysing fluids: ............................................................................................................... 5

Methods of analysis: ........................................................................................................................... 5

Control mass: .................................................................................................................................. 5

Control volume: .................................................................................................................................. 5

Way to answer questions: ...................................................................................................................... 6

Idealised 1 dimensional fluid flow .......................................................................................................... 6

Continuity equation: ........................................................................................................................... 6

Reference frames: Lagrangian ................................................................................................................ 8

Eulerian: .................................................................................................................................................. 8

Pressure, density and continuum ........................................................................................................... 8

Viscosity: ................................................................................................................................................. 9

Dynamics viscosity .................................................................................................................................. 9

Kinematic viscosity: ......................................................................................................................... 9

Surface tension: .................................................................................................................................... 11

gases: .................................................................................................................................................... 12

Standard atmosphere: .......................................................................................................................... 12

Mamometers: ................................................................................................................................... 13

Hydrostatic force on a submerged surface ........................................................................................... 14

Submerged plane surface ................................................................................................................. 14

Centre of pressure ................................................................................................................................ 15

Moment areas of standard shapes ....................................................................................................... 16

Hydrostatic forces on submerged surfaces .......................................................................................... 18

Example ......................................................................................................................................... 18

Buoyancy: .............................................................................................................................................. 19

Example ............................................................................................................................................. 19

Trapezoidal rule: ............................................................................................................................... 20

Stability analysis .................................................................................................................................... 21

Unstable: ........................................................................................................................................... 21

Unconditionally/neutral stable ......................................................................................................... 21

G above B: ......................................................................................................................................... 22

Example ......................................................................................................................................... 23

Fundamentals of fluid dynamics ........................................................................................................... 24

Integral form of fluid dynamics: ........................................................................................................ 24

Conservation of mass ............................................................................................................................ 24

Special forms of flow: ....................................................................................................................... 24

Volume flow rate: ............................................................................................................................. 25

Mass flow rate: ................................................................................................................................. 25

Example ......................................................................................................................................... 25

Conservation of linear momentum ....................................................................................................... 25

Special forms of flow: ....................................................................................................................... 26

Example ......................................................................................................................................... 26

Conservation of angular momentum .................................................................................................... 27

Differential forms of flow ...................................................................................................................... 28

Conservation of mass/ continuity equation ..................................................................................... 28

Cylindrical coordinates: ................................................................................................................. 28

Conservation of linear momentum ................................................................................................... 28

Shear stress in 3D: ......................................................................................................................... 29

Naviar stokes equation: ........................................................................................................................ 29

Fully developed flow: ........................................................................................................................ 29

Euler equation, inviscid fluids: (easier to solve) ................................................................................... 29

Dimensional analysis ............................................................................................................................. 31

Buckingham pi theorem: ....................................................................................................................... 31

To determine pi groups:.................................................................................................................... 32

Significant Π groups: ......................................................................................................................... 32

Reynolds number: ................................................................................................................................. 32

Euler number: ....................................................................................................................................... 33

Froude number: .................................................................................................................................... 33

Weber number: ..................................................................................................................................... 33

Flow similarity: ...................................................................................................................................... 33

Incomplete similarlity: .......................................................................................................................... 33

Inviscid flow .......................................................................................................................................... 35

Euler equation ................................................................................................................................... 35

Inviscid flow over wings: ............................................................................................................... 35

Euler equation along a streamline: ................................................................................................... 35

Streamline: .................................................................................................................................... 35

Euler equation along streamline steady flow: .............................................................................. 36

Bernoulli equation ............................................................................................................................ 37

Example ......................................................................................................................................... 37

Static, stagnation and dynamic pressure .......................................................................................... 38

Static pressure ...................................................................................................................................... 38

Stagnation pressure: ......................................................................................................................... 38

Measuring velocity: ........................................................................................................................... 38

Module 5: potential flow theory ........................................................................................................... 39

Stream function: ............................................................................................................................... 39

Definition of stream function ........................................................................................................... 39

Stream function in polar coordinates: .......................................................................................... 40

Potential function: ................................................................................................................................ 42

Definition of Potential function 𝜙𝑥, 𝑦, 𝑡: .......................................................................................... 42

Potential function polar coordinates ............................................................................................ 42

Laplave’s equation: ............................................................................................................................... 42

Laplace equation: .............................................................................................................................. 42

Example: ........................................................................................................................................ 43

Elementary plane flow: ......................................................................................................................... 43

Uniform flow: .................................................................................................................................... 43

Source flow: ...................................................................................................................................... 43

Sink: ................................................................................................................................................... 44

Irrotational vortex: ............................................................................................................................ 45

Doublet ............................................................................................................................................. 45

Superposition of elementary plane flows: ............................................................................................ 45

Direct method (simple approach) ..................................................................................................... 46

Flow past a bluff body: ...................................................................................................................... 46

Rankine body (source, sink, uniform) ............................................................................................... 47

Example: flow over a cylinder ....................................................................................................... 47

Turbomachinary of inviscid fluids ......................................................................................................... 48

Pumps/fans/ blowers/compressors: ............................................................................................. 48

Positive displacement pumps: .......................................................................................................... 48

Dynamic pumps: ............................................................................................................................... 49

Comparison of pump types ........................................................................................................... 50

Euler turbomachine equations: ........................................................................................................ 50

Torque: .......................................................................................................................................... 50

Power: ........................................................................................................................................... 51

Head rise/drop: ............................................................................................................................. 51

Radial flow turbomachiary: ............................................................................................................... 51

Viscous flow: ......................................................................................................................................... 52

Internal flow development: .............................................................................................................. 52

For laminar flow: ........................................................................................................................... 53

Turbulent flow: ............................................................................................................................. 53

Transition to turbulence: .............................................................................................................. 53

Fully developed laminar flow in a pipe: ............................................................................................ 53

Laminar pipe flow equations: ........................................................................................................... 54

Reduced naviar stokes: ................................................................................................................. 54

Velocity distribution: ..................................................................................................................... 54

Shear stress: .................................................................................................................................. 54

Volumetric flow rate: .................................................................................................................... 54

Pressure gradient: ......................................................................................................................... 54

Mean velocity: ............................................................................................................................... 54

Max velocity: ................................................................................................................................. 54

Introduction to external viscous flow: .............................................................................................. 56

Pitch, roll, yaw/side,lift,drag ......................................................................................................... 57

Coefficient of lift and drag ................................................................................................................ 57

Drag: .................................................................................................................................................. 57

Drag force ...................................................................................................................................... 58

Drag on a sphere: .......................................................................................................................... 58

Steamlining: ...................................................................................................................................... 59

Common drag coefficients: ............................................................................................................... 60

Lifting bodies: Wing .......................................................................................................................... 62

Lift and drag as a function of angle of attack.................................................................................... 63

Flaps .............................................................................................................................................. 63

Module 1: Introduction to fluid properties

Definition of fluids: - A solid can resist shear stresses, and will undergo static deflection (up to a point) if a stress is

applied

- A fluid cannot resist shear stress, and will translate (move) if stress applied

o Liquids: are incompressible and will retain their volume

o Gases: can be compressed and will take the volume of their container

o Fluids are not elastic, but have viscosity

Methods of analysing fluids: 1. Analytical analysis

o Uses equations such as:

Conservation of mass

Newton’s equations

Conservation of angular momentum

1st and 2nd laws of thermodynamics

To analysis and give exact answers to fluid analysic

2. Computational/numerical (CFD)

3. Experimental

o Partial image velocimetry

o Streak/smoke lines

Methods of analysis: 1st step: define the system involved (boundaries, forces, ect)

Usually either control mass or control volme:

Control mass: - Fixed mass of fluid, fluid mass does not cross boundaries, mass doesn’t change, volume can

- Eg: piston is control mass

Control volume: - Fixed volume, mass can change but volume constant; mass flux across boundaries

- Eg: pipe junction

-

Way to answer questions: 1. Diagram, labelling control surface, control volume/mass, inlets/outlets, forces ect

2. Write assumptions: eg constant densit

3. Start with fundamental equations

4. Simplify, finally add numbers

Idealised 1 dimensional fluid flow

Continuity equation: If flow is a ‘continuum’ (the difference between the fluid little volumes is ‘smooth’, can be called a

continuum

𝜌1𝐴1𝑉1 = 𝜌2𝐴2𝑉2

Eg 1.1:

ANSWER:

𝐴𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛𝑠: 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 (𝑤𝑎𝑡𝑒𝑟 𝑖𝑠 𝑢𝑠𝑢𝑎𝑙𝑙𝑦 𝑎 𝑔𝑜𝑜𝑑 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑡ℎ𝑖𝑠)

�̇�1 = �̇�2 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑜𝑖𝑛𝑡 1 𝑎𝑛𝑑 2)

𝐶𝑜𝑛𝑖𝑛𝑢𝑖𝑡𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛:

𝜌1𝐴1𝑣1 = 𝜌2𝐴2𝑉2

∴ 𝑉2 =𝐴1𝑉1

𝐴2 (𝜌1 = 𝜌2 (𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛))

=𝜋 (

𝐷12

4 ) 𝑉1

𝜋 (𝐷2

2

4 )

(𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒)

= (𝐷1

𝐷2)

2

𝑉2

= (50

30)

2

(2.5)𝑚

𝑠

= 6.9 𝑚𝑠−1

Average speed: is the average speed of all the particles along the inlet/outlet (speed of 0 in contact

of pipe, higher speed in centre)

2 1

Control system

Control volume 𝑣2 𝑣1

Reference frames: Lagrangian

- Considers elemental globs of fluid (control masses), and forces are solved for each glob

- - Moving reference frame

-

- This is very time costly though, solving ∑�⃗� = 𝑚�⃗�; 𝑡𝑜 𝑔𝑒𝑡 �⃗⃗�(𝑡) 𝑎𝑛𝑑 𝑟(𝑡) for every particle.

- Can be used to analyse discrete phases (eg- a water spray)

Eulerian: - Fixed reference frame

- Make a grid, and monitor the flow through each section of grid

�⃗⃗� = 𝑉(𝑟, 𝑡 )

Pressure, density and continuum Fluids are aggregations of molecules, and the distance between molecules can be very large

compared to molecular diameter

- Density on a small scale does not have much meaning, due to microscopic uncertainty.

- Microscopic uncertainty diminishes when you increase the volume and is large compared to

molecular spacing ≈ 10−9𝑚 (most problems will be above microscopic uncertainty)

- For very large observations, there can be smooth variations in density too, called

macroscopic uncertainty (eg: density difference in a room, slightly higher on the floor than

the ceiling)

- This fluid is called a continuum, where the variation in fluid property is smooth enough to

perform calculus on. (at very low pressure (eg- atmosphere renty), the molecular spacing

can become too large, as the spacing is comparable to the system size, and so molecular

theory of rarefied gas flow must be used)

Viscosity:

Dynamics viscosity - Measure of a fluid’s resistance to shear stress

- If a stress of 𝜏 is applied, in Newtonian fluids, there is a linear relationship between shear

and resulting strain rate. The top surface moved 𝑑𝑢, and the bottom surface is static

𝜏 ∝𝑑𝜃

𝑑𝑡 =

𝜇𝑑𝜃

𝑑𝑡

∴

∴ 𝝉 = 𝝁𝒅𝜽

𝒅𝒕= 𝝁

𝒅𝒖

𝒅𝒚

(𝑢 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑛 𝑠𝑡𝑟𝑒𝑠𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛; 𝑦 𝑖𝑠 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 ℎ𝑒𝑖𝑔ℎ𝑡) 𝑤ℎ𝑒𝑟𝑒 𝜇 𝑖𝑠 𝑑𝑦𝑛𝑎𝑚𝑖𝑐𝑠 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦

𝜇 𝑖𝑠 𝑖𝑛: 𝑘𝑔. 𝑚−1. 𝑠−1 𝑜𝑟 𝑃𝑎. 𝑠

Kinematic viscosity: Ratio of dynamics viscosity to density

𝜈 =𝜇

𝜌, 𝑖𝑠 𝑖𝑛

𝑚2

𝑠

Example 1.2

𝐴𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛𝑠: 𝑐𝑜𝑢𝑒𝑡𝑡𝑒 𝑓𝑙𝑜𝑤 (𝑎𝑠 𝐷 ≫ ℎ, 𝑖𝑡 𝑐𝑎𝑛 𝑏𝑒 𝑚𝑜𝑑𝑒𝑙𝑙𝑒𝑑 𝑎𝑠 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑛𝑔 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙𝑙𝑦)

𝜏 = 𝜇𝑑𝑢

𝑑𝑦= 𝜇

𝑢

ℎ

𝑎𝑠 𝐶𝑜𝑢𝑒𝑡𝑡𝑒 𝑓𝑙𝑜𝑤: 𝑢𝑝𝑝𝑒𝑟 𝑝𝑙𝑎𝑡𝑒 𝑚𝑜𝑣𝑖𝑛𝑔, 𝑏𝑜𝑡𝑡𝑜𝑚 𝑖𝑠 𝑠𝑡𝑎𝑖𝑡𝑜𝑛𝑎𝑟𝑦 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡𝑜 𝑖𝑡

𝑢 = 𝜔 (𝐷

2) = (

2𝜋𝜔

60) (

𝐷

2) =

𝜋𝜔𝐷

60

∴ 𝜏 =𝜇 (

𝜋𝜔𝐷60 )

ℎ

𝜏 =𝐹

𝐴=

(𝑇𝐷2

)

𝐴

𝑇 = 𝜏𝐴 (𝐷

2) = 𝜏(𝜋𝐷𝐿) (

𝐷

2) =

1

2𝜋𝜏𝐷2𝐿

𝐷

2

ℎ

𝑈𝑥

𝑈𝑥

=1

2(

𝜇 (𝜋𝜔𝐷

60 )

ℎ ) 𝜋𝐷2𝐿

=𝜋2𝜇𝜔𝐷3𝐿

120

Non newtonima fluids do not have linear relationships between 𝜏 and deformation rate

𝜏 = 𝜂𝑑𝑢

𝑑𝑦; 𝜂 = 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦

Surface tension: Fluid behaves like elastic membraine in tension: between a fluid and another fluid or solid

𝐹𝑜𝑟𝑐𝑒 𝑜𝑓 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑡𝑒𝑛𝑠𝑖𝑜𝑛: 𝐹𝑡 = 𝜎(𝑙)

𝜎 = 𝑓𝑜𝑟𝑐𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ; 𝑙 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑎𝑟𝑒𝑎 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑖𝑛𝑒𝑟𝑓𝑎𝑐𝑒

fluid statics:

dP

dz= −ρg = −γ (specific weight)

P − P0 = −ρg h

gases:

𝑃 = 𝑃0𝑒−𝑔ℎ𝑅𝑇

Standard atmosphere:

Pressure is function of depth, and does not depend on geometry

Hydrolic jack:

𝐹1

𝐹2=

𝐴1

𝐴2

Mamometers:

𝑃3 = 𝑃𝑎𝑡𝑚 + 𝜌𝑔ℎ

𝑃4 = 𝑃𝑎𝑡𝑚 + 𝜌𝐻20𝑔ℎ − 𝜌𝑎𝑖𝑟𝑔ℎ2

𝑊 = 𝑃1𝐴 = 𝜌𝑔ℎ

∴ 𝐹𝑜𝑟𝑐𝑒 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 = 𝑃2𝐴 − 𝑊 = 𝜌𝑔(ℎ2 − ℎ1)

Hydrostatic force on a submerged surface

Submerged plane surface

𝐹𝑅 = ∫ 𝑑𝐹𝐴

= ∫ 𝑃𝑑𝐴𝐴

= ∫ (𝑃0 + 𝜌𝑔ℎ)𝑑𝐴 = ∫ (𝑃0 + 𝜌𝑔𝑠𝑖𝑛𝜃𝑦)𝑑𝐴

= 𝑃0𝐴 + 𝜌𝑔 𝑠𝑖𝑛𝜃 ∫ 𝑦𝐴

𝑑𝐴

(∫ 𝑦𝐴

𝑑𝐴 = 𝑦𝑐𝐴 = 1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑟𝑒𝑎)

𝐹𝑅 = (𝑃0 + 𝜌𝑔𝑠𝑖𝑛𝜃𝑦𝑐)𝐴 = 𝑃𝑐𝐴 (𝑃𝑐 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑)

Note: if air was on both sides of the surface (eg, at a gate): 𝑃𝑐 =

𝜌𝑔𝑠𝑖𝑛𝜃𝑦𝑐

Eg:

𝐹𝑟 = ∫𝑝𝑑𝐴𝐴

= ∫𝜌𝑔(𝐷 + 𝜂 sin 30)𝑤𝑑𝜂𝐴

(𝑛𝑜𝑡𝑒: 𝑃𝑎𝑡𝑚𝑐𝑎𝑛𝑐𝑒𝑙𝑠 𝑎𝑠 𝑜𝑛 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓 𝑔𝑎𝑡𝑒)

= ∫ 𝜌𝑔(𝐷 + 𝜂 sin 30)𝑤𝑑𝜂𝐿

0

= 𝜌𝑔 (𝐷𝜂 +𝜂2

4)

0

𝐿

Centre of pressure Even though 𝐹𝑅 is calculated with

𝑦𝑐 (𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎𝑟𝑒𝑎), 𝐹𝑅 𝑎𝑐𝑡𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑦′ (𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒)

To find centre of pressure:

∑𝑀0 = 0

∴

𝑦′𝐹𝑅 = ∫ 𝑦𝑃𝑑𝐴𝐴

=

= ∫ 𝑦(𝑃0 + 𝜌𝑔ℎ)𝑑𝐴𝐴

= ∫ 𝑦(𝑃0 + 𝜌𝑔𝑦𝑠𝑖𝑛𝜃)𝑤𝑑𝐿𝐴

= 𝑃0 ∫ 𝑦𝑑𝐴𝐴

+ 𝜌𝑔𝑠𝑖𝑛𝜃 ∫ 𝑦2𝑑𝐴𝐴

(1𝑠𝑡 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑟𝑒𝑎) + (2𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎𝑏𝑜𝑢𝑡 𝑥: 𝐼𝑥𝑥)

𝑤𝑒 𝑐𝑎𝑛 𝑐ℎ𝑎𝑛𝑔𝑒 𝐼𝑥𝑥𝑖𝑛𝑡𝑜 𝑎 𝑚𝑜𝑚𝑒𝑛𝑡 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑖𝑛𝑠𝑡𝑒𝑒𝑑 𝑜𝑓 𝑎𝑏𝑜𝑢𝑡 𝑂

∴ 𝐼𝑥𝑥 = 𝐼�̂��̂� + 𝐴𝑦𝑐2

…

∴ 𝑦′ = 𝑦𝑐 +𝜌𝑔𝑠𝑖𝑛𝜃𝐼𝑥𝑥

𝐹𝑟

Moment areas of standard shapes

Hydrostatic forces on submerged surfaces 𝑑𝑭 = −𝑃𝑑𝑨

𝑭𝑹 = − ∫ 𝑃𝑑𝑨𝒙𝑨𝒙

− ∫ 𝑃𝑑𝑨𝒚𝑨𝒚

− ∫ 𝑃𝑑𝑨𝒛𝑨𝒛

𝑭𝑹 = 𝐹𝑥𝒊 + 𝐹𝑦𝒋 + 𝐹𝑧𝒌

the resultant:

𝐹𝑅 = 𝐹𝑣 + 𝐹𝐻

Example

↻ +∶ ∑𝑀0 = 0 (𝑖𝑓 𝑔𝑎𝑡𝑒 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑)

∴ 𝐹𝑎𝑙 − 𝐹𝐻𝑦′ − 𝐹𝑉𝑥′ = 0

𝐹𝐻 = 𝑃𝑐𝐴 = 𝜌𝑔𝐷

2× 𝐷𝑤 = 396 𝑘𝑁

𝑦′ = 𝑦𝑐∗ +

𝐼𝑥�̂�

𝐴𝑦𝑐∗

=𝐷

2+

𝑤𝐷3

12

𝐷𝑤𝐷2

= 2.67𝑚

∴ 𝑦′ = 4 − 𝑦′ = 1.33

𝐹𝑣 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 = 𝜌𝑔 ∫ (𝑃 − 𝑦)𝑤𝑑𝑥

𝐷2

4

0

= 𝜌𝑔𝑤

𝑥′ = 1.2

Buoyancy:

Buoyancy is the net vertical pressure acting on the object

𝑑𝐹𝑏 = 𝜌𝑔(ℎ2 − ℎ1)𝑑𝐴 = 𝜌𝑔𝑑ℎ𝑑𝐴

𝑑𝐹𝑏 = 𝜌𝑔𝑑𝑉

𝐹𝑏 = 𝜌𝑔𝑉 (𝑤ℎ𝑒𝑟𝑒 𝑉 𝑖𝑠 𝑡ℎ𝑒 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 𝑣𝑜𝑙𝑢𝑚𝑒)

Example A hot air balloon is to lift a basket and payload weighing 270 kg. The balloon may be approximated

as a sphere of diameter of 16 m. To what temperature must the air be heated in order to achieve

lift-off?

𝐹𝑏

𝑊𝑔𝑎𝑠 𝑊𝑏𝑎𝑠𝑘𝑒𝑡

If in equilibrium:

𝐹𝑏 = 𝑊𝑏 + 𝑊𝑔

∴ 𝜌ℎ𝑜𝑡 𝑎𝑖𝑟𝑔𝑉 = 𝑚𝑏𝑎𝑠𝑘𝑒𝑡𝑔 + 𝑚𝑔𝑎𝑠𝑔 = 𝑔(𝑚𝑏 + 𝜌𝑎𝑡𝑚𝑉)

𝜌ℎ = 𝜌𝑎𝑡𝑚 +𝑚𝑏

𝑉

As ideal gas:

𝜌1𝑉1

𝑇1=

𝜌2𝑉2

𝑇2

∴ 𝑇2 =𝜌2𝑇1

𝜌1

In ship design and operation, buoyancy is a critical parameter. Will the ship float? How much more

load can be added? Ships’ hulls are rarely nice simple shapes. There is a tradition of designing ships

with “fair lines” for hydrodynamic (e.g. drag) and aesthetic reasons. Example lines plan on the next

page, and example table of offsets is on the next after that. It is generally not possible to analytically

integrate the buoyancy force over the irregular shaped volume that is floating. Numerical integration

(quadrature) is required. Simple quadrature methods are Simpson’s Rule and the Trapezoidal Rule.

Trapezoidal rule:

𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙 ≈ ℎ (𝑓(𝑥0)

2+ 𝑓(𝑥1) + ⋯ + 𝑓(𝑥𝑛−1) +

𝑓(𝑥𝑛)

2)