Ship Stability for Masters and Mates 5 Episode 2 ppt

Since 1000 kilograms of fresh water occupyone cubic metre, the volume of the stem between X and Y is equal to ˆMyÿ Mx 1000 L sq mNow let the hydrometer ¯oat in water of density d kg/m3 w

The variable immersion hydrometer

The variable immersion hydrometer is an instrument, based on the Law ofArchimedes, which is used to determine the density of liquids The type ofhydrometer used to ®nd the density of the water in which a ship ¯oats isusually made of a non-corrosive material and consists of a weighted bulbwith a narrow rectangular stem which carries a scale for measuring densitiesbetween 1000 and 1025 kilograms per cubic metre, i.e 1.000 and 1.025 t/

m3

The position of the marks on the stem are found as follows First let thehydrometer, shown in Figure 4.4, ¯oat upright in fresh water at the mark X.Take the hydrometer out of the water and weigh it Let the mass be Mx

kilograms Now replace the hydrometer in fresh water and add lead shot inthe bulb until it ¯oats with the mark Y, at the upper end of the stem, in the

Fig 4.4

waterline Weigh the hydrometer again and let its mass now be Mykilograms.

The mass of water displaced by the stem between X and Y is thereforeequal to Myÿ Mx kilograms Since 1000 kilograms of fresh water occupyone cubic metre, the volume of the stem between X and Y is equal to

ˆMyÿ Mx

1000 L sq mNow let the hydrometer ¯oat in water of density d kg/m3 with thewaterline `x' metres below Y

Volume of water displaced ˆ My

Volume of water displaced ˆ Mass of water displaced

Density of water displaced



In this equation, My, Mx and L are known constants whilst d and x arevariables Therefore, to mark the scale it is now only necessary to selectvarious values of d and to calculate the corresponding values of x

Laws of ¯otation 25

Tonnes per Centimetre Immersion (TPC)

The TPC for any draft is the mass which must be loaded or discharged tochange a ship's mean draft in salt water by one centimetre, where

TPC ˆwater-plane area

100  density of water

; TPC ˆWPA

100  rWPA is in m2

r is in t/m3

Consider a ship ¯oating in salt water at the waterline WL as shown inFigure 4.5 Let `A' be the area of the water-plane in square metres.Now let a mass of `w' tonnes be loaded so that the mean draft isincreased by one centimetre The ship then ¯oats at the waterline W1 L1.Since the draft has been increased by one centimetre, the mass loaded isequal to the TPC for this draft Also, since an extra mass of water equal tothe mass loaded must be displaced, then the mass of water in the layerbetween WL and W1 L1 is also equal to the TPC

Mass ˆ Volume  Density

WPA100TPC in dock water

Note When a ship is ¯oating in dock water of a relative density other than1.025 the weight to be loaded or discharged to change the mean draft by 1centimetre (TPCdw) may be found from the TPC in salt water (TPCsw) bysimple proportion as follows:

TPCdw

TPCsw ˆrelative density of dock water …RDdw†

relative density of salt water …RDsw†or

TPCdw ˆRDdw

1:025 TPCsw

Fig 4.5

Reserve buoyancy

It has already been shown that a ¯oating vessel must displace its ownweight of water Therefore, it is the submerged portion of a ¯oating vesselwhich provides the buoyancy The volume of the enclosed spaces abovethe waterline are not providing buoyancy but are being held in reserve Ifextra weights are loaded to increase the displacement, these spaces abovethe waterline are there to provide the extra buoyancy required Thus, reservebuoyancy may be de®ned as the volume of the enclosed spaces above thewaterline It may be expressed as a volume or as a percentage of the totalvolume of the vessel

Example 1

A box-shaped vessel 105 m long, 30 m beam, and 20 m deep, is ¯oating upright

in fresh water If the displacement is 19 500 tonnes, ®nd the volume of reserve buoyancy.

Volume of water displaced ˆDensityMass ˆ 19 500 cu: m

Volume of vessel ˆ 105  30  20 cu: m

ˆ 63 000 cu: m Reserve buoyancy ˆ Volume of vessel N volume of water displaced Ans Reserve buoyancy ˆ 43 500 cu m

Example 2

A box-shaped barge 16 m  6 m  5 m is ¯oating alongside a ship in fresh water at a mean draft of 3.5 m The barge is to be lifted out of the water and loaded on to the ship with a heavy-lift derrick Find the load in tonnes borne by the purchase when the draft of the barge has been reduced to 2 metres.

Note By Archimedes' Principle the barge suffers a loss in mass equal to the mass of water displaced The mass borne by the purchase will be the difference between the actual mass of the barge and the mass of water displaced at any draft, or the difference between the mass of water originally displaced by the barge and the new mass of water displaced.

Mass of the barge ˆ Original mass of water displaced

ˆ Volume  density

ˆ 16  6  3:5  1 tonnes Mass of water displace at 2 m draft ˆ 16  6  2  1 tonnes

; Load borne by the purchase ˆ 16  6  1  …3:5 ÿ 2† tonnes Ans ˆ 144 tonnes

Example 3

A cylindrical drum 1.5 m long and 60 cm in diameter has mass 20 kg when empty Find its draft in water of density 1024 kg per cu m if it contains 200

Laws of ¯otation 27

litres of paraf®n of relative density 0.6, and is ¯oating with its axis dicular to the waterline (Figure 4.6).

perpen-Note The drum must displace a mass of water equal to the mass of the drum plus the mass of the paraf®n.

Density of the paraffin ˆ SG  1000 kg per cu m

ˆ 600 kg per cu m Mass of the paraffin ˆ Volume  density ˆ 0:2  600 kg

ˆ 120 kg Mass of the drum ˆ 20 kg

Total mass ˆ 140 kg Therefore the drum must displace 140 kg of water.

Volume of water displaced ˆDensityMass ˆ1024140 cu m

Volume of water displaced ˆ 0:137 cu m

Let d ˆ draft, and r ˆ radius of the drum, where r ˆ 602 ˆ 30 cm ˆ 0:3 m.

Fig 4.6

Volume of water displaced (V) ˆ pr 2 d or

Homogeneous logs of rectangular section

The draft at which a rectangular homogeneous log will ¯oat may be found

as follows:

Mass of log ˆ Volumedensity

ˆ L  B  D  SG of log1000 kgMass of water displaced ˆ Volumedensity

ˆ L  B  d  SG of water1000 kgBut Mass of water displaced ˆ Mass of log

; L  B  d  SG of water  1000 ˆ L  B  D  SG of log  1000or

d  SG of water ˆ D  SG of log

DraftDepthˆ

If BM is equal to12 db2 determine if this log will ¯oat with two of its sides parallel to the waterline.

Note The centre of gravity of a homogeneous log is at its geometrical centre See Figure 4.7

Draft Depthˆ

Relative density of log Relative density of water Draft ˆ0:6  0:81

Laws of ¯otation 29

Laws of ¯otation 31

Exercise 4

1 A drum of mass 14 kg when empty, is 75 cm long, and 60 cm in diameter Find its draft in salt water if it contains 200 litres of paraf®n of relative density 0.63.

2 A cube of wood of relative density 0.81 has sides 30 cm long If a mass of

2 kg is placed on the top of the cube with its centre of gravity vertically over that of the cube, ®nd the draft in salt water.

3 A rectangular tank (3 m  1.2 m  0.6 m) has no lid and is ¯oating in fresh water at a draft of 15 cm Calculate the minimum amount of fresh water which must be poured into the tank to sink it.

4 A cylindrical salvage buoy is 5 metres long, 2.4 metres in diameter, and

¯oats on an even keel in salt water with its axis in the water-plane Find the upthrust which this buoy will produce when fully immersed.

5 A homogeneous log of rectangular cross-section is 30 cm wide and 25 cm deep The log ¯oats at a draft of 17 cm Find the reserve buoyancy and the distance between the centre of buoyancy and the centre of gravity.

6 A homogeneous log of rectangular cross-section is 5 m long, 60 cm wide,

40 cm deep, and ¯oats in fresh water at a draft of 30 cm Find the mass of the log and its relative density.

7 A homogeneous log is 3 m long, 60 cm wide, 60 cm deep, and has relative density 0.9 Find the distance between the centres of buoyancy and gravity when the log is ¯oating in fresh water.

8 A log of square section is 5 m  1 m  1 m The relative density of the log

is 0.51 and it ¯oats half submerged in dock water Find the relative density

of the dock water.

9 A box-shaped vessel 20 m  6 m  2.5 m ¯oats at a draft of 1.5 m in water

of density 1013 kg per cu m Find the displacement in tonnes, and the height of the centre of buoyancy above the keel.

10 An empty cylindrical drum 1 metre long and 0.6 m in diameter has mass

20 kg Find the mass which must be placed in it so that it will ¯oat with half of its volume immersed in (a) salt water, and (b) fresh water.

11 A lifeboat, when fully laden, displaces 7.2 tonnes Its dimensions are 7.5 m  2.5 m  1 m, and its block coef®cient 0.6 Find the percentage of its volume under water when ¯oating in fresh water.

12 A homogeneous log of relative density 0.81 is 3 metres long, 0.5 metres square cross-section, and is ¯oating in fresh water Find the displacement

of the log, and the distance between the centres of gravity and buoyancy.

13 A box-shaped barge 55 m  10 m  6 m is ¯oating in fresh water on

an even keel at 1.5 m draft If 1800 tonnes of cargo is now loaded,

®nd the difference in the height of the centre of buoyancy above the keel.

14 A box-shaped barge 75 m  6 m  4 m displaces 180 tonnes when light If

360 tonnes of iron are loaded while the barge is ¯oating in fresh water,

®nd her ®nal draft and reserve buoyancy.

15 A drum 60 cm in diameter and 1 metre long has mass 30 kg when empty.

If this drum is ®lled with oil of relative density 0.8, and is ¯oating in fresh water, ®nd the percentage reserve buoyancy.

Mass ˆ Volume  Density

If the density of the water increases, then the volume of water displacedmust decrease to keep the mass of water displaced constant, and vice versa

The effect on box-shaped vessels

New mass of water displaced ˆ Old mass of water displaced

; New volume  new density ˆ Old volume  Old density

New volumeOld volume ˆ

Old densityNew densityBut volume ˆ L  B  draft

;L  B  New draft

L  B  Old draft ˆ

Old densityNew density

Old draft ˆ

Old densityNew density

Example 1

A box-shaped vessel ¯oats at a mean draft of 2.1 metres, in dock water of density 1020 kg per cu m Find the mean draft for the same mass displacement

in salt water of density 1025 kg per cubic metre.

New draft Old draft ˆ

Old density New density New draft ˆNew densityOld density  Old draft

ˆ10201025 2:1 m

ˆ 2:09 m Ans New draft ˆ 2:09 m

Example 2

A box-shaped vessel ¯oats upright on an even keel as shown in fresh water of density 1000 kg per cu m, and the centre of buoyancy is 0.50 m above the keel Find the height of the centre of buoyancy above the keel when the vessel is

¯oating in salt water of density 1025 kg per cubic metre.

Note The centre of buoyancy is the geometric centre of the underwater volume and for a box-shaped vessel must be at half draft, i.e KB ˆ 1

Old density New density New draft ˆ Old draft New densityOld density

ˆ 1 10001025 New draft ˆ 0:976 m New KB ˆ 1

2 new draft Ans New KB ˆ 0:488 m, say 0.49 m.

Fig 5.1

The effect on ship-shaped vessels

It has already been shown that when the density of the water in which avessel ¯oats is changed the draft will change, but the mass of water in kg ortonnes displaced will be unchanged i.e

New displacement ˆ Old displacementor

New volume  new density ˆ Old volume  old density

; New volumeOld volume ˆ

Old densityNew densityWith ship-shapes this formula should not be simpli®ed further as it was inthe case of a box-shape because the underwater volume is not rectangular

To ®nd the change in draft of a ship-shape due to change of density aquantity known as the `Fresh Water Allowance' must be known

The Fresh Water Allowance is the number of millimetres by which themean draft changes when a ship passes from salt water to fresh water,

or vice versa, whilst ¯oating at the loaded draft It is found by theformula:

FWA (in mm) ˆDisplacement (in tonnes)

4  TPCThe proof of this formula is as follows:

To show that FWA (in mm) ˆDisplacement (in tonnes)4  TPC

Consider the ship shown in Figure 5.2 to be ¯oating at the load Summerdraft in salt water at the waterline WL Let V be the volume of salt waterdisplaced at this draft

Now let W1L1 be the waterline for the ship when displacing the samemass of fresh water Also, let `v' be the extra volume of water displaced infresh water

Effect of density on draft and displacement 35

Fig 5.2

The total volume of fresh water displaced is then V ‡ v.

Mass ˆ Volume  density

; Mass of SW displaced ˆ 1025 V

and mass of FW displaced ˆ 1000 …V ‡ v†

but mass of FW displaced ˆ mass of SW displaced

; 1000 …V ‡ v† ˆ 1025 V

1000 V ‡ 1000 v ˆ 1025 V

1000 v ˆ 25 V

v ˆ V=40Now let w be the mass of salt water in volume v, in tonnes and let W bethe mass of salt water in volume V, in tonnes

; w ˆ W=40but w ˆFWA

10  TPCFWA

10  TPC ˆ W=40or

4  TPC mmwhere

W ˆ Loaded salt water displacement in tonnes

Figure 5.3 shows a ship's load line marks The centre of the disc is at adistance below the deck line equal to the ship's Statutory Freeboard Then

540 mm forward of the disc is a vertical line 25 mm thick, with horizontallines measuring 230  25 mm on each side of it The upper edge of theone marked `S' is in line with the horizontal line through the disc andindicates the draft to which the ship may be loaded when ¯oating in saltwater in a Summer Zone Above this line and pointing aft is another linemarked `F', the upper edge of which indicates the draft to which the shipmay be loaded when ¯oating in fresh water in a Summer Zone If loaded

to this draft in fresh water the ship will automatically rise to `S' when shepasses into salt water The perpendicular distance in millimetres betweenthe upper edges of these two lines is therefore the ship's Fresh WaterAllowance

When the ship is loading in dock water which is of a density betweenthese two limits `S' may be submerged such a distance that she willautomatically rise to `S' when the open sea and salt water is reached The

distance by which `S' can be submerged, called the Dock Water Allowance, isfound in practice by simple proportion as follows:

Let x ˆ The Dock Water AllowanceLet rDWˆ Density of the dock waterThen

x mm

1025 rDW

1025 1000or

Dock Water Allowance ˆ FWA …1025 rDW†

Let x ˆ The change in draft in millimetres Then x

FWAˆ

1025 1010 25

x ˆ 150 1525

x ˆ 90 mm Ans Draft will decrease by 90 mm, i.e 9 cm

Note This ship is obviously listed to port and if brought upright the lower edge of the `S' load line on each side would be 25 mm above the waterline Also, it is the upper edge of the line which indicates the `S' load draft and, since the line is 25 mm thick, the ship's draft must be increased by 50 mm to bring her to the `S' load line in dock water In addition `S' may be submerged by

x mm.

x FWAˆ

1025 rDW25

x ˆ 62:5 20

25

x ˆ 50 mm

; Total increase in draft required ˆ 100 mm or 10 cm

and cargo to load ˆ Increase in draft  TPC

ˆ 10  15 Ans Cargo to load ˆ 150 tonnes

Effect of density on displacement when the draft

is constant

Should the density of the water in which a ship¯oats be changed withoutthe shipaltering her draft, then the mass of water displaced must have

Fig 5.5

changed The change in the mass of water displaced may have beenbrought about by bunkers and stores being loaded or consumed during asea passage, or by cargo being loaded or discharged.

In all cases:

New volume of water displaced ˆ Old volume of water displacedor

New displacementNew density ˆ

Old displacementOld densityor

New displacementOld displacement ˆ

New densityOld density

Example 1

A ship displaces 7000 tonnes whilst ¯oating in fresh water Find the ment of the ship when ¯oating at the same draft in water of density 1015 kg per cubic metre, i.e 1.015 t/m 3

displace-New displacement

Old displacement ˆ

New density Old density New displacement ˆ Old displacement New densityOld density

ˆ 7000 10151000Ans New displacement ˆ 7105 tonnes

Example 2

A ship of 6400 tonnes displacement is ¯oating in salt water The ship has to proceed to a berth where the density of the water is 1008 kg per cu m Find how much cargo must be discharged if she is to remain at the salt water draft.

New displacement Old displacement ˆ

New density Old density or

New displacement ˆ Old displacement New density

Old density

ˆ 6400 10081025New displacement ˆ 6293:9 tonnes

Old displacement ˆ 6400:0 tonnes

Ans Cargo to discharge ˆ 106:1 tonnes

be loaded to remain at the same draft in salt water.

Old displacement ˆ L  B  draft  C b  density

ˆ 120  17  7:2  0:800  1000 tonnes Old displacement ˆ 11 750 tonnes

New displacement

Old displacement ˆ

New density Old density New displacement ˆ Old displacement New densityOld density

ˆ 11; 750 1025

1000 New displacement ˆ 12 044 tonnes

Old displacement ˆ 11 750 tonnes

Ans Cargo to load ˆ 294 tonnes

Note This problem should not be attempted as one involvingTPC and FWA.